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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Volume 10, Issue 1 New
DNC Volume 1, Issue 1
Front/Back Materials
Heavy-tailed Distributions in Some Stochastic Dynamical Models
pp. 1-40 | DOI:10.5890/DNC.2012.02.001
Ph. Blanchard, T. Kr¨uger, D. Volchenkov
Abstract | Access Full Text
Heavy-tailed distributions are found throughout many naturally occurring phenomena. We have reviewed some models of stochastic dynamics that lead to heavy-tailed distributions (and power law distributions, in particular) including the multiplicative noise models, the models subjected to the Degree-Mass-Action principle (the generalized preferential attachment principle), the intermittent behavior occurring in complex physical systems near a bifurcation point, queuing systems, and the models of Self-organized criticality. Heavy-tailed distributions appear in them as the emergent phenomena sensitive for coupling rules essential for the entire dynamics.
Sources and Sinks of Energy Balance for Nonlinear Atmospheric Motion Perturbed by West-to-east Winds Progressing on a Surface of a Rotating Spherical Shell
pp. 41-55 | DOI:10.5890/DNC.2012.02.002
Ranis N. Ibragimov, Michael Dameron and Chamath Dannangoda
We study the asymptotic behavior of sources and sinks associated with the effects of rotation and nonlinearity of the energy balance of atmospheric motion perturbed by west-to-east winds progressing on the surface of a rotating spherical shell. The model uses nonlinear viscous and nonviscous incompressible fluid flows on a rotating spherical domain with infinitely small thickness. The energy density and associated sources and sinks were determined and visualized by means of elementary functions that provide the exact solutions of the nonviscous baratropic vorticity equation on the rotating sphere. It is shown that there exists a particular form of west-to east flows for which the source and sink terms associated with the energy balance vanishes providing thus the energy conservation law. Moreover, this particular form of atmospheric perturbation preserves the exact solutions for the case of viscous flows.
Secure Communication Based on Chaotic Cipher and Chaos Synchronization
Maricela Jim´enez-Rodr´ıguez, Rider Jaimes-Reategui, Alexander N. Pisarchik
We proposed a secure communication system, which combines two different techniques of chaotic cryptography: chaotic cipher based on the logistic map for information diffusion, and chaos synchronization of two coupled R¨ossler oscillators for information confusion. The system contains two channels, one for synchronization and another for information transmission. Using the chaotic logistic map, a message is encrypted into one of the variables of the R¨ossler oscillator in the transmitter and sent to the receiver via the communication channel, while the R¨ossler oscillators are synchronized via the synchronization channel by another variable. Due to excellent confusion and diffusion properties, both exploiting important properties of chaos, the proposed communication system is extremely secure and fast enough to provide communication in real time.
Regularization of Map-based Neuron Models Using Phase Control
Javier Used, Alexandre Wagemakers, Miguel A.F. Sanju´an
Recently discrete dynamical systems, maps, have been also used as valid phenomenological neuron models, and are able to furnish an advantageous alternative to continuous dynamical systems for the modelling of the spiking behavior of single neurons and of neuronal networks as well. Periodic and chaotic spiking, phasic and tonic bursting, subthreshold oscillations and many more specific features of the activity of real neurons can be reproduced by maps with a minimum of analytical complexity. As an external stimulation is applied to the neuron, its response can be of two different natures: periodic or erratic. We present a simple method of control that allows to choose one of the possible responses when the perturbation is periodic. The phase difference between the periodical driving and the control plays a decisive role.
Statistical Mechanics of Fragmentation-advection Processes and Nonlinear Measurements Problem. I
pp. 79-112 | DOI:10.5890/DNC.2012.03.002
Vladimir V. Uchaikin
The random fragmentation-advection process is described in terms of generating functional and multiparticle densities. The aim of this article is to connect characteristics of a device performing the measurements of the process at a fixed time with characteristics of elementary fragmentation processes. The measurand is a random number being a symmetrical function of particles distribution at the moment of measurement. This function is assumed to be non-linear, and this part of the article is devoted to development of the general approach to computing of the mathematical expectation of such detector's reading whereas the next one will discuss the adjoint function approach in connection with solving the inverse problem: defining characteristics of elementary processes by using results of non-linear measurements.
Unstable and Stable Period-m Motions in a Twin-well Potential Duffing Oscillator
pp. 113--145 | DOI:10.5890/DNC.2012.03.001
Albert C. J. Luo, Jianzhe Huang
In this paper, unstable and stable period-m motions in the periodically forced Duffing oscillator are predicted analytically through the generalized harmonic balance method. Period-3, period-5 and period-7 periodic motions are investigated as examples for the Duffing oscillator with a twin-well potentials. The Hopf bifurcation of periodic motions yields the onset of period-doubling periodic motions. With increasing period number of periodic motions, there are too many co-existing stable and unstable periodic motions, and such stable periodic motions are much less than the corresponding unstable periodic motions. This investigation provides a complete picture of unstable and stable periodic motions rather than stable motions only. For any unstable periodic motion, if there is at least one co-existing stable periodic motion, then such unstable periodic motion will reach the stable periodic motion through transient motion.
Modelling of Synaptic STDP and Analysis in a Two-Neuron Model
V. K. Menz, S. Popovych, T.Kupper
A mathematical model is developed to describe the behaviour of spiking neurons and adaptation of synaptic weights in the framework of spike-timing-dependent plasticity (STDP) by modifying the model of STDP suggested by Gorchetchnikov, Versace, and Hasselmo [1]. As a result an STDP curve similar to that found experimentally by Bi and Poo [2] is produced. This approach is applied to a system of two integrate-and-fire neurons interacting via adapting synapses and stimulated by a constant external current. The dynamics of the considered system over long time periods is examined for both permanent and short initial stimulations. The obtained results are then compared with real data for in vivo and in vitro neurons.
One Kink Solution for a Variety of Nonlinear Fifth-order Equations
pp. 161–170 | DOI:10.5890/DNC.2012.05.001
Abdul-MajidWazwaz
In this work, we derive one kink solution for each equation of a variety of nonlinear fifth-order equations. Each equation involves a third order temporal derivative and a mixed fifth-order derivative. The simplified form of Hirota's method will be used to derive these solutions. The proposed work presents reliable approach compared to some existing techniques.
Statistical Mechanics of Fragmentation-advection Processes and Nonlinear Measurements Problem. II.
The continuation of the article [1] contains statement of the adjoint transport theory, derivation and comparatively discussion backward and adjoint equations and their solutions, the theory of statistical fluctuations, the theory of perturbations and the method variational interpolations.
On the Existence of Stationary Solutions for Some Systems of Non-Fredholm Integro-Differential Equations
Vitaly Volpert, Vitali Vougalter
We prove the existence of stationary solutions for certain systems of reaction-diffusion type equations in the corresponding H2 spaces. Our method relies on the fixed point theorem when the elliptic problem involves second order differential operators with and without Fredholm property.
A Method for Generating Lie Algebras and Applications
pp. 211-224 | DOI:10.5890/DNC.2012.05.004
Yufeng Zhang
A way for generating Lie algebras from a Lie subalgebra of the Lie algebra A1 is proposed for which a few enlarged Lie algebras are constructed. By establishing their loop algebras and introducing Lax pairs, we obtain two integrable Hamiltonian hierarchies of evolution type, one of them reduces to the well-known nonlinear Schr¨odinger equation, another is a nonlinear integrable coupling of the Kaup-Newell (KN) hierarchy,which reduces to a coupled nonlinear integrable model with Hamiltonian structure.
Existence of Solutions to Weakly Generalized Vector F-implicit Variational Inequalities
Salahuddin, Ahmad M.K., Agarwal R.P.
In this paper we studied a new class of weakly generalized vector F-implicit variational inequalities in real topological vector spaces. We use the Ferro minimax theorem to discuss the existence of strong solutions for generalized vector F-implicit variational inequalities.
Heat Conduction in Anisotropic Media
Nail H. Ibragimov, Elena D. Avdonina
Nonlinear self-adjointness of the anisotropic nonlinear heat equation is investigated. Mathematical models of heat conduction in anisotropic media with a source are considered and a class of self-adjoint models is identified. Conservation laws corresponding to the symmetries of the equations in question are computed.
Global Synchronization of Large Ensembles of Pulse Oscillators with Time-Delay Coupling
Vladimir V.Klinshov, Vladimir I.Nekorkin
In this paper we study the global synchronization in the ensembles of time-delay coupled pulse oscillators. We prove that the stability of the synchronization regime depends on the fulfillment of the simple inequality for the derivative of the phase reset curve. This finding is checked by numerical simulations which confirm the above inequality to be the sufficient condition for the synchronization. In the case of sin-shape phase reset curve it results in a stripped structure of the parameter space in which the areas with and without synchronic state alternate with the growth of the coupling delay.
The Dynamical Relationship Between Vegetation and Sediment in Arid and Semiarid Areas
Wei Tang, Huayong Zhang, Tousheng Huang, Liming Dai
The main goal of this research is to show the effects of the interaction among sediment, Aeolian sand and vegetation growth. The critical condition is found when determining the existence of stable interior equilibrium point. Intuitive phase graphs are described for three cases of the consequences of the interaction. The systematic perspective analysis provides an explanation for the dynamic interaction between wind erosion and vegetation growth in arid and semiarid areas.
Vectorial Inequalities for Integral Operators Involving Ratios of Functions and Convexity
George A. Anastassiou
Here we present vectorial integral inequalities for products of multivariate convex and increasing functions applied to vectors of ratios of functions. As applications we derive a wide range of vectorial fractional inequalities of Hardy type. They involve the left and right Riemann-Liouville fractional integrals and their generalizations, in particular the Hadamard fractional integrals. Also inequalities for Riemann-Liouville, Caputo, Canavati and their generalizations fractional derivatives. These application inequalities are of Lp type, p ≥ 1, and exponential type.
Fractional Maps and Fractional Attractors. Part I: α-Families of Maps
M. Edelman
In this paper we present a uniform way to derive families of maps from the corresponding differential equations describing systems which experience periodic kicks. The families depend on a single parameter - the order of a differential equation α > 0. We investigate general properties of such families and how they vary with the increase in α which represents increase in the space dimension and the memory of a system (increase in the weight of the earlier states). To demonstrate general properties of the α -families we use examples from physics (Standard α -Family of Maps) and population biology (Logistic α -Family of Maps). We show that with the increase in α systems demonstrate more complex and chaotic behavior.
Fractional Electromagnetic Wave
J.J. Rosales, M. Guía, J.F. Gómez, V.I. Tkach
In this article we propose an alternative procedure for constructing fractional differential equations. The order of the derivative being considered is 0 < γ ≤ 1. In particular, we will consider the propagation of electromagnetic waves in an infinitely extended homogeneous media at rest, characterized by the permittivity ε and permeability μ. Two types of the fractional differential equations will be examined separately; with fractional time derivative and the spatial fractional derivative. The parameters σt and σx are introduced which characterize the existence of the fractional time and space components, respectively. It is shown that in the first case there is a relation between σt and the period T0 of the wave given by the order γ of the fractional differential equation, and in the second case the relation is between σx and the wavelength λ . Due to these relations the solutions of the corresponding fractional differential equations are given in terms of the Mittag-Leffler function depending only on the parameter γ .
Exponentially Dichotomous Linear systems of Differential Equations with Piecewise Constant Argument
M. Akhmet
We consider differential equations with piecewise constant argument of generalized type. It is the first time, an attention is given to the exponential dichotomy of linear systems. Bounded, almost periodic and periodic solutions and their stability are discussed. The study is made in such a way that further construction of the theory will follow for ordinary differential equations. The results are illustrated by examples.
Method of Conservation Laws for Constructing Solutions to Systems of PDEs
N. H. Ibragimov
In the present paper, a new method is proposed for constructing exact solutions for systems of nonlinear partial differential equations. It is called the method of conservation laws. Application of the method to the Chaplygin gas allowed to construct new solutions containing several arbitrary parameters. It is shown that these solutions cannot be obtained, in general, as group invariant solutions.
Chaos Generation in Hyperbolic Systems
M.U. Akhmet, M.O. Fen
In the present paper, we consider extension of chaos in hyperbolic systems with arbitrary large dimensions. Our investigations comprise chaos in the sense of both Devaney and Li-Yorke. We provide a mechanism for unidirectionally coupled systems through the insertion of chaos from one system to another, where the latter is initially nonchaotic. In our procedure for the chaos extension, we take advantage of chaotic sets of functions to provide mathematically approved results. The theoretical results are supported through the simulations for the extension of chaos generated by a Duffing's oscillator. A control procedure for the extended chaos is demonstrated numerically in the paper.
Nonlinear Dynamic and Chaotic Saddle in Rectifier Circuit
L.F.R. Turci, E.E.N. Macau, T. Yoneyama
Chaotic systems are recognized by presenting an evolved dynamics in which sophisticated phenomena as crisis, metamorphoses and transitions may take places. In this work we show that these phenomena are present even in very simple system. Here we analyze a simple rectifier electronic circuit and show the mechanisms that mediate its transition from a simple to a rich dynamics. Furthermore, we identified the presence of a chaotic saddle in the system, which also implies the occurrence of transient chaos even for parameters for which the systems eventually sets in a regular behavior.
Invariants in 3D for Classical Superintegrable Systems in Complex Phase Space
J. S. Virdi, S. C. Mishra
Physical dynamical systems in higher dimensions are always interesting. In this context we present here the possibility of its three-dimensional complex dynamical invariant in extended complex phase space(ECPS). Lie algebraic method is used to study three-dimensional classical superintegrable system on the extended complex phase space. Such complex invariants play an important role in the analysis of complex trajectories, also study of non-hermitian Hamiltonian systems.
An Efficient Deformable Image Registration Method with Mexican Hat Wavelets
M.A. Akinlar, R. Ibragimov
An efficient deformable image registration method using Mexican hat wavelets is presented. Image registration algorithm is expressed as an optimization problem by the minimization of a summation of a similarity measure and a regularization term. A discrete L2-norm type similarity measure is associated with probabilistic wavelet coefficient regularization term. Greedy algorithm is used to solve the aforementioned optimization problem. Computational applications indicate that the present method is quite fast and efficient in the registration of brain MR images.
Establishing the Onset, and Parameter and Spatial Domains, of Chaos in a Smooth Quadratic Autonomous System Using Sil'nikov Criteria and Competitive Modes Analysis
pp. 1--19 | DOI: 10.5890/DNC.2012.10.001
Thomas K. Vogel and S. Roy Choudhury
In this paper we examine the chaotic regimes of a recently discovered hyperchaotic system in greater depth. Towards that end, we first analyze the numerical observations of horseshoe-type chaotic behavior in this system in detail by the use of Shilnikov analysis. Subsequently,we also employ the technique of Competitive Modes analysis to identify "possible chaotic parameter regimes" for this multi-parameter system. We find that the Competitive Modes conjectures may in fact be interpreted and employed slightly more generally than has usually been done in recent investigations, with negative values of the squared mode frequencies in fact being admissible in chaotic regimes, provided that the competition among them persists. This is somewhat reminiscent of, but of course not directly correlated to, "stretching (along unstable manifolds) and folding (due to local volume dissipation)" on chaotic attractors. This new feature allows for the system variables (which also define the position ON the attractor) to grow exponentially during time intervals when mode frequencies are imaginary and comparable, while oscillating at instants when the frequencies are real and locked in or entrained. Finally, in a novel twist, we re-interpret the components of the Competitive Modes analysis as simple geometric criteria to map out the spatial location and extent, as well as the rough general shape, of the system attractor for any parameter sets corresponding to chaos. The accuracy of this mapping adds further evidence to the growing body of recent work on the correctness and usefulness of the Competitive Modes conjectures.
Transient Dynamics in Complex Systems: Heteroclinic Sequences with Multidimensional Unstable Manifolds
pp. 21--41 | DOI: 10.5890/DNC.2012.11.001
Valentin Afraimovich , Irma Tristan, Pablo Varona and Mikhail Rabinovich
We formulate the basic principles of multi-agents complex system dynamics following the lessons from experimental neuro- and cognitive science: 1) the cognitive dynamics in a changing environment is transient and can be considered as a temporal sequence of metastable states; 2) the available resources for the information processing are limited; 3) the transient dynamics is robust against noise and at the same time sensitive to information signals. We suggest the basic dynamical models that describe the evolution of cooperative modes. We focus on two limit cases: a) the unstable manifold of metastable states has one leading direction and many others that are characterized by small positive eigenvalues (system on the edge of instability), and b) the unstable manifold is characterized by small number of positive eigenvalues having the same range (integration of different flows - binding).
Relaxation Processes in Many Particle Systems -- Recurrence Relations Approach
Anatolii V. Mokshin
The general scheme for the treatment of relaxation processes and temporal autocorrelations of dynamical variables for many particle systems is presented in framework of the recurrence relations approach. The time autocorrelation functions and/or their spectral characteristics, which are measurable experimentally (for example, due to spectroscopy techniques) and accessible from particle dynamics simulations, can be found by means of this approach, the main idea of which is the estimation of the so-called frequency parameters. Model cases with the exact and approximative solutions are given and discussed.
Who Replaces Whom? Local versus Non-local Replacement in Social and Evolutionary Dynamics
Sven Banisch and Tanya Araújo
In this paper, we inspect well–known population genetics and social dynamics models. In these models, interacting individuals, while participating in a self-organizing process, give rise to the emergence of complex behaviors and patterns. While one main focus in population genetics is on the adaptive behavior of a population, social dynamics is more often concerned with the splitting of a connected array of individuals into a state of global polarization, that is, the emergence of speciation. Applying computational and mathematical tools we show that the way the mechanisms of selection, interaction and replacement are constrained and combined in the modeling have an important bearing on both adaptation and the emergence of speciation. Differently (un)constraining the mechanism of individual replacement provides the conditions required for either speciation or adaptation, since these features appear as two opposing phenomena, not achieved by one and the same model. Even though natural selection, operating as an external, environmental mechanism, is neither necessary nor sufficient for the creation of speciation, our modeling exercises highlight the important role played by natural selection in the interplay of the evolutionary and the self–organization modeling methodologies.
Breathing Instability in Biological Cells, Patterns of Membrane Proteins
M. Leonetti, G. Boëdec, M. Jaeger
The activity of biological cells involves often the electric activity of its membranes which exhibit various spatiotemporal dynamics, from pulse, oscillatory bifurcation to stationary spatial modulation. This last kind of patterns appears on a typical diffusive time. A model has been proposed implying a coupling between the current flowing through membrane proteins and their electrophoretic motions in the case of mobile proteins. Here, we study the stability of the pattern in a 2D circular model cell versus the appearance of standing waves, the so-called breathing secondary instability.
Exact Solutions and Analysis for a Class of Extended Stokes' Problems
pp. 85--102 | DOI: 10.5890/DNC.2012.12.003
L. Z. Zhang, H. S. Tang, Jerome P. -Y. Maa, and G. Q. Chen
This paper studies a class of unsteady flows as extensions of the classic Stokes' problems to consider influence of solid walls, effect of pressure gradients, and situation of two-layer fluids. The flows are solved using the method of separating variables and the eigenfunction expansion method. With simplifications, the derived solutions will degenerate to solutions to the classic Stokes' problems, the Couette flow, and the Poiseuille flow. The exact solutions of these flows clearly illustrate the complexity of the involved physics including evolution of flow velocity profiles and energy transferring at fluid boundaries. For a single-layer flow driven by a plate moving at a constant speed, energy transferred from the plate decreases with time and tends to a non-zero constant as a result of wall effect. In a single-layer flow with an oscillatory boundary, negative energy input may appear at the boundary. For an air-water flow with a finite depth, the interface velocity is proportional to the air velocity, which is a well-known observation in physical oceanography. In addition, there is no energy transferring at the interface between the two fluids in a purely pressure driven two-layer flow.
Fractional Fourier Detection of L´evy Flights: Application to Hamiltonian Chaotic Trajectories
pp. 103--114 | DOI: 10.5890/DNC.2013.04.001
Francoise Briolle, Xavier Leoncini, and Benjamin Ricaud
A signal processing method designed for the detection of linear (coherent) behaviors among random fluctuations is presented. It is dedicated to the study of data recorded from nonlinear physical systems. More precisely the method is suited for signals having chaotic variations and sporadically regular linear patterns, possibly impaired by noise. We use time-frequency techniques and the Fractional Fourier transform in order to make it robust and easily implementable. The method is illustrated with an example of application: the analysis of chaotic trajectories of advected passive particles. The signal has a chaotic behavior and encounters L´evy flights (straight lines). The method allows to detect and quantify these ballistic transport regions, even in noisy situations.
Cosmic Evolution in Fractional Action Cosmology
Victor K. Shchigolev
For the fractional action cosmological model, derived earlier by the author from the variational principle for a fractional action functional, the exact solutions are obtained. The case of a quasi - vacuum state of matter that fills the universe is considered. Moreover, on the basis of specific ansatz proposed in this paper for the cosmological term, the class of exact solutions of the model equations is obtained. Examples for some given laws of the cosmological term evolution are provided. Besides, a formula for the effective equation of state is derived, and the deceleration parameter of the obtained models is studied.
High Degree Multivariate Fuzzy Approximation by Quasi-Interpolation Neural Network Operators
Here are considered in terms of multivariate fuzzy high approximation to the multivariate unit sequences of multivariate fuzzy quasiinterpolation neural network operators. These operators are multivariate fuzzy analogs of earlier considered multivariate real ones. The derived results generalize earlier real ones into the fuzzy setting. Here the high degree multivariate fuzzy pointwise and uniform convergence with rates to the multivariate fuzzy unit operator are given through multivariate fuzzy inequalities involving the multivariate fuzzy moduli of continuity of the N th order (N ≥ 1) H-fuzzy partial derivatives, of the involved multivariate fuzzy number valued function.
Conservation Laws in Thomas's Model of Ion Exchange in a Heterogeneous Solution
N.H. Ibragimov and Raisa Khamitova
Physically significant question on calculation of conservation laws of the Thomas equation is investigated. It is demonstrated that the Thomas equation is nonlinearly self-adjoint. Using this property and applying the theorem on nonlocal conservation laws the infinite set of conservation laws corresponding to the symmetries of the Thomas equation is computed. It is shown that the Noether theorem provides only one of these conservation laws.
Solvability Conditions for Some Systems of Nonlinear Non-Fredholm Elliptic Equations
Vitali Vougalter
We establish the existence and uniqueness of standing solitary wavelike solutions in H2 for certain systems of nonlocal nonlinear equations. These problems involve second order differential operatorswithout Fredholm property.
Parameter Characteristics of Projective Synchronization of two Gyroscope Systems with Different Dynamical Behaviors
Fuhong Min and Albert C.J. Luo
In this paper, parameter characteristics of the projective synchronization for two gyroscopes with different dynamical behaviors are investigated. The projective synchronization conditions are presented from the theory of discontinuous dynamical systems. From such synchronization conditions, the parameter characteristics for partial and full projective synchronizations for two gyroscope systems are studied. The full projective synchronization can be achieved exactly in finite time instead of asymptotic synchronization in the traditional projective synchronization. The scaling factors in such synchronization are observed through numerical simulations.
Synchronization of Two Identical Restricted Planar Isosceles Three-Body-Problem and a Study on Possible Chaos Control
Ayub Khan and Rimpi Pal
In this paper, we have investigated Complete Synchronization, Anti- Synchronization and all possible cases of Hybrid Synchronization of two restricted planar isosceles three-body-problem evolving from different initial conditions using Active Control Technique. Further, the possible role of control functions in the control of Chaos is analyzed. Finally, numerical simulations are performed to illustrate the effectiveness of the proposed control techniques.
Through the Looking-Glass of the Grazing Bifurcation: Part I - Theoretical Framework
James Ing, Sergey Kryzhevich, and Marian Wiercigroch
It is well-known for vibro-impact systems that the existence of a periodic solution with a low-velocity impact (so-called grazing) may yield complex behavior of the solutions. In this paper we show that unstable periodic motions which pass near the delimiter without touching it may give birth to chaotic behavior of nearby solutions. We demonstrate that the number of impacts over a period of forcing varies in a small neighborhood of such periodic motions. This allows us to use the technique of symbolic dynamics. It is shown that chaos may be observed in a two-sided neighborhood of grazing and this bifurcation manifests at least two distinct ways to a complex behavior. In the second part of the paper we study the robustness of this phenomenon. Models of impact Particularly, we show that the same effect can be observed in "soft" models of impacts.
Pulse Phenomena for Impulsive Dynamical Systems
Shasha Zheng and Xilin Fu
In this paper, we investigate the pulse phenomena for a class of impulsive dynamical systems. Some sufficient conditions that guarantee the absence or presence of pulse phenomena are obtained, without the boundeness requirement on impulse surfaces. Besides, we also utilize methods of the flow theory, focus on the dynamical behavior in the normal direction to the switching boundary and generalize several known results to apply to an important example. Then, we study the stability of a nontrivial solution in a class of functional differential equations with pulse phenomena by using the concept of quasistability and method of comparison.
Spectrum of Dimensions for Escape Time
Valentin Afraimovich and Rosendo Vazquez
We introduce a new notion-the spectrum of dimensions of escape time- and study its properties. The escape time was defined for an initial point of the trajectory according to its ability to reach a hole in the phase space. In the article we generalize this notion onto "spots" of initial points making the escape time to be a function of a set (spot). Then we apply the Caratheodory-Pesin machinary of fractal dimensions to define the spectrum. For dynamical systems generated by maps of the interval or the circle we obtain explicit formulas in the case where an element of Markov partition is chosen as a hole.
Period-m Motions and Bifurcation Trees in a Periodically Excited, Quadratic Nonlinear Oscillator
Albert C.J. Luo and Bo Yu
In this paper, analytical bifurcation trees of period-1 motions to chaos in the periodically forced, quadratic nonlinear oscillator are discussed from the generalized harmonic balance method. The analytical solutions for stable and unstable periodic motions in such quadratic nonlinear oscillator are achieved, and the corresponding stability and bifurcation were discussed. The analytical bifurcation trees from period-1 motions to period-4 motions in such quadratic oscillator are presented, and numerical illustrations of stable and unstable periodic motions are carried out by the numerical and analytical solutions. This investigation provides a comprehensive picture of complex periodic motion in the periodically excited, quadratic nonlinear oscillator.
Markov Chain Scaffolding of Real World Data
D. Volchenkov
Discrete time scale dependent random walks on a graph model of a relational database can be generated by a variety of equivalence relations imposed between walks (i.e.composite functions, inheritance, property relations, ancestor-descendant relations, data queries, address allocation and assignment polices). The Green function of diffusion process induce by the random walks allows to define scale dependent geometry. Geometric relations on databases can guide the datainterpretation. In particular, first passage times in a urban spatial network help to evaluate the tax assessment value of land.
Stochastic Patterns and the Role of Crowding
pp. 301–319 | DOI: 10.5890/DNC.2013.11.001
Claudia Cianci and Duccio Fanelli
A stochastic variant of the Brusselator model is investigated. The model accounts for a long range coupling among constituents, as well as for the finite capacity of the embedding medium. The mean field limit of the model is studied and the condition for Turing and wave instability obtained. A degenerate, cusp like transition that separates the domains of Turing and wave order can take place. The point of transition is worked out analytically. Interestingly, the region of Turing instability, as delimited by such transition point, can set in also if the inhibitor diffuses slower then the activator. This is a consequence of the generalized diffusion scheme here analyzed and which originates from having imposed an effect of spatial competition. Beyond the deterministic, mean field picture, we elaborate on the role of stochastic corrections. Granularity, endogenous to the system, can eventually materialize in waves or Turing like patterns, that we here categorized in distinct classes.
A Special Type of Invariant Solutions and its Connection with Dispersion Relations
Nail H. Ibragimov and Ranis N. Ibragimov
The concept of dispersion relations is widely used in physics and ap- plied mathematics in investigating wave type solutions of differential equations. On the other hand, Lie group analysis provides another useful method for constructing exact solutions of linear and nonlinear differential equations via the concept of invariant solutions. We show in the present paper that for certain types of differential equations there is a remarkable connection between these two concepts. Namely, the idea of dispersion relations leads to a special type of invariant solu- tions.
Synchronization Attack to Chaotic Communication Systems
Massimiliano Zanin, J. Ricardo Sevilla-Escoboza, Rider Jaimes-Reátegui, J. Hugo Garcıa-Lopez, Guillermo Huerta-Cuellar, and Alexander N. Pisarchik
Synchronization of chaotic oscillators has an important application in cryptography. When two identical oscillators are coupled, they can be completely synchronized and the chaotic output of the transmit- ter oscillator can be used to mask a message. Although the oscilla- tor parameters are usually used as secret keys, the sensitivity of such cryptosystems to parameter changes has never been systematically an- alyzed. To cryptanalyze a communication system based on synchro- nization of chaotic oscillators, we use a synchronization attack that allows estimating all unknown parameters by minimizing the synchro- nization error. Using this attack we cryptanalyze popular communica- tion systems based on the Ro ̈ssler and Chua chaotic electronic circuits. We suggest to include this attack as a standard security test for crypt- analysis of chaotic communication systems.
Parallel Computation of Reliable Chaotic Solutions of Saltzman's Equations by Means of the Clean Numerical Simulation
Peng Yang, Zhiliang Lin and Shijun Liao
The method of the so-called "Clean Numerical Simulation" (CNS) is applied to gain reliable chaotic solutions of Saltzman's dynamic system, a simplified model for convection flows of fluid. Based on the high-order Taylor series method with data in multiple precision library and a validation check of global reliability of result, the CNS provides us a practical way to gain reliable, accurate enough solutions of chaotic dynamic systems in a finite but long enough time interval. The parallel computation is used to greatly increase the computational efficiency. The numerical noises of the CNS can be controlled to be so small that even the influence of the micro-level inherent uncertainty of initial conditions can be investigated in details. It is found that the micro-level inherent physical uncertainty (i.e. the unavoidable statistical fluctuation of temperature and velocity of fluid) of initial conditions of chaotic Saltzman's system transfers into macroscopic randomness. This suggests that chaos might be a bridge between micro-level inherent physical uncertainty and macroscopic randomness. The current work illustrates that the above conclusion holds not only for Lorenz equation with three ODEs but also for Saltzman's equation with up-to nine ODEs, and thus has general meanings.
Three Compartmental Model for Propofol Diffusion During General Anesthesia
Clara M. Ionescu, Dana Copot, and Robin De Keyser
This paper presents the initial steps towards the development of a compartmental model for drug diffusion in the human body, using fractional calculus. The model presented here preserves the mass balance, therefore it maintains the link between physiological and mathematical concepts. The final purpose of this model is to predict drug pharma-cokinetics and pharmacodynamics during general anesthesia. However, in this case the model is derived for a general class of drugs, therefore it can be employed in many biomedical applications.
Almost Periodic Solutions of Second Order Neutral Differential Equations with Functional Response on Piecewise Constant Argument
We consider second order functional differential equations with generalized piecewise constant argument. Conditions for existence, uniqueness and stability of Bohr almost periodic solutions are defined. Appropriate examples which illustrate the results are provided.
Solvability in the Sense of Sequences for Some Non-fredholm Elliptic Problems
Vitaly Volpert, and Vitali Vougalter
We establish solvability in the sense of sequences in the appropriate H 2 spaces for certain linear nonhomogeneous elliptic problems involving Schro ̈dinger type operators without Fredholm property using the technique developed in our preceding work [23]. We show the existence of bounded solutions for certain nonlinear Lippmann-Schwinger equa- tions.
A Novel Moist Carbon Dioxide Generation Enhanced Oil Recovery Technology
Geylani M. Panahov, Eldar M. Abbasov, Sayavur I. Bakhtiyarov, Gulshan R. Agayeva, Gabil A. Aliyev, and Sevinj R. Rasulova
A new technological technique based on the "moist" carbon dioxide generation in the wellbore zone of the oil formations or at the wellhead is proposed. In this new technology a carbon dioxide is generated as a result of the chemical reaction between an aqueous acid solution and a natural calcium carbonate rock (limestone) as a "gas yielding" component. The mechanism of the process allows controlling a generation rate and a volume of the gas, the thermobaric conditions of the oil formation, mineral and acidity levels of water and a phase state of carbon dioxide. An industrial application of this technology in a large scale has a significant ecological importance as this technology utilizes the wastes of processing plants of the natural minerals, which is considered as a major source of the pollution of atmosphere by greenhouse gas emissions.
Treasure Hunting in Virtual Environments Self-Organized Criticality in Searches Amid Uncertainty
pp. 1–17 | DOI: 10.5890/DNC.2014.03.001
Dimitri Volchenkov, Jonathan Helbach, Marko Tscherepanow, and Sina Ku ̈hne
Searching experiments conducted in different virtual environments over a gender balanced group of people revealed a gender irrelevant scale-free spread of searching activity on large spatiotemporal scales. The better per- formance of men in virtual environments can be associated with the reg- ularly renewed computer game experience, essentially in games played through a first-person perspective. We suggested a simple self-organized critical model of search, in which the experimentally observed scale-free behavior can be interpreted as a trade-off between the value of exploitation versus exploration amid uncertainty.
Synchronization of the Cardiac Pacemaker Model with Delayed Pulse-coupling
pp. 19–31 | DOI: 10.5890/DNC.2014.03.002
We reconsider the C. Peskin model of the cardiac pacemaker assuming that pulse-couplings are delayed. Sufficient conditions for synchronization of identical and non-identical oscillators are obtained. The results are demon- strated with numerical simulations.
Boundary Value Problems for Impulsive Fractional Evolution Integrodifferential Equations with Gronwall's Inequality in Banach Spaces
Dimplekumar N. Chalishajar and K. Karthikeyan
In this paper, we study boundary value problems for impulsive fractional evolution integrodifferential equations with Caputo derivative in Banach spaces. A generalized singular type Gronwall inequality is given to obtain an important priori bounds. Some sufficient conditions for the existence solutions are established by virtue of fractional calculus and fixed point method under some mild conditions. An example is given to illustrate the results.
Treatment and Vertical Transmission in a HIV-TB Co-infection Model
Carla MA Pinto and Ana Carvalho
In this paper, it is proposed a mathematical model for co-infection of HIV/AIDS and tuberculosis. The model includes treatment and vertical transmission for HIV/AIDS. The treatment for tuberculosis is not included. The disease-free equilibrium of the model is computed and local stability is proved. The reproduction numbers of the full model and of its two sub- models, concerning single infection by HIV/AIDS and single infection by tuberculosis, are also calculated. Numerical simulations show the effect of the variation of the recruitment rate, of the movement rate, and of the tuberculosis infection rate on the variables of the model. Results are as expected. Namely an increase in the recruitment rate increases the suscep- tible population. As the movement rate is decreased, the individuals single infected with HIV decrease. Moreover, an increase in the tuberculosis in- fection rate translates in an increase of the single infected with tuberculosis and dually-infected with tuberculosis and HIV/AIDS individuals. Future work will consider the inclusion of treatment of tuberculosis.
Solvability Conditions For Some Non Fredholm Operators in a Layer in Four Dimensions
Vitaly Volpert and Vitali Vougalter
We study solvability in H2 of certain linear nonhomogeneous elliptic prob- lems involving the sum of the periodic Laplacian and a Schro ̈dinger oper- ator without Fredholm property and prove that under reasonable technical conditions the convergence in L2 of their right sides implies the existence and the convergence in H2 of the solutions. We generalize the methods of spectral and scattering theory for Schro ̈dinger type operators from our preceding work
Properties of a Periodic Ansatz for the Coarsening of Soliton-lattice
Simon Villain-Guillot
Soliton lattices are periodic solutions of Ginzburg-Landau equation which can be useful tools to explore the coarsening process (or Ostwald ripening) which takes place during a Cahn-Hilliard dynamics.They can be used to identify the stationary solutions of the dynamics and how these intermedi- ate states are destroyed by fluctuations. The coarsening process drives the systems from a stationary solution to the next one which is of period double and of lower energy. Using another family of soliton lattices, this process can be described continuously via a phase field equation. We present here properties of these two families, including the Fourier series decomposition of the non symmetric soliton lattice which we use as building block of our ansatz.
Bifurcation Trees of Period-m Motions to Chaos in a Time-Delayed, Quadratic Nonlinear Oscillator under a Periodic Excitation
pp. 87–107 | DOI: 10.5890/DNC.2014.03.007
Albert C. J. Luo and Hanxiang Jin
In this paper, analytical solutions of periodic motions in a periodi- cally excited, time-delayed, quadratic nonlinear oscillator are obtained through the Fourier series, and the stability and bifurcation of such pe- riodic motions are discussed by eigenvalue analysis. The analytical bifurcation tree of period-1 motion to chaos in such a time-delayed, quadratic oscillator is presented through period-1 to period-8 motion. Numerical illustrations of stable and unstable periodic motions are given by numerical and analytical solutions. Compared to dynami- cal systems without time-delay, the time-delayed dynamical systems possess different periodic motions and the bifurcation trees of periodic motions to chaos are also distinguishing.
Variational Iteration Method for Generalized Pantograph Equation with Convergence Analysis
Mohsen Alipour, Dumitru Baleanu, Kobra Karimi, and Sunil Kumar
In this paper, we solve generalized pantograph equation by changing the problem to a system of ordinary equations and using the variational iteration method. We discuss convergence of the proposed method to the exact solution. Finally, illustrative examples are given to demonstrate the efficiency of the method.
Mathematical Modelling and Simulation of the Bifurcational Wobblestone Dynamics
Jan Awrejcewicz and Grzegorz Kudra
The Celtic stone, sometimes also called wobbles tone or rattleback usually is a semi-ellipsoidal solid with a special mass distribution. Most celts lied on aflat and horizontal base, set in rotational motion around a vertical axis can rotate in only one direction. In this work the dynamics of the celt is simulated numerically, but the solid is forced untypically, i.e. it is situated on a harmonically vibrating base. Essential part of the model are approximate functions describing the contact forces, i.e. dry friction forces and rolling resistance. They are based on previous works of the authors, but some modifications of friction model are made, which can be described as a generalization of the earlier used Pad´e approximants. Periodic, quasiperiodic and chaotic dynamics of a harmonically forced rattleback is found and presented by the use of Poincar´e maps and bifurcation diagrams.
The Solvability and Optimal Controls for Some Fractional Impulsive Equation with Finite Delay
Xianghu Liu and Yanfang Li
This paper is concerned with the existence and uniqueness of mild solution of some fractional impulsive equations withfinite delay. Firstly,we introduce the fractional calculus,Gronwall inequality, leray-schauder'sfixed point theorem,Secondly with the help of them, the sufficient condition for the existence and uniqueness of solutions is presented. Finally we give an example to illustrate our main results.
I Dress Like Everyone, I Dress Like No Other
D. Volchenkov, E. Banaszak, K.Kocjan ,and T.Kruger
By a successive reduction of complexity in the appearance data collected in situ, we have shown that the way people maintain their appearance constitutes a multi-level dynamical process evolving in several incomparable time scales. The slowest component of this process represents a gradual diffusion of features (with a measurable pace of growing variance) from a proto-costume, which is likely has the 19th century military uniform as a prototype for men, and a robe de soir´ee - for women. The rapidly varying component of the process is statistically reminiscent of adiabatic processes in thermodynamics which evolve rapidly, without exchange of heat of a system with its environment. The appearance of other people can be considered as being in a state of "thermal equilibrium" between the almost imperceptible, subtle modifications of a traditional costume and the one day gusts of unpredictable fashion.
Extended Mixed AKNS-Lund-Regge Model and Its Self-similarity Reduction
D.V. Ruy and G.R. de Melo
We discuss the relation between the self-similarity reduction of the generalized mixed mKdV-sinh-Gordon model and the fourth-order equation obtained by Kudryashov inJ. Phys. A: Math. Theor.35(2002) 93-99. Also, it is shown two particular solutions for this equation. Then, we extend the mixed AKNS-Lund-Regge model and study its self-similarity reduction. We obtain thefifth Painlev´ e equation as a particular case of this reduction and a fourth-order second-degree equation otherwise. The relation between a integrable model and a fourth-order second-degree equation is interesting because the general solution of this equation must have the Painlev´ e property due the Ablowitz, Ramani and Segur conjecture.
Improving Accuracy of Complex Network Modeling Using Maximum Likelihood Estimation and Expectation-Maximization
Ehsan Jahanpour and Xin Chen
Structure of a complex network provides important information about its performance and may be used to predict changes in network performance. Degree distributions are used to model the network structure. Four degree distributions, including the power law, Weibull, Poisson, and negative binomial, are applied in this research to three complex networks, the Krebs, HIV, and Power Grid networks. To improve accuracy of network modeling, the maximum likelihood estimation method and expectation-maximization algorithm are used to estimate parameters of the four degree distributions. Several statistical analyses and a simulation study are conducted to determine which degree distribution best describes the network structure. The results show that the degree distributions with two descriptive parameters, Weibull and negative binomial, provide better estimations than the onedescriptive-parameter degree distributions, power law and Poisson.
Mathematics of Multi-Level Complex Systems
This special issue of the Journal "Discontinuity, Nonlinearity, and Complexity" contains the proceedings of the joint workshop "Mathematics of Complex Systems" that took place at the Center for Interdisciplinary Research of the University of Bielefeld on the 7-9 October, 2013 supported by the Work program of FP7: FET Proactive Intiative of the European Comission: "Dynamics of Multi-Level Complex Systems (DyM-CS)".
Q-analysis Based Clustering of Online News
David M.S. Rodrigues†
With online publication and social media taking the main role in dissemination of news, and with the decline of traditional printed media, it has become necessary to devise ways to automatically extract meaningful information from the plethora of sources available and to make that information readily available to interested parties. In this paper we present a method of automated analysis of the underlying structure of online newspapers based onQ-analysis and modularity optimisation. We show how the combination of the two strategies allows for the identification of well defined news clusters that are free of noise (unrelated stories) and provide automated clustering of information on trending topics on news published online.
Balanced Growth in the Structural Dynamic Economic Model SDEM-2
Dmitry V. Kovalevsky
The Structural Dynamic Economic Model SDEM-2, a follow-up of the model SDEM developed earlier, is essentially an actor-based, systemdynamic model of a closed economy evolving under conditions of conflict of interests of two powerful aggregated actors: entrepreneurs and wageearners. We derive the model equations applicable to both the balanced and unbalanced growth paths, and then study the balanced growth (with neither idle physical capital nor unemployment). Wefirst consider an inflexible control strategy of entrepreneurs for deterministic and stochastic cases, and then turn to a more sophisticated nonlinear control strategy. We also solve a simple optimization problem by calculating the (time-independent) value of model control parameter maximizing the discounted dividend of entrepreneurs. In view of simplicity of model equations, exact analytical solutions can be obtained in many cases, other cases being studied semianalytically. Even the simplest versions of SDEM-2 are able to produce rather versatile trajectories of the economy, dependent on the values of model parameters and initial conditions.
Path Integral Distance for the Automated Data Interpretation
The process of data interpretation is always based on the implicit introduction of equivalence relations on the set of walks over the database. Every equivalence relation on the set of walks specifies a Markov chain describing the transitions of a discrete time random walk. In order to geometrize and interpret the data, we propose the new distance between data units defined as a "Feynman path integral", in which all possible paths between any two nodes in a graph model of the data are taken into account, although some paths are more preferable than others. Such a path integral distance approach to the analysis of databases has proven its efficiency and success, especially on multivariate strongly correlated data where other methods fail to detect structural components (urban planning, historical language phylogenies, music, street fashion traits analysis, etc. ). We believe that it would become an invaluable tool for the intelligent complexity reduction and big data interpretation.
The Probabilistic Structure of Discrete Agent-Based Models
Sven Banisch
This paper describes a formalization of agent-based models (ABMs) as random walks on regular graphs and relates the symmetry group of those graphs to a coarse-graining of the ABM that is still Markovian. An ABM in whichNagents can be inδdifferent states leads to a Markov chain with δ N states. In ABMs with a sequential update scheme by which one agent is chosen to update its state at a time, transitions are only allowed between system configurations that differ with respect to a single agent. This characterizes ABMs as random walks on regular graphs. The non-trivial automorphisms of those graphs make visible the dynamical symmetries that an ABM gives rise to because sets of micro configurations can be interchanged without changing the probability structure of the random walk. This allows for a systematic loss-less reduction of the state space of the model
Standardization of Agent-based Modeling in Economic System
Xihao Li
This work combines generic complexity of economic system and economic agents with methodologies of multi-agent system analysis and development. This combination results in an integrative framework that serves as communication protocol for delivering and transmitting agent-based model for economic system. The integrative framework provides guidance for analyzing economic system in micro-level, which embeds with properties of complexity in structure, heterogeneity in agents' beliefs, and interactions among agents' behaviors. It provides routines on developing standardized agent-based model for economic system that can be used and reused among interdisciplinary research.
A Hierarchy of Out-of-Equilibrium Actor-Based System-Dynamic Nonlinear conomic Models
Dmitry V. Kovalevsky, Klaus Hasselmann
The actor-based system-dynamic approach to macroeconomic modeling is illustrated for a simple model hierarchy consisting of a basic twodimensional model with several alternative three-dimensional extensions. The hierarchy is based on an out-of-equilibrium approach: market clearing is not assumed, supply is not equal to demand, and there exists a stock of unsold goods. Depending on actor behaviour, the models exhibit stable exponential growth or instabilities leading to nonlinear oscillations or economic collapse. In most cases, the simplicity and tractability of the models enables analytical solutions. The examples serve as illustration of more realistic models developed within the Multi-Actor Dynamic Integrated Assessment Model System (MADIAMS) to assess the long-term impacts of climate mitigation policies.
Critical Phase in Complex Networks: a Numerical Study
Takehisa Hasegawa, Tomoaki Nogawa, and Koji Nemoto
We compare phase transition and critical phenomena of bond percolation on Euclidean lattices, nonamenable graphs, and complex networks. On a Euclidean lattice, percolation showsa phase transition between the nonpercolating phase and percolating phase at the critical point. The critical point is stretched to a finite region, called the critical phase, on nonamenable graphs. To investigate the critical phase, we introduce a fractal exponent, which characterizes a subextensive order of the system. We perform the Monte Carlo simulations for percolation on two nonamenable graphs – the binary tree and the enhanced binary tree. The former shows the nonpercolating phase and the critical phase, whereas the latter shows all three phases. We also examine the possibility of critical phase in complex networks. Our conjecture is that networks with a growth mechanism have only the critical phase and the percolating phase. We study percolation on a stochastically growing network with and without a preferential attachment mechanism, and a deterministically growing network, called the decorated flower, to show that the critical phase appears in those models. We provide afinite-size scaling by using the fractal exponent, which would be a powerful method for numerical analysis of the phase transition involving the critical phase.
Random Parametric Resonance in Time-Dependent Networks of Adaptive Frequency Oscillators
We consider a network of interacting phase oscillators endowed with adaptive mechanisms, leading the collective motion to a consensual dynamical state. Specifically, for a given network topology (i.e. an adjacency matrix) governing the mutual interactions, the adaptive mechanisms enable all oscillators to ultimately adopt a consensual frequency. Once reached, the consensual frequency subsists even if interactions between the oscillators are switched off. For the class of models we consider, the consensual frequency is independent of the network topology. Even though this independence might suggest that extension totime-dependent networks is straightforward, this is not true here. For time-dependent networks and spectra of the underlying Laplacian matrices, one may observe the emergence of more complex dynamics. Due to their high degree of complexity, these dynamics generally offer little hope for analytical tractability. In this paper, we focus on connected time-dependent networks with circulant adjacency matrices. The simple spectral structures and commutativity properties enjoyed by circulant matrices enable an analytical stability analysis of the consensus state. Ultimately, we are able to reduce the stability analysis to a dissipative harmonic oscillator with parametric pumping.
Adaptive Synchronization of Delayed Chen Chaotic System
Tao Fan, Chang-Zhong Chen, Xiao-Hong Ren, and Ping He
In this paper, we focus on the adaptive synchronization of delayed Chen chaotic systems with unknown parameters. An adaptive synchronization controller and the adaptive updating law are designed. At last, the numerical simulation is shown to prove the effectiveness of the proposed synchronization controller schemes.
Ulam-Hyers-Rassias Stability for Semilinear Equations
Jinrong Wang, and Michal Feckan
We study the Ulam-Hyers-Rassias stability for linear and semilinear equations on Banach spaces from a functional analysis point of view with several illustrative examples. More precisely, surjective linear equations on Banach spaces, linear equations on Banach spaces with closed ranges and surjective semilinear equations between Banach spaces are investigated one by one.
Filaments-nets Structure of the Phase Space of Coin Tossing: Mechanism for Sensitivity and Complexity
Zengyuan Yue
Detailed analysis of trajectories reveals a filaments-nets structure of the phase space of coin tossing, leading to a simple and unified explanation for the extremely sensitive dependence of the outcome, head or tail, on the initial state, for the extremely complex geometry of the cross sections of basins of attraction for heads and tails, and for the big difference between the transitional region and the "completely random region". A "GDGC" (Great Differentiation & Great Combination) condition is proposed for the stability of statistical regularity, which can also be summarized by the following "Compensation Principle": The more sensitive, i.e. the more unstable, the deterministic process is, the more stable, i.e. the more insensitive, the associated statistical regularity would be.
Automatic Recognition and Tagging of Topologically Different Regimes in Dynamical Systems
Jesse J. Berwald, Marian Gidea, and Mikael Vejdemo-Johansson
Complex systems are commonly modeled using nonlinear dynamical systems. These models are often high-dimensional and chaotic. An important goal in studying physical systems through the lens of mathematical models is to determine when the system undergoes changes in qualitative behavior. A detailed description of the dynamics can be difficult or impossible to obtain for high-dimensional and chaotic systems. Therefore, a more sensible goal is to recognize and mark transitions of a system between qualitatively different regimes of behavior. In practice, one is interested in developing techniques for detection of such transitions from sparse observations, possibly contaminated by noise. In this paper we develop a framework to accurately tag different regimes of complex systems based on topological features. In particular, our framework works with a high degree of success in picking out a cyclically orbiting regime from a stationary equilibrium regime in high-dimensional stochastic dynamical systems.
An (2+1)-dimensional Expanding Model of the Davey-Stewartson Hierarchy As Well As Its Hamiltonian Structure
Yufeng Zhang, Wenjuan Rui, and Hon-Wah Tam
Introducing a new 6-dimensional Lie algebra aims at generating a Lax pair whose compatibility condition gives rise to (1+1)-dimensional integrable hierarchy of equations which can reduce to the nonlinear Schr¨odinger equation and two sets of nonlinear integrable equations by taking various parameters. The Hamiltonian structure of the (1+1)-dimensional hierarchy is also obtained by using the trace identity. The reason for generating the above (1+1)-dimensional integrable hierarchy lies in obtaining (2+1)-dimensional equation hierarchy. That is to say, with the hep of the higher dimensional Lie algebra, we introduce two 4 × 4 matrix operators in an associative algebra A [ ξ ] for which a new (2+1)-dimensional hierarchy of equations is derived by using the TAH scheme and the Hamiltonian operator in the case of 1+1 dimensions , which generalizes the results presented by Tu, that is, the reduced case of the hierarchy obtained by us can be reduced to the Davey-Stewartson (DS) hierarchy. Finally, the Hamiltonian structure of the (2+1)-dimensional hierarchy is produced by the trace identity used for 2+1 dimensions, which was proposed by Tu. As we have known that there is no paper involving such the problem on generating expanding models of (2+1)-dimensional integrable hierarchy.
Nonlinear Electron Distribution Function in a Plasma
D. Benisti, A. Friou, and L. Gremillet
In this paper, we revisit the distribution function derived by making use of the adiabatic approximation, for electrons acted upon by a slowly varying electrostatic wave. This allows us to resolve an apparent paradoxical discrepancy between the results of two published papers, namely, V.B. Krapchev and A.K. Ram (1980), Phys. Rev. A 22, 1229-12242, and, D. B´enisti and L. Gremillet (2007), Phys. Plasmas, 14, 042304. We then briefly recall the relevance of the adiabatic approximation as well as its limitations, and further indicate how to go beyond this approximation to derive very accurate electron distribution functions.
On Stationary Solutions of the Reduced Gardner–Ostrovsky Equation
Maria Obregon and Yury Stepanyants
The detailed analysis of stationary solutions of the reduced Gardner– Ostrovsky (GO) equation is presented. The GO equation (ut + c0ux + αuux + α1u2ux + βuxxx )x = γu is the popular model for the description of large-amplitude internal oceanic waves affected by Earth's rotation. Its reduced version in which the small-scale dispersion is neglected ( β = 0 ) is used when very long internal waves are considered. The equation is also applicable to other types of nonlinear waves in various media (plasma, optical media, relaxing media, etc.) when the large-scale dispersion ∼ γplays a dominant role in comparison with the small-scale dispersion ∼ β. Balancing the nonlinear effect such dispersion gives rise to existence of stationary waves, both periodic and non-periodic. It is shown that only smooth periodic waves make physical sense. Systematic analysis of stationary solutions to the GO equation and their categorisation is presented.
Solvability Relations For Some Diffusion Equations With Convection Terms
Vitali Vougalter and Vitaly Volpert
Linear second order elliptic equations containing the sum of the two Laplace operators with convection terms or a free Laplacian and a Laplacian with drift are considered in Rd. The corresponding operator L may be non Fredholm, such that solvability conditions for the equation Lu = f are unknown. We obtain solvability conditions in H2 (Rd ) for the non selfadjoint problem via relating it to a self-adjoint Schr¨odinger type operator, for which solvability relations are derived in our preceding work.
How to Resist Synchronization Attacks
pp. 1–9 | DOI: 10.5890/DNC.2015.03.001
A.N. Pisarchik, M. Jimenez-Rodriguez, and R. Jaimes-Reategui
Conventional synchronization-based chaotic communication is vulnerable to synchronization attacks enable to recuperate system parameters. However, it is possible to make these attacks inefficient. The simple way to resist synchronization attacks is to change a parameter of the master system faster than the time needed for the system to synchronize. To verify this idea we construct a hybrid communication system composed of two chaotic R¨ossler oscillators and the chaotic logistic map. The latter is used for fast variation of the most sensitive system parameter when the R¨ossler oscillators synchronize. The algorithm is robust to noise in the communication channel.
Lattice Model with Nearest-Neighbor and Next-Nearest-Neighbor Interactions for Gradient Elasticity
Vasily E. Tarasov
Lattice models for the second-order strain-gradient models of elasticity theory are discussed. To combine the advantageous properties of two classes of second-gradient models, we suggest a new lattice model that can be considered as a discrete microstructural basis for gradient continuum models. It was proved that two classes of the second-gradient models (with positive and negative sign in front the gradient) can have a general lattice model as a microstructural basis. To obtain the second-gradient continuum models we consider a lattice model with the nearest-neighbor and next-nearestneighbor interactions with two different coupling constants. The suggested lattice model gives unified description of the second-gradient models with positive and negative signs of the strain gradient terms. The sign in front the gradient is determined by the relation of the coupling constants of the nearest-neighbor and next-nearest-neighbor interactions.
Impact of Nonlinearity of Climate Damage Functions on Long-term Macroeconomic Projections under Conditions of Global Warming
Dmitry V. Kovalevsky, Svetlana I. Kuzmina, and Leonid P. Bobylev
A weakly nonlinear climate damage function used by Nordhaus in his DICE model and a strongly nonlinear climate damage function recently proposed by Weitzman are incorporated in the Kellie-Smith— Cox Integrated Assessment model of coupled climate—economic dynamics, the initial version of which contained only linear climate damages. Long-term projections of the dynamics of coupled climate— economic system are computed with the modified versions of the model. Simulation results for Weitzman function demonstrate pronounced nonlinear emergent dynamics and suggest extremely dramatic long-term economic scenarios for high values of background economic growth rates.
Integrability of a Coupled Harmonic Oscillator in Extended Complex Phase Space
Ram Mehar Singh
With in the frame work of extended complex phase space characterized by x = x1 + ip3, y = x2 + ip4, px = p1 + ix3 and py = p2 + ix4, we investigate the exact invariants for a coupled harmonic oscillator along with PT-symmetric version in two dimensions. For this purpose rationalization method is employed and the invariants obtained in this work play an important role to study the complex trajectories of the concerned classical system.
The Models with Impact Deformations
M. U. Akhmet and A. Kıvılcım
We consider mechanisms with impact deformations such that colliding parts are deformable and the Newton's coefficient of restitution is variable. It is shown how a system with impact deformations can replace the KelvinVoigt viscoelastic model in analysis of a mechanism with contact motion. The suggested impact deformations are compared with the experimental data. By applying deformable surfaces of contacts and non-constant coeffi- cients of restitution, we suppress the chattering in two different mechanical models. We have investigated the existence and stability of periodic solutions in mechanisms with contacts. To actualize the theoretical results, extended examples with simulations are presented.
Unpredictability of Coin Tossing in the Most Sensitive Regions of the Phase Space
Detailed calculations reveal that the sensitivity of the dependence of the result of coin toss (head or tail) on the initial state in the phase space is not only very inhomogeneous but also fractal. In the most sensitive regions, the number of turns of the coin is still fluctuating as the initial height changes within the atomic scale. Thus, the predictability of the toss of a real coin fails in these regions. The portion of such unpredictable regions with sub-atomic sensitivity becomes dominant in the phase space with the improvement of the elasticity of the surface. This offers one unique example in the macroscopic physics that macroscopic determinism fails due to the extreme sensitivity. This also helps to understand why the long-time accurate weather report is not possible.
Operator-theoretic Identification of Closed Sub-systems of Dynamical Systems
Oliver Pfante and Nihat Ay
A central problem of dynamical systems theory is to identify a reduced description of the dynamical process one can deal easier. In this paper we present a systematic method of identifying those closed sub-systems of a given discrete time dynamical system in the frame of operator theory. It is shown that this problem is closely related to finding invariant sigma algebras of the dynamics.
Poincare Recurrences in the Circle Map: Fibonacci Stairs
V.S. Anishchenko N.I. Semenova, and T.E. Vadivasova
We show that the dependence of the mimimal Poincar′e return time on the vicinity size is universal for the golden and silver ratios in the circle map and can be referred to as the "Fibonacci stairs". The rigorous result for the Afraimovich-Pesin dimension equality αc = 1 is confirmed for irrational rotation numbers with the measure of irrationality μ = 2. It is shown that some transcendental number are Diophantine and have the measure μ = 2. It is also confirmed that the gauge function 1/t cannot be applied for Liouvillian numbers. All the obtained features hold for both the linear and the nonlinear circle map.
A Semi-analytical Prediction of Periodic Motions in Duffing Oscillator Through Mapping Structures
Albert C.J. Luo and Yu Guo
In this paper, periodic motions in the Duffing oscillator are investigated through the mapping structures of discrete implicit maps. The discrete implicit maps are obtained from differential equation of the Duffing oscillator. From mapping structures, bifurcation trees of periodic motions are predicted analytically through nonlinear algebraic equations of implicit maps, and the corresponding stability and bifurcation analysis of periodic motion in the bifurcation trees are carried out. The bifurcation trees of periodic motions are also presented through the harmonic amplitudes of the discrete Fourier series. Finally, from the analytical prediction, numerical simulation results of periodic motions are performed to verify the analytical prediction. The harmonic amplitude spectra are also presented, and the corresponding analytical expression of periodic motions can be obtained approximately. The method presented in this paper can be applied to other nonlinear dynamical systems for bifurcation trees of periodic motions to chaos.
Topology of Delocalization in the Nonlinear Anderson Model and Anomalous Diffusion on Finite Clusters
A.V. Milovanov and A. Iomin
This study is concernedwith destruction of Anderson localization by a nonlinearity of the power-law type. We suggest using a nonlinear Schr¨odinger model with random potential on a lattice that quadratic nonlinearity plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localization-delocalization transition with unlimited spreading already at the delocalization border. For super-quadratic nonlinearity the borderline spreading corresponds to diffusion processes on finite clusters. We have proposed an analytical method to predict and explain such transport processes. Our method uses a topological approximation of the nonlinearAnderson model and, if the exponent of the power nonlinearity is either integer or half-integer, will yield the wanted value of the transport exponent via a triangulation procedure in an Euclidean mapping space. A kinetic picture of the transport arising from these investigations uses a fractional extension of the diffusion equation to fractional derivatives over the time, signifying non-Markovian dynamics with algebraically decaying time correlations.
A Method for Solving Nonlinear Differential Equations: An Application to λφ4 Model
Danilo V. Ruy
Recently, it has been great interest in the development of methods for solving nonlinear differential equations directly. Here, it is shown an algorithm based on Pad′e approximants for solving nonlinear partial differential equations without requiring a one-dimensional reduction. This method is applied to the λφ4 model in 4 dimensions and new solutions are obtained.
Synchronization of Micro-Electro-Mechanical-Systems in Finite Time
Hadi Delavari1, Ayyob Asadbeigi, and Omid Heydarnia
Finite time synchronization of chaotic Micro-Electro-Mechanical Sys-tems (MEMS) is considered. In particular, a Lyapunov-based adaptive controller is developed such that convergence of synchronization error is guaranteed globally in the presence unknown perturbations. The system under consideration suffers from bounded parametric uncertainties, additive external disturbances as well as dead zone input nonlinearities. We establish the controller on being resistance against hard nonlinearities by a novel scheme which can be developed to general chaotic systems even. We provide rigorous stability analysis to come up with sufficient conditions that guarantee finite time error convergence of perturbed system. Several simulation scenarios are carried out to verify the effectiveness of obtained theoretical results.
Scaling Modeling of the Emitted Substance Dispersion Transported by Advection Caused by Non-homogeneousWind Field and by Isotropic and Anisotropic Diffusion in Vicinity of Obstacles
Ranis N. Ibragimov, Andrew Barnes, Peter Spaeth, Radislav Potyrailo, and Majid Nayeri
DetailedA simple two-dimensional mathematical approach for source localization of contaminants in the vicinity of individual simple two-dimensional obstacles is proposed. The approach consists of scaling analysis of advectiondiffusion potential flows that can be used in the vicinity of two-dimensional cylindrical obstacles. Three different modeling scenarios are developed in order to simulate the effects of wind. Particularly, the model incorporates the cases of anisotropic diffusion and spatially and temporary inhomogeneous airflow speeds.
Coarse-Graining and Master Equation in a Reversible and Conservative System
Felipe Urbina, Sergio Rica, and Enrique Tirapegui
A coarse graining process is applied to a Ising like model with a conservative and a reversible dynamics. It is shown that, under some assumptions, this coarse graining leads to a tractable probability transfer matrix of finite size which provides a master equation for a coarse graining probability distribution. Some examples are discussed.
On Selective Decay States of 2D Magnetohydrodynamic Flows
Mei-Qin Zhan
The selective decay phenomena has been observed by physicists for many dynamic flows such as Navier-Stoke flows, barotropic geophysical flows, and magnetohydrodynamic (MHD) flows in either actual physical experiments or numerical simulations. Rigorous mathematical works have been carried out for both Navier-Stoke and barotropic geophysical flows. In our previous work, we have rigorously showed the existence of selective states for 2D MHD flows. In this paper, we present a partial result on instability of the selective states.
Modeling Fluid Dynamics in the Ocean and Atmosphere
S.V. Prants
This Special Issue collects together works on analytic solutions and numerical simulation of fluid dynamics in the ocean and atmosphere. The contributed papers address a variety of problems in geophysical fluid dynamics including formation of coherent structures in random hydrodynamic flows, hyperbolicity in the ocean, mesoscale surface and deep vortices in the ocean, formation of vocalized atmospheric vortices and motion of tropical cyclones, convective instability and nonlinear structures in systems with a multi-component convection, instability development in shear stratified flows and others.
Clustering of a Positive Random Field –What is This?
V.I. Klyatskin
It is shown that, in parametrically excited stochastic dynamic systems described by partial differential equations, spatial structures (clusters) can appear with probability one, i.e., almost in every system realization, due to rare events happened with probability approaching to zero. The problems of such type arise in hydrodynamics, magnetohydrodynamics, physics of plasma, astrophysics, and radiophysics.
Equilibrium Distributions for Hydrodynamic Flows
This paper deals with the problem of stochastic structure formation in random hydrodynamic flows. In particular, starting from an analysis of the steady-state probability density, it considers coherent structures of vortex formation (vortex genesis) in stochastic quasi-geostrophic flows, which are related to rotation and random topography of the bottom.
Hyperbolicity in the Ocean
S.V. Prants1, M.V. Budyansky1, M.Yu. Uleysky1, J. Zhang2
Some manifestations of hyperbolicity in the ocean, the important concept in dynamical systems theory, are discussed. It is shown how to identify hyperbolic points, hyperbolic trajectories and their stable and unstable manifolds solving advection equations for passive scalars in a satellite-derived AVISO velocity field and computing finite-time Lyapunov exponents by the singular-value decomposition method. To validate our simulation we use available tracks of oceanic drifters following near surface currents in some areas in the Northwestern Pacific Ocean. The tracks illustrate how drifters "feel" the presence of hyperbolic points, hyperbolic trajectories and stable and unstable manifolds and change abruptly their trajectories when approaching a hyperbolicity region.
Application of the Hydromechanical Model for a Description of Tropical Cyclones Motion
Boris Shmerlin, Mikhail Shmerlin
Within the framework of the hydromechanical model (HMM), proposed by one of the authors, a tropical cyclone (TC) motion is defined by a largescale wind field and a TC intensity. The model contains parameters describing TC and its interaction with wind field. The diagnostic, quasi-prognostic and prognostic calculations of TC movement are carried out. Diagnostic and quasi-prognostic calculations mean that an objective analysis of a large scale wind field and an objective analysis of a TC intensity is used during a TC whole lifetime. In case of diagnostic calculations, model parameters (constants for each TC) are defined from the best coincidence between the real and calculated track of a TC during a TC whole lifetime; for quasiprognostic calculations they are defined during the preliminary "preprognostic" period. Diagnostic calculations show that the HMMrather correctly describes peculiarities of a TC motion. Quasi-prognostic calculations show that model parameters may be rather correctly defined during a preliminary "preprognostic" period. The results of the diagnostic, quasi-prognostic and prognostic calculations are presented.
Influence of Deep Vortices on the Ocean Surface
Daniele Ciani, Xavier Carton1, Igor Bashmachnikov2, Bertrand Chapron3, Xavier Perrot4
We study the influence of deep vortices on the ocean surface in terms of sea-surface elevation, a quantity related to a fluid stream function. We use several mathematical and numerical models, from the most idealized configurations (point vortices) to realistic ones (finite volume vortices). We determine analytically the surface influence of vortices at rest (steady signature) and in motion (dynamical signature). Then, using a nonlinear, numerical hydrodynamicmodel for oceanic vortices, we determine the growth with time of a dynamical signature for drifting vortices without steady signature. We conclude on the possibility to detect several types of oceanic vortices with surface measurements, using the results from our theory and experiments.
The Formation of Localized Atmospheric Vortices of Different Spatial Scales and Ordered Cloud Structures
Boris Shmerlin1, Maxim Kalashnik1,2, Mikhail Shmerlin1
The classical Rayleigh theory of convective instability of a viscous and heat conductive rotating atmospheric layer is generalized to the case of phase transitions of water vapor both for the precipitation convection (PC) and for the nonprecipitation (NPC) one. A principal difference is stated between moist convection and Rayleigh convection, on the one hand, and PC and NPC, on the other hand. In particular, the instability region on the plane of model parameters turned out to generally consist of two subregions, in one of which the localized axisymmetric disturbances with a tropical cyclone (hurricane) structure have the highest growth rate. In case of PC the ascending motions on the axis of symmetry correspond to such disturbances, in case of NPC a spontaneous growth of localized vortices both with ascending and descending motions on the axis is possible. Under other parameters values in case of PC spatially periodic cloud structures (convective rolls or closed cloud cells) have the highest growth rate and in case of NPC–mesoscale systems of convective rolls or mesoscale cloud clusters with annular cloud structures.
An Approach to the Modeling of Nonlinear Structures in Systems with a Multi-component Convection
Sergey Kozitskiy
We consider 3D multi-component convection in a horizontally infinite layer of an uncompressible fluid slowly rotating around a vertical axis. A family of CGLE type amplitude equations is derived by multiple-scaled method in the neighborhood of Hopf bifurcation points. We numerically simulate a case of the three-mode convection at large Rayleigh numbers. It was shown that the convection typically takes a form of hexagonal structures for a localized initial conditions. The rotation of the system prevents the spread of the convective structures on the entire area. The approach to the modeling of the Saturn's polar hexagon on the basis of amplitude equations is discussed.
Instability Development in Shear Flow with an Inflection–Free Velocity Profile and Thin Pycnocline
S.M. Churilov
Weakly stratified flows of the class under study have a wide 3D spectrum of the most unstable waves with very close growth rates and phase velocities so that their individual critical layers merge into a common one. The analysis of evolution equations for those waves has shown that throughout a weakly nonlinear stage of development their amplitudes grow explosively. During the first (three-wave) phase, the most rapidly growing are low-frequency waves whereas at the next phase, when numerous and diverse higher-order wave interactions come into play, the growth of highfrequency waves is accelerated and they overtake low-frequency waves. The results obtained are illustrated by numerical calculations for some ensembles of waves.
Transient Free Surface Flow Past a Two-dimensional Flat Stern
Osama Ogilat1,2, Yury Stepanyants2
A transient free surface flow past a two-dimensional semi-infinite flat plate in the fluid of a finite depth is considered in the linear approximation. It is assumed that the fluid is inviscid and incompressible and the flow is irrotational. The plate is suddenly submerged at relatively small depth below the free surface into the fluid uniformly moving with a constant velocity. The linearized problem is solved for relatively small Froude numbers F < 1 using the Laplace and Fourier transforms, as well as the Wiener– Hopf technique. It is shown that eventually at large time, the transient solution approaches asymptotically the steady-state solution. Peculiarities of the solution obtained are discussed and illustrated graphically.
Cavitating Flow between Two Shear Moving Parallel Plates and Its Control
Yan Liu1, Sheng Ren2, Jiazhong Zhang3
Two parallel plates with shear moving velocity in opposite direction is introduced as external excitations to induce cavitating flow between them, and a developed scheme based on Lattice Boltzmann method is used to simulate and analyze the evolution of the cavitation or phase transition. First, the principles and simulation process of Lattice Boltzmann method and potential models for single component multiphase flow are introduced, including a special model to the moving boundary conditions. Then, the numerical simulations of evolution of phase transition, induced by shear motions of two parallel plates, are carried out in detail, and the complicated pattern formation of cavitating flows are analyzed in such micro- and multiphase dynamic system and some new results are obtained. In particular, the influences of main parameters, such as initial density and moving velocity, on the cavitation and flow pattern are studied further. The results show that the shear moving motion of two parallel plates could induce the cavitation, and the cavitation and cavitating flow pattern could be controlled availably and efficiently by the main parameters listed above. Further, the method and analysis could be extended to flowing liquid, and an idea of drag reduction utilizing the cavitation due to phase transition in such liquid is proposed.
Fractional Dynamics and Systems with Power-Law Memory
M. Edelman, J.A. Tenreiro Machado
The special issue of DNC "Fractional Dynamics and Systems with Power- Law Memory" contains papers related to presentations given at the 5th Conference on Nonlinear Science and Complexity, NSC'14, held on August 4-9, 2014 in Xi'an Jiaotong University, Xi'an, P. R. of China. Within this conference the authors proposed two mini-symposia: "Nonlinear Fractional Dynamics and Systems with Memory" and "Fractional Calculus Applications".
Fractional Calculus: Models, Algorithms, Technology
J.A. Tenreiro Machado
In the last three decades Fractional Calculus (FC) became an area of intense research and development. The accompanying poster illustrates the present day major achievements in the application of FC in physics, engineering and biology.
Fractional Maps and Fractional Attractors. Part II: Fractional Difference Caputo α- Families of Maps
In this paper we extend the notion of an α-family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator. The equations considered are equivalent to maps with falling factorial-law memory which is asymptotically power-law memory. We introduce the fractional difference Universal, Standard, and Logistic α- Families of Maps and propose to use them to study general properties of discrete nonlinear systems with asymptotically power-law memory.
Analysis of Terrorism Data-series by means of Power Law and Pseudo Phase Plane
Antonio M. Lopes, J.A. Tenreiro Machado
Terrorist attacks are catastrophic events often accompanied by a large number of human losses. The statistics of these casualties can be approximated by Power Law (PL) distributions. In this paper we analyze a dataset of terrorist events by means of PL distributions and Pseudo Phase Plane (PPP) technique. We consider worldwide events grouped into 13 geographical regions. First, for each region, we approximate the empirical data by PL functions and we analyze the emerging PL parameters. Second, we model the dataset as time-series and interpret the data as the output of a dynamical system. For each region, we compute the correlation coefficient to find the optimal time delay for reconstructing the PPP. Third, we compare the PPP curves using clustering tools in order to unveil relationships among the data.
Adaptive Memory Identification of Fractional Order Systems
Yang Zhao, Yan Li, Fengyu Zhou
This paper deals with a previously ignored problem that how to find the memory (initialization function) of fractional order system by using the recent sampled input-output data. A novel and practical strategy is proposed to estimate the initialization function, which adapts to all system parameters but fractional order. To implement this method, a P-type order learning approach is introduced to identify the system order separably and accurately, thanks to the fractional order sensitivity function. The initialization response is computed through an iterative learning identification strategy that guarantees the accuracy and adaptiveness simultaneously. Along with the estimations of order and initialization response, a practical piecewise identification criterion of initialization function is established by using the least squares and instrumental variable methods. The above strategy is available for both Caputo and Riemann-Liouville fractional order systems, where the initial values are applied rather than the initial conditions. Two illustrated examples are provided to support the conclusions.
The Optimal Control Problem for Linear Systems of Non-integer Order with Lumped and Distributed Parameters
V.A. Kubyshkin, S.S. Postnov
The optimal control problem for linear dynamic systems of fractional order with lumped and distributed parameters is investigated. This problem is reduced to the classical moment problem. This paper validates the conditions making possible to formulate and resolve the obtained moment problem. Some particular cases of fractional-order systems are discussed. The explicit solutions for the problems of optimal control were obtained in case of systems with lumped parameters. In case of system with distributed parameters an approximate solution analyzed for moment problem. In particular, this paper studies the problem to minimize the norm of control for the assigned time interval and the problem of control with the minimal time of the object transition into the desirable state with the given limitation of the norm of control.
Sliding Mode Control of Fractional Lorenz-Stenflo Hyperchaotic System
Jian Yuan, Bao Shi
This paper proposes sliding mode control for the 4-D fractional order Lorenz-Stenflo hyperchaotic system. Two methods are utilized: one is based on the frequency distributed model of fractional integral operator; and the other is based on the Mittag-Leffler stability theorem and the Caputo operator property. Both of the two methods involve two steps: firstly, constructing a fractional order sliding surface; secondly, designing a single sliding control law for suppression of the nominal plant. Numerical simulations are carried out to verify the efficiency of the theoretical results.
Hybrid Projective Synchronization in Mixed Fractional-order Complex Networks with Different Structure
Li-xin Yang, Jun Jiang, Xiao-jun Liu
In this paper, a fractional-order drive-response complex network model with different order nodes is proposed for the first time. To achieve the hybrid projective synchronization (HPS) of drive-response complex network with different orders, a general strategy is proposed and effective controllers for hybrid projective synchronization are designed. The fractional operators are introduced into the controller to transform problem into synchronization problem between drive-response complex network with identical orders. Numerical simulation results which are carried show that the method is easy to implement and reliable for synchronizing the driveresponse fractional-order complex networks.
Nonlinear Four-point Impulsive Fractional Differential Equations with p-Laplacian Operator
Fatma Tokmak Fen, Ilkay Yaslan Karaca
In this paper, we investigate the existence of solutions for a four-point nonlocal boundary value problem of nonlinear impulsive differential equations of fractional order α ∈ (2,3]. By using some well known fixed point theorems, sufficient conditions for the existence of solutions are established. Some illustrative examples are also discussed.
About Utility of the Simplified Gr¨unwald-Letnikov Formula Equivalent Horner Form
Dariusz W. Brzezinski, Piotr Ostalczyk
First we discuss some crucial factors that determine numerical calculations accuracy of the Gr¨unwald-Letnikov formula and its equivalent Horner form. Then we introduce simplified variants of both formulas and the concept of the calculation tail. We analyze the utility of its length for mitigation of a time and a memory shortages influence on the accuracy in realtime microprocessor calculations. Credibility of the conclusions is lent by the comparison of the results obtained on a PC and on a real-time DSP system.
The Double Exponential Formula as a Gauss Quadratures Replacement for Numerical Integration
We propose to replace the Gauss Quadratures with a numerical integration method known as the Double Exponential (DE) Formula. The numerical quadrature built upon it is at least equivalently accurate and much simpler to customize and apply in situations when tabulated values of the Gauss Quadratures' nodes and weights can not be applied. The DE Formula was developed for integrals with endpoint singularities. However, we confirm that it can be successfully applied to any integral and interval, for which the Gauss Quadratures have been usually selected. To remain compact, the following presentation focuses only on the most difficult integrals, e.g. the improper integrals and the integrals with endpoint singularities. The main part of the paper consists of the calculations accuracy comparison between numerical quadrature based upon the DE Formula and the Gauss-Laguerre, the Gauss-Hermite or Gauss-Chebyshev Quadratures.
Introducing Increasing Returns to Scale and Endogenous Technological Progress in the Structural Dynamic Economic Model SDEM-2
Two nonlinear modifications of the Structural Dynamic Economic Model SDEM-2 are developed and studied analytically and numerically. In the first model version described in the present paper the production function is assumed to be nonlinear that leads to increasing returns to scale, while the second model version proposed describes endogenous technological progress by treating the technology parameter of the production function as an additional state variable. Dependent on the values of model parameters and on initial conditions, both modifications of SDEM considered demonstrate two different dynamic regimes: either an explosive economic growth or the collapse of the economy.
The Existence of Optimal Control for Semilinear Distributed Degenerate Systems
M. Plekhanova
Optimal control problems for a class of semilinear distributed systems unsolved with respect to the times derivative are studied. Two types of initial condition for the system state and various cost functionals are considered in the problems. Abstract results are illustrated by examples of the start control problems for the quasistationary system of phase field equations.
Spin-transfer Torque and Topological Changes of Magnetic Textures
Alberto Verga
The electric manipulation of magnetic textures in nanostructures, important for applications in spintronics, can be realized through the spin-transfer torque mechanism: a spin-polarized current can modify the magnetization of skyrmions and magnetic vortices, and eventually change the topology of the magnetization. The spin-transfer torque and the intrinsic space and time scales of the topological changes are essentially quantum mechanical. We model the interaction between itinerant and fixed spins with a simple tightbinding hamiltonian in a square lattice. The dynamics is described by the Schr¨odinger equation for the electrons and the Landau-Lifshitz equation for the evolution of the magnetic texture. We investigate the phenomenology of the topological change of a Belavin-Polyakov skyrmion under the action of a spin-polarized current and show that adding an exchange dissipation term, regularizes the transition towards a ferromagnetic state.
Equilibrium States Under Constraint in a Variational Problem on a Surface
Panayotis Vyridis, M.K. Christophe Ndjatchi, Fernando Garcıa Flores, Julio Cesar Flores Urbina
We study the equilibrium states for an energy functional with a parametric force field on a region of a surface under a constraint of geometrical character. We use an improved method, based in Skrypnik's variational theories [10]. In local coordinates, equilibrium points satisfy an elliptic boundary value problem. This model can be described as the deformation of the elastic medium and membranes.
Stability of Hopfield Neural Networks with Delay and Piecewise Constant Argument
M.U. Akhmet, M. Karacaoren
In this paper, by using the concept of differential equations with piecewise constant argument, the model of Hopfield neural networks with constant delay is developed. Sufficient conditions for the existence of an equilibrium as well as its global exponential stability by means of Lyapunov functionals and a linear matrix inequality (LMI) are obtained. An example is given to illustrate our results.
Vibrational Resonance in a System with a Signum Nonlinearity
K. Abirami, S. Rajasekar, M.A.F. Sanjuan
We present our investigation on vibrational resonance in a system with a signum nonlinearity. We construct an exact analytical solution of the system in the presence of an external biharmonic force with two frequencies ω and Ω, Ω ω and use it for the computation of the response amplitude Q at the low-frequencyω. We analyse the effect of the strength of the signum nonlinearity on vibrational resonance for the cases of the potential with a single-well, a double-well and a single-well with a double-hump. An interesting feature of vibrational resonance in the system is that Q does not decay to zero for g (the amplitude of the high-frequency force) → ∞. We compare the features of the vibrational resonance of these two systems, since the potential of the system with the signum nonlinearity and that of the Duffing oscillator show ssimilar forms. The strength of the nonlinearity in these two systems is found to give rise distinct effects on resonance.
Controllability of Nonlinear Fractional Delay Integrodifferential Systems
R. Joice Nirmala, K. Balachandran
In this paper we establish the sufficient conditions for controllability of nonlinear fractional delay integrodifferential systems. The results are obtained by using the solution representation of fractional delay differential equations and the application of Schauder's fixed point theorem. Examples are provided to illustrate the results.
Existence of Stationary Solutions for some Systems of Integro-Differential Equations
Vitali Vougalter, Vitaly Volpert
The article deals with the existence of solutions of a system of nonlocal reaction-diffusion equations which appears in population dynamics. The proof relies on a fixed point technique. Solvability conditions for elliptic operators in unbounded domains which fail to satisfy the Fredholm property are being used.
Regular and Singular Pulse and Front Solutions and Possible Isochronous Behavior in the Extended-Reduced Ostrovsky Equation: Phase-Plane, Multi-Infinite Series and Variational Formulations
U. Tanriver, G. Gambino, S. Roy Choudhury
In this paper we employ three recent analytical approaches to investigate several classes of traveling wave solutions of the so-called extendedreduced Ostrovsky Equation (exROE). A recent extension of phase-plane analysis is first employed to show the existence of breaking kink wave solutions and smooth periodic wave (compacton) solutions. Next, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of the travelingwave equations for the exROE equation. These correspond to pulse solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddle equilibrium points of the corresponding traveling-wave equations, as well as ensure simultaneous convergence and continuity of the multi-infinite series solutions for the homoclinic orbits anchored by these saddle points. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. And finally, variational methods are employed to generate families of both regular and embedded solitary wave solutions for the exROE PDE. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and it is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the assumed ansatz for the trial functions). Thus, a direct error analysis is performed, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that not much is known about solutions of the family of generalized exROE equations considered here, the results obtained are both new and timely.
Exogenous Versus Endogenous for Chaotic Business Cycles
Marat Akhmet1, Zhanar Akhmetova2, Mehmet Onur Fen1
We propose a novel approach to generate chaotic business cycles in a deterministic setting. Rather than producing chaos endogenously, we consider aggregate economic models with limit cycles and equilibriums, subject them to chaotic exogenous shocks and obtain chaotic cyclical motions. Thus, we emphasize that chaotic cycles, which are inevitable in economics, are not only interior properties of economic models, but also can be considered as a result of interaction of several economical systems. This provides a comprehension of chaos (unpredictability, lack of forecasting) and control of chaos as a global economic phenomenon from the deterministic point of view. We suppose that the results of our paper are contribution to the mixed exogenous-endogenous theories of business cycles in classification by P.A. Samuelson [1]. Moreover, they demonstrate that the irregularity of the extended chaos can be structured, and this distinguishes them from the generalized synchronization. The advantage of the knowledge of the structure is that by applying instruments, which already have been developed for deterministic chaos, one can control the chaos, emphasizing a parameter or a type of motion. For the globalization of cyclic chaos phenomenon we utilize new mechanisms such as entrainment by chaos, attraction of chaotic cycles by equilibriums and bifurcation of chaotic cycles developed in our earlier papers.
Non-Abelian Bell Polynomials and Their some Applications for Integrable Systems
The noncommutative Bell polynomials and their dual Bell polynomials are presented, respectively, which are extensively applied to mathematics and physics. We make use of them to exhibit a method for generating integrable hierarchies of evolution equations. As applications, we obtain the Burgers hierarchy and a convection-diffusion equation which can be applied to fluid mechanics, specially, be used to represent mass transformations in fluid systems under some constrained conditions. As reduced cases, the Burgers equation which has extensive applications in physics is followed to produce. Furthermore, we obtain a set of nonlinear evolution equations with four potential functions which reduces to a new nonlinear equation similar to the Calogero-Degasperis-Fokas equation. Finally, we discrete the convection-diffusion equation and obtain its three kinds of finite-difference schemes, that is,the weighted implicit difference scheme and the Lax difference scheme. Their some properties including truncation errors, compatibilities and stabilities based on the Von Neumann condition are discussed in detail.
Review on Finite Difference Method for Reaction-Diffusion Equation Defined on a Circular Domain
pp. 133-144 | DOI: 10.5890/DNC.2016.06.003
Walid Abid1, R. Yafia2 †, M.A. Aziz-Alaoui3, H. Bouhafa1, A. Abichou1
In this paper, a finite difference method for a non-linear reaction diffusion equation defined on a circular domain is presented. A simple second-order finite difference treatment of polar coordinate singularity for Laplacian operator, the centered difference approximations, the treatments for Neumann boundary problems are used to discretize this equation. By using this method, numerical solutions can be computed. In the end, we give two applications of reaction diffusion predator-prey models with modified Leslie-Gower and Holling type II functional responses.
Asymptotic Behavior of Solutions of Singular Integro-differential Equations
M.K. Dauylbaev, A.E. Mirzakulova,
We study the asymptotic behavior of the two-point integral boundary value problem for third order integro-differential equations with the small parameter at two highest derivatives. The asymptotic estimations of the solution of the integral boundary value problem is obtained. The obtained results shown that the solution of integral boundary value problem on both sides of given segment has the initial jumps with different orders.
On the Solvability of Nonlocal Boundary Value Problem for the Systems of Impulsive Hyperbolic Equations with Mixed Derivatives
A.T. Assanova
A nonlocal boundary value problem for a system of impulsive hyperbolic equations at the fixed times is considered. The questions of existence, uniqueness, and construction of algorithms for finding the solutions to this problem are studied. By introducing the additional parameters as values of solutions on specific lines the considered problem is reduced to the problem consisting of the Goursat problem for a system of hyperbolic equations and the Cauchy problem for ordinary differential equations. The algorithms for finding the approximate solutions of latter problem are obtained and their convergence to the solution of original problem is proved. Conditions for existence of a unique solution to the nonlocal boundary value problem with impulse effects are set in the terms of initial data.
Nonlinear Dissipation for Some Systems of Critical NLS Equations in Two Dimensions
We prove the global well-posedness in H1(R2,CN) for certain systems of the critical Nonlinear Schrodinger equations coupled linearly or nonlinearly with nonlinear supercritical dissipation terms, generalizing the previous result of [1] obtained for a single equation of this kind.
Dynamical Systems Generated by a Gonosomal Evolution Operator
Utkir A. Rozikov, Richard Varro
In this paper we consider discrete-time dynamical systems generated by gonosomal evolution operators of sex linked inheritance. Mainly we study dynamical systems of a hemophilia, which biologically is a group of hereditary genetic disorders that impair the body's ability to control blood clotting or coagulation, which is used to stop bleeding when a blood vessel is broken. We give an algebraic model of the biological system corresponding to the hemophilia. The evolution of such system is studied by a nonlinear (quadratic) gonosomal operator. In a general setting, this operator is considered as a mapping from Rn, n ≥ 2 to itself. In particular, for a gonosomal operator at n = 4 we explicitly give all (two) fixed points. Then limit points of the trajectories of the corresponding dynamical system are studied. Moreover we consider a normalized version of the gonosomal operator. In the case n = 4, for the normalized gonosomal operator we show uniqueness of fixed point and study limit points of the dynamical system.
Exact Analytic Solutions of Pochammer-Chree and Boussinesq Equations by Invariant Painlev´e Analysis and Generalized Hirota Techniques
Matthew Russo, S. Roy Choudhury
Combinations of truncated Painlev′e expansions, invariant Painlev′e analysis, and generalized Hirota series are used to solve ('partially reduce to quadrature') the integrable Boussinesq and the cubic and quintic generalized Pochammer-Chree (GPC) equation families. Although the multisolitons of the Boussinesq equation are very well-known, the solutions obtained here for all the three NLPDEs are novel, and non-trivial. All of the solutions obtained via invariant Painlev′e analysis are complicated rational functions, with arguments which themselves are trigonometric functions of various distinct traveling wave variables. This is reminiscent of doublyperiodic elliptic function solutions when nonlinear ODE systems are reduced to quadratures. The solutions obtained using recently-generalized Hirota-type expansions are closer in functional form to conventional hyperbolic secant solutions, although with non-trivial traveling-wave arguments which are distinct for the two GPC equations.
Modeling of Complex Rheological Fluids with Fractal Structures
Rakiz M. Sattarov1†, Ilham R. Sattarzada1, Sayavur I. Bakhtiyarov2, and Ranis N. Ibragimov3
An unsteady flow of rheological complex fluid with fractal structure in various pipe geometries was studied when the pressure is a harmonic function of time at a given frequency at the initial cross section of the pipe. An integral type rheological equation by fractional derivative is applied for the first time to model the thixotropic rheological type oils with a high content of wax, resins and asphaltene. The obtained results show that, the relaxation time and the fractal parameters can significantly influence the process of damping pressure along the pipe. It is shown that depending on these parameters the attenuation process may increase and decrease compared to the processes in a viscous damping fluid pressure. The results of simulations are compared to those for resin-asphalt-paraffin oils of two types.
Analytical Prediction of Homoclinic Bifurcations Following a Supercritical Hopf Bifurcation
Tanushree Roy1, Roy Choudhury1†, and Ugur Tanriver2
An analytical approach to homoclinic bifurcations at a saddle fixed point is developed in this paper based on high-order, high-accuracy approximations of the stable periodic orbit created at a supercritical Hopf bifurcation of a neighboring fixed point. This orbit then expands as the Hopf bifurcation parameter(s) is(are) varied beyond the bifurcation value, with the analytical criterion proposed for homoclinic bifurcation being the merging of the periodic orbit with the neighboring saddle. Thus, our approach is applicable in any situation where the homoclinic bifurcation at any saddle fixed point of a dynamical system is associated with the birth or death of a periodic orbit. We apply our criterion to two systems here. Using approximations of the stable, post-Hopf periodic orbits to first, second, and third orders in a multiple-scales perturbation expansion, we find that, for both systems, our proposed analytical criterion indeed reproduces the numerically-obtained parameter values at the onset of homoclinic bifurcation very closely.
Analysis of Stochastic Diffusive Predator PreyModel with Hyperbolic Mortality Rate
M. Suvinthra1†, K. Balachandran1, M. Sambath2
In this work, we establish a Freidlin-Wentzell type large deviation principle for a diffusive predator-prey modelwith hyperbolic mortality rate perturbed by multiplicative type Gaussian noise. We implement the variational representation developed by Budhiraja and Dupuis to establish the large deviation principle for the solution processes.
Nonlinear Parametrizations of Outgoing Longwave Radiation in Zero-Dimensional Energy Balance Models
Dmitry V. Kovalevsky†
A one-layer and two-layer zero-dimensional (0D) energy balance models (EBMs) of the global climate system with different approximations for parametrization of outgoing longwave radiation (OLR) are considered. Three alternative approximations for parametrizing the OLR are explored in detail: (i) the (conventional) linear approximation, (ii) the quadratic approximation, and (iii) the 'exact' (power 4) model. In case of one-layer 0D EBM, exact analytical solutions are derived in closed form for all three alternative approximations for parametrizing the OLR. In the numerical examples provided, the deviations of the linear approximation from the 'exact' model are visible, while the quadratic approximation is virtually indistinguishable from the 'exact' model.
Exact Analytical Solutions of Selected Behaviourist Economic Growth Models with Exogenous Climate Damages
Capital dynamics are calculated for (i) the AK model with output reduced by climate damages, (ii) the AK model with climate-dependent depreciation rate, and (iii) the Solow–Swan model with output given either by the Cobb–Douglas production function or by the constant elasticity of substitution (CES) production function and reduced by climate damages. The climate projections used as model inputs are exogenous. Simple analytical parametrisations for temperature dynamics are assumed (either linear or exponential temperature growth). The quadratic and the Nordhaus climate damage functions are considered. Exact analytical solutions for capital dynamics are derived in closed form (with the exception of the Solow–Swan model with CES production function). Numerical examples are provided for illustrative purposes. As the unabated climate change with unlimited temperature growth is assumed, the long-runmodel dynamics are dramatic: the capital converges to zero at infinite time, and the economy collapses.
On Symmetric Strictly non-Volterra Quadratic Stochastic Operators
U.U. Jamilov
For a symmetric strictly non-Volterra quadratic stochastic operator on the three-dimensional simplex it is proved that this operator has a unique fixed point. A sufficient condition of attractiveness for the unique fixed point is found. For such operators we describe the set of ω− limit points. We proved that some classes of such operators have infinitely many periodic points. Also it is shown that there are trajectories which are asymptotically cyclic with period two.
On the Existence of Stationary Solutions for Some Systems of Integro-Differential Equations with Anomalous Diffusion
Vitali Vougalter1†, Vitaly Volpert2
The article is devoted to the proof of the existence of solutions of a system of integro-differential equations appearing in the case of anomalous diffusion when the negative Laplacian is raised to some fractional power. The argument relies on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains along with the Sobolev inequality for a fractional Laplace operator are being used.
Existence of Semi Linear Impulsive Neutral Evolution InclusionsWith Infinite Delay in Frechet Spaces
Dimplekumar N. Chalishajar†, K. Karthikeyan, A. Anguraj
In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractivemultivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We study a new phase space for impulsive system with infinite delay.
Perspectives on Multi-Level Dynamics
Fatihcan M. Atay1†, Sven Banisch2, Philippe Blanchard3, Bruno Cessac4, Eckehard Olbrich5, Dima Volchenkov6
As Physics did in previous centuries, there is currently a common dream of extracting generic laws of nature in economics, sociology, neuroscience, by focalising the description of phenomena to a minimal set of variables and parameters, linked together by causal equations of evolution whose structure may reveal hidden principles. This requires a huge reduction of dimensionality (number of degrees of freedom) and a change in the level of description. Beyond the mere necessity of developing accurate techniques affording this reduction, there is the question of the correspondence between the initial system and the reduced one. In this paper, we offer a perspective towards a common framework for discussing and understanding multi-level systems exhibiting structures at various spatial and temporal levels. We propose a common foundation and illustrate it with examples from different fields. We also point out the difficulties in constructing such a general setting and its limitations. delay.
Further Results on the Stability of Neural Network for Solving Variational Inequalities
Mi Zhou1†, Xiaolan Liu2,3‡
This paper analyzes and proves the global Lyapunov stability of the neural network proposed by Yashtini and Malek when the mapping is continuously differentiable and the Jacobian matrix of the mapping is positive semi-definite. Furthermore, the neural network is shown to be exponentially stable under stronger conditions. In particular, the stability results can be applied to the stability analysis of variational inequalities with linear constraints and bounded constraints. Some examples show that the proposed neural network can be used to solve the various nonlinear optimization problems. The new results improve the existing ones in the literature.
How the Minimal Poincar´e Return Time Depends on the Size of a Return Region in a Linear Circle Map
N. Semenova, E. Rybalova, V. Anishchenko†
It is found that the step function of dependence of the minimal Poincar´e return time on the size of a return region τinf(ε) for the linear circle map with an arbitrary rotation number can be approximated analytically. All analytical results are confirmed by numerical simulation.
Reversible Mixed Dynamics: A Concept and Examples
S.V. Gonchenko†
We observe some recent results related to the new type of dynamical chaos, the so-called, "mixed dynamics" which can be considered as an intermediate link between "strange attractor" and "conservative chaos". We propose a mathematical concept of mixed dynamics for two-dimensional reversible maps and consider several examples.
We Speak Up the Time, and Time Bespeaks Us
Dimitri Volchenkov†, Anna Cabigiosu, MassimoWarglien
We have presented the first study integrating the analysis of temporal patterns of interaction, interaction preferences and the local vs. global structure of communication in networks of agents. We analyzed face-to-face interactions in two organizations over a period of three weeks. Data on interactions among ca 140 individuals have been collected through a wearable sensors study carried on two start-up organizations in the North-East of Italy. Our results suggest that simple principles reflecting interaction propensities, time budget and institutional constraints underlie the distribution of interaction events. Both data on interaction duration and those on intervals between interactions respond to a common logic, based on the propensities of individuals to interact with each other, the cost of interrupting other activities to interact, and the institutional constraints over behavior. These factors affect the decision to interact with someone else. Our data suggest that there are three regimes of interaction arising from the organizational context of our observations: casual, spontaneous (or deliberate) and institutional interaction. Such regimes can be naturally expressed by different parameterizations of our models.
On Quasi-periodic Perturbations of Duffing Equation
A.D. Morozov†, T.N. Dragunov
Quasi-periodic two-frequency perturbations are studied in a system which is close to a nonlinear two-dimensional Hamiltonian one. The example of Duffing equation with a saddle and two separatix loops is considered. Several problems are studied: dynamical behavior in a neighborhood of a resonance level of the unperturbed system, conditions for the existence of resonance quasi-periodic solutions (two-dimensional resonance tori), global behavior of solutions inside domains separated from the unperturbed separatrix. In a neighborhood of the unperturbed separatrix the problem of relative position of stable an unstable separatrix manifolds is studied, conditions for the existence of doubly asymptotic solutions are found.
A Study of the Dynamics of the Family f λ ,μ = λsinz+μ/(z−kπ) where λ ,μ ∈ R\{0} and k ∈ Z\{0}
Patricia Dom´ınguez†, Josu´e V´azquez, Marco A. Montes de Oca
In this article we investigate the dynamics of the meromorphic family f λ ,μ (z) = λ sin z+ μ/( z−kπ) , λ ,μ ∈ R \ {0} and k ∈ Z \ {0}. We show that for some parameters λ ,μ the Stable set contains an attracting component which is multiply connected and completely invariant. We give a definition of a cut of the space of parameters, with μ and kπ fixed, and show examples of a cut and the Stable and Chaotic sets related to the cut, for some λ given.
New Results on Exponential Stability of Fractional Order Nonlinear Dynamic Systems
Tianzeng Li1,2†, Yu Wang1,3, Yong Yang
In this letter stability analysis of fractional order nonlinear systems is studied. An extension of Lyapunov direct method for fractional order systems is proposed by using the properties of Mittag-Leffler function and Laplace transform. Some new sufficient conditions which ensure local exponential stability of fractional order nonlinear systems are proposed firstly. And we apply these conditions to the Riemann-Liouville fractional order systems by using fractional comparison principle. Finally, three examples are provided to illustrate the validity of the proposed approach.
Robust Exponential Stability of Impulsive Stochastic Neural Networks with Markovian Switching and Mixed Time-varying Delays
Haoru Li1, Yang Fang2, Kelin Li2†
This paper is concerned with the robust exponential stability problem for a class of impulsive stochastic neural networks with Markovian switching, mixed time-varying delays and parametric uncertainties. By construct a novel Lyapunov-Krasovskii functional, and using linear matrix inequality (LMI) technique, Jensen integral inequality and free-weight matrix method, several novel sufficient conditions in the form of LMIs are derived to ensure the robust exponential stability in mean square of the trivial solution of the considered system. The results obtained in this paper improve many known results, since the parametric uncertainties have been taken into account, and the derivatives of discrete and distributed time-varying delays need not to be 0 or smaller than 1. Finally, three illustrative examples are given to show the effectiveness of the proposed method.
Slowing Down of So-called Chaotic States: "Freezing" the Initial State
M. Belger1, S. De Nigris†2, X. Leoncini‡1,3
The so-called chaotic states that emerge on the model of XY interacting on regular critical range networks are analyzed. Typical time scales are extracted from the time series analysis of the global magnetization. The large spectrum confirms the chaotic nature of the observable, anyhow different peaks in the spectrum allows for typical characteristic time-scales to emerge. We find that these time scales τ (N) display a critical slowing down, i.e they diverge as N →∞. The scaling law is analyzed for different energy densities and the behavior τ (N) ∼ √ N is exhibited. This behavior is furthermore explained analytically using the formalism of thermodynamicequations of the motion and analyzing the eigenvalues of the adjacency matrix.
Relaxation Oscillations and Chaos in a Duffing Type Equation: A Case Study
L. Lerman1†, A. Kazakov2,1, N.Kulagin3
Results of numerical simulations of a Duffing type Hamiltonian system with a slow periodically varying parameter are presented. Using theory of adiabatic invariants, reversibility of the system and theory of symplectic maps, along with thorough numerical experiments, we present many details of the orbit behavior for the system. In particular, we found many symmetric mixed mode periodic orbits, both being hyperbolic and elliptic, the regions with a perpetual adiabatic invariant and chaotic regions. For the latter region we present details of chaotic behavior: calculation of homoclinic tangles and Lyapunov exponents.
Potential Symmetries, Lie Transformation Groups and Exact Solutions of Kdv-Burgers Equation
XiaoMin Wang, Sudao Bilige†, YueXing Bai
In this paper, the classical symmetries and the potential symmetries of KdVBurgers equation are calculated based on differential characteristic set algorithm, and the corresponding Lie transformation groups and invariant solutions of the potential symmetry are derived. Moreover a series of new exact solutions for KdV-Burgers equation are obtained by acting Lie transformation groups on the invariant solutions. It is important that these solutions can not be obtained from the classical symmetries of KdV-Burgers equation.
Conservation Laws in Group Analysis of Gas Filtration Model
S.V. Khabirov†
One-dimensional gas filtration was described nonlinear parabolic equation as the conservation law. The potential of the conservation law satisfies a equation as the conservation law. The introduce of the second potential gives a system of equations which admits 6-dimensional Lie algebra. This extends group properties of initial model. With the help of optimal system of subalgebras are classified all invariant and partial invariant solutions which are reduced to invariant solutions of the initial model. Some time it is possible to find integral of invariant submodel.
Existence of Semi Linear Impulsive Neutral Evolution Inclusions with Infinite Delay in Frechet Spaces
Dimplekumar N. Chalishajar†, K. Karthikeyan‡, A. Anguraj§
Wave Collision for the gKdV-4 equation. Asymptotic Approach
Georgy Omel'yanov†
We consider an approach which allows to describe uniformly in time the process of collision of solitary waves. Next we apply it to the KdV-type equation with nonlinearity u4 for three interacting waves assuming that all wave trajectories intersect at the same point. The constructed asymptotic solution satisfies the equation in a weak sense and it can be treated as a classical asymptotics in the sense that it satisfies some conservation and balance laws associated with the gKdV-4 equation. Results of numerical simulation confirm the theoretical conclusion about the elastic type of the wave interaction.
P-Moment Exponential Stability of Caputo Fractional Differential Equations with Random Impulses
Ravi Agarwal1†, Snezhana Hristova2, Donal O'Regan3
Fractional differential equations with random impulses arise in modeling real world phenomena where the state changes instantaneously at uncertain moments. Using queuing theory and the usual distribution for waiting time, we study the case of exponentially distributed random variables between two consecutive moments of impulses. The p-moment exponential stability of solutions is defined and studied when the waiting time between two consecutive impulses is exponentially distributed. The argument is based on Lyapunov functions. We discuss both continuous and differentiable Lyapunov functions and Caputo fractional Dini derivatives and Caputo derivatives are applied. Some examples are given to illustrate our results.
An Analytic Technique for the Solutions of Nonlinear Oscillators with Damping Using the Abel Equation
A Ghose-Choudhury1†, Partha Guha2†
Using the Chiellini condition for integrability we derive explicit solutions for a generalized system of Riccati equations $\ddot{x}+\alpha x^{2n+1}\dot{x}+x^{4n+3}=0$ by reduction to the first-order Abel equation assuming the parameter $\alpha \ge w\sqrt{(2n+1)}$. The technique, which was proposed by Harko et al, involves use of an auxiliary system of first-order differential equations sharing a common solution with the Abel equation. In the process analytical proofs of some of the conjectures made earlier on the basis of numerical investigations in [1] is provided.
On the Existence of Stationary Solutions for Some Systems of Non-Fredholm Integro-Differential Equations with Superdiffusion
Vitali Vougalter1†, Vitaly Volpert2†
We establish the existence of stationary solutions for certain systems of reaction-diffusion equations with superdiffusion. The corresponding elliptic problem involves the operators with or without Fredholm property. The fixed point technique in appropriate H2 spaces of vector functions is employed.
Defensive Driving Strategy for Autonomous Ground Vehicle in Mixed Traffic
Xiang Li1 , Jian-Qiao Sun2†
One of the challenges of autonomous ground vehicles (AGVs) is to interact with human driven vehicles in the traffic. This paper develops defensive driving strategies for AGVs to avoid problematic vehicles in the mixed traffic. A multi-objective optimization algorithm for local trajectory planning is proposed. The dynamic predictive control is used to derive optimal trajectories in a rolling horizon. The intelligent driver model and lanechanging rules are employed to predict the movement of the vehicles. Multiple performance objectives are optimized simultaneously, including traffic safety, transportation efficiency, driving comfort and path consistency. The multi-objective optimization problem is solved with the cell mapping method. Different and relatively simple scenarios are created to test the effectiveness of the defensive driving strategies. Extensive experimental simulations show that the proposed defensive driving strategy is promising and may provide a new tool for designing the intelligent navigation system that helps autonomous vehicles to drive safely in the mixed traffic.
An Impact Oscillator with A Grazing Cycle
M. U. Akhmet†, A. Kıvılcım
An oscillator which impacts against a rigid barrier is taken into account. A cycle with zero impact velocity is discussed. The main result of this article concerns stability of the grazing cycle. A significant attention to a model with the variable coefficient of restitution depending on velocity is paid. The mechanical reasons for that are provided as well as new theoretical advantages have been discovered for the investigation of dynamics near a grazing cycle. The W-map which reduces the system with variable moments of impacts to that with fixed moments and simplifies the analysis, is defined. A new type of linearization system with two compounds is applied to investigate the stability of the grazing cycle whose existence is easily examined. A new approach to suppress a singularity, caused by the tangency, in linearization has been developed. Simulations are provided to visualize the stability of the grazing cycle.
Partially Integrable &weierp T -Symmetric Hierarchies of the KdV and Burgers' Equations in (1+1) and (2+1)
Keri Pecora, S. Roy Choudhury†
In this paper, we generalize the work of Bender and co-workers to derive new partially-integrable hierarchies of various &weierp T -symmetric, nonlinear partial differential equations. The possible integrable members are identified employing the Painlev´e Test, a necessary but not sufficient, integrability condition, and are indexed by the integer n, corresponding to the negative of the order of the dominant pole in the singular part of the Painlev´e expansion for the solution. For the &weierp T -symmetric Korteweg-de Vries (KdV) equation, as with some other hierarchies, the first or n = 1 equation fails the test, the n = 2 member corresponds to the regular KdV equation, while the remainder form an entirely new, possibly integrable, hierarchy. B¨acklund Transformations and analytic solutions of the n = 3 and n = 4 members are derived. The solutions, or solitary waves, prove to be algebraic in form. The &weierp T -symmetric Burgers' equation fails the Painlev´e Test for its n = 2 case, but special solutions are nonetheless obtained. Also, a &weierp T - Symmetric hierarchy of the (2+1) Burgers' equation is analyzed. The Painlev´e Test and invariant Painlev´e analysis in (2+1) dimensions are utilized, and BTs and special solutions are found for those cases that pass the Painlev´e Test.
On Quadratic Stochastic Operators Corresponding to Cyclic Groups
U.A. Rozikov , U.U. Jamilov†
We introduce a new class of quadratic stochastic operators corresponding to cyclic groups. We study the set of fixed points and prove that almost all (w.r.t. Lebesgue measure) trajectories of such operators converge to the center of the simplex. For the cyclic groups of order 2n we show that for any subgroup corresponding quadratic stochastic operator is a regular operator.
Group Analysis of the Generalized Hunter-Saxton System
Yuri Bozhkov1,†, Valter Aparecido Silva Junior2,3
We find the Lie point symmetries of the generalized two-component Hunter-Saxton system. Then we show that it is nonlinearly self-adjoint and establish the corresponding conservation laws using a recent theorem of Nail Ibragimov which enables one to determine conservation laws for problems without variational structure. Finally we obtain some invariant solutions.
Existence of Mild Solutions of Abstract Fractional Differential Equations with Fractional Non-Instantaneous Impulsive Conditions
A. Anguraj1†, S. Kanjanadevi1† , Juan J. Trujillo2†
We study the existence and uniqueness of a mild solution of fractional impulsive differential equations with nonlocal conditions. Here we consider fractional derivative in the non-instantaneous impulsive conditions. We use fixed point techniques and resolvent operators to prove our existence results.
Studying the Dynamcis of Neuronal Membrane Using a Numerical Model
Marina Gonza´lez†, Marco Aure´lio Santos Granero , Mariana Pelissari Monteiro Aguiar Baroni
The Hodgkin and Huxley model describes the electrophysiology of the membrane of the giant squid's axon. This model was developed from measurements of passive and active electrical behavior of the nerve cell. The four-coupled nonlinear ordinary differential equations, which describe the model, are based on the behavior of sodium and potassium channels. The aim of this work is to study the dynamic behavior of a neuronal physiological system described by the Hodgkin-Huxley (H-H) model through the analysis and interpretation of numerical simulations.
Genealogical Tree of Russian schools on Nonlinear Dynamics
S. V. Prants†, M. Yu. Uleysky
One of the most prominent feature of research in Russia and the former Soviet Union is so-called scientific schools. It is a collaboration of researchers with a common scientific background working, as a rule, together in a specific city or even at an institution. The genealogical tree of scientific schools on nonlinear dynamics in Russia and the former Soviet Union is grown. We use these terminology in a broad sense including theory of dynamical systems and chaos and its applications in nonlinear physics. In most cases we connect two persons if one was an advisor of the Doctoral thesis of another one. It is an analogue of the Candidate of Science thesis in Russia. If the person had no official advisor or we don't know exactly who was an advisor, we fix that person who was known to be an informal teacher and has influenced on him/her very much.
Asymptotic Stability of Nonzero Solutions of Discontinuous Systems of Impulsive Differential Equations
K. G. Dishlieva†
Discontinuous systems of nonlinear non-autonomous differential equations with impulsive effects are the main object of investigation in the paper. These systems consist of two basic parts: (i) A set of non-linear nonautonomous systems of ordinary differential equations that define the continuous parts of the solutions. The right-hand sides of the systems are elements of the set of functions f = { f1, f2, ...} ; (ii) The conditions which consistently determine "the switching moments". The structural change (discontinuity) of the right-hand side and impulsive perturbations take place at the moments of switching. In these moments, the trajectory meets the "switching sets". They are parts of the hyperplanes, situated in the phase space of the system considered. Sufficient conditions are found so that the nonzero solutions of the studied discontinuous system with impulsive effects are asymptotically stable.
A Dynamic Systems Approach to the Representation of Policy Implementation Processes in a Multi-Actor World
Dmitry V. Kovalevsky1,2†, Richard Hewitt3,4, Cheryl de Boer5, Klaus Hasselmann6,7
Implementation of any policy involves negotiation between multiple actors, and is therefore susceptible to a dynamic systems modelling approach. In this paper, we make an attempt to develop a family of dynamic systems models of policy implementation in such a multi-actor world by translating an existing, semi-quantitative, application of two theoretical approaches, Contextual Interaction Theory (CIT) and Participatory Action Research (PAR), into a quantitative dynamic framework. We explore various alternative actor-based dynamic systems for this proposed Participatory Contextual Interaction Theory (PCIT), including linear, piecewise linear, and strongly nonlinear models. Analytical results are supplemented with results of numerical simulations. One of the goals of the modelling exercise is to advance the actor dynamics module in the APoLUS land use cellular automata model; illustrative examples of the incorporation of actor dynamics models developed in the present paper into the computation of APoLUS transition potentials are provided.
Simultaneous Stabilization using Set Stabilizing Gain Generation
Venkatesh S. Deshmukh†
A set stabilizing gain generator (SSGG) framework is developed along with dynamic controllers and observers to regulate and estimate the state trajectories of an arbitrary set of three or more open loop unstable linear time invariant dynamic system models of variable dimensions in the state space form. The individual systems in the set are assumed to be controllable and observable. An algorithm is suggested for Single Input (SI) simultaneous stabilization and Single Output (SO) simultaneous observation. This is then extended to a set of Multiple Input (MI) and Multiple Output (MO) models. A dynamic observer based dynamic controller is also formulated for SISO and MIMO systems. The algorithm is then extended to accommodate simultaneous stabilization and observation of system models of variable dimensions. It is finally shown that the dynamic controllers and observers guarantee a certain stability radius against bounded time-varying uncertainty for the entire set which their static counterparts fail to assign. Suitable SI, SO, SISO and MI, MO, MIMO examples are provided.
On Large Deviations of Stochastic Integrodifferential Equations with Brownian Motion
A. Haseena1 , M. Suvinthra2 , N. Annapoorani2†
In this paper, a Freidlin-Wentzell type large deviation principle is established for the stochastic integrodifferential equation driven by finite dimensional Brownian motion. Both the additive and multiplicative noise cases are considered here. Large deviation principle for additive noise case is established via contraction principle whilst weak convergence approach is employed to obtain the same for the multiplicative noise case.
Abundant Solutions of Distinct Physical Structures for Three Shallow Water Waves Models
Abdul-Majid Wazwaz
In this work, we investigate three completely integrable model equations used to describe shallow water waves. A variety of techniques will be sued to determine abundant solutions, of distinct physical structures, for each model. The three models give soliton solutions, periodic solutions, rational hyperbolic functions and rational solutions as well.
Estimates from Below for Blow Up Time for Two Species Chemotaxis System
V. Bhuvaneswari†, K. Balachandran
This paper deals with the two species chemotaxis system in a bounded domain R3under different boundary conditions. Explicit lower bounds for blow up time are obtained by using differential inequality technique under suitable assumptions on the data.
Particle Dynamics in a Viscous Fluid Under the Action of Acoustic Radiation Force
Hassan K. Hassan1 , Lev A. Ostrovsky2 , Yury A. Stepanyants1
Complex dynamics of small particles under the action of acoustic radiation force is considered. This work extends the previous studies of Ostrovsky & Sarvazyan (2009) and Ostrovsky (2015) in two aspects. Firstly, here the particle material can have an arbitrary compressibility and sound speed. Secondly, in addition to the viscosity effect described by the Stokes drag force, the Boussinesq–Basset drag force and the inertial force which includes added mass effect are accounted for. Although, as was assumed in the earlier publications, the latter effects are usually small, nevertheless their influence can be noticeable in specific cases considered in this paper. The control of particle motion by switching of acoustic modes is also studied for particles of different properties. Quantitative estimates are given for particles made of different materials.
Bifurcation Trees of Period-1 Motions to Chaos of a Nonlinear Cable Galloping
Bo Yu1 and Albert C. J. Luo2†
In this paper, period-m motions on the bifurcation trees of peiod-1 to chaos for nonlinear cable galloping are studied analytically, and the analytical solutions of the period-m motions in the form of the finite Fourier series are obtained through the generalized harmonic balance method, and the corresponding stability and bifurcation analyses of the period-m motions in the galloping system of nonlinear cable are carried out. The bifurcation trees of period-m motions to chaos are presented through harmonic frequency-amplitudes. Numerical illustrations of trajectories and amplitude spectra are given for periodic motions in nonlinear cables. From such analytical solutions of periodic motions to chaos, galloping phenomenon in flow-induced vibration can be further understood.
Bifurcation and Stability Analysis of a Diffusive Predator-Prey Model with Ratio-Dependent Type III Functional Response
M. Sivakumar†, K. Balachandran
This paper is concerned with a diffusive Leslie-Gower predator-prey system with ratio dependent Holling type III functional response subject to Neumann boundary conditions. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation in detail, local stability, existence of a Hopf bifurcation at the coexistence of the equilibrium and stability of bifurcating periodic solutions of the system in the absence of diffusion are studied. Furthermore, Turing instability and Hopf bifurcation analysis for the system with diffusion are studied. Finally, numerical simulations are provided in order to verify our theoretical results.
Controllability of Nonlinear Stochastic Fractional Systems with L´evy Noise
R. Mabel Lizzy∗,K. Balachandran† and M. Suvinthra‡
In this paper we study the controllability of linear and nonlinear stochastic fractional systems driven by L´evy noise. Here we use the L´evy-Itˆo decomposition of an arbitrary L´evy process into Brownian and Poisson parts. The necessary and sufficient conditions for controllability of the linear system is obtained. Also, the nonlinear system is shown controllable under the assumption that the corresponding linear system is controllable and using the Banach contraction principle.
Modelling, Analysis and Control of Nonlinear Discrete and Continuous Mechanical Structures Dedicated for Mechatronic Applications
pp. 421–423 | DOI: DOI:10.5890/DNC.2017.12.001
Jan Awrejcewicz, Dariusz Grzelczyk
This is the Special Issue dedicated to the 13th International Conference on "Dynamical Systems - Theory and Applications" (DSTA-2015) which was held on December 7-10, 2015 in Lodz Poland. The main aim of the conference was to provide a platform for researchers and engineers to present and discuss the current state and contemporary investigations in different disciplines of science, bioscience, and engineering. This issue presents eight selected manuscripts related to the modeling, analysis and control of nonlinear discrete and continuous mechanical structures met in mechatronic applications.
Dynamics of Waves in the Cubically Nonlinear Model for Mutually Penetrating Continua
Vjacheslav Danylenko, Sergii Skurativskyi†
In this report we study the mathematical model for mutually penetrating continua. This model consists of the wave equation describing the carrying medium and the equation for oscillators forming the oscillating inclusion. Prescribing the constitutive equation of the carrying medium and kinetics of oscillator's dynamics for model in question, the cubic nonlinearity is accounted. We are interested in the structure of wave solutions obeying the dynamical system of Hamiltonian type. This allows us to determine the peculiarities of the phase space of dynamical system, namely, the relation describing the homoclinic trajectories, the division of phase plane into the parts with equivalent orbits' behavior, the conditions of bifurcations. To simulate the wave dynamics, we construct the three level finite-difference numerical scheme and study the evolution of solitary waves, their pair interactions and stability. Propagation of periodic waves is modeled as well.
Design of a Tracking Controller for Object Interception in Space
Elzbieta Jarzebowska, Bartlomiej Pilarczyk
The paper presents a model-based tracking controller design for a freefloating space robot for a mission scenario of intercepting an object. Such missions are of interest due to a growing number of objects needed to be removed from space. The free-floating mode requires spacecraft thrusters to be off and linear and angular momentum are conserved then. Momentum conservation generates holonomic and nonholonomic constraint equations, respectively. The free-floating mode implicates underactuation, so the robot becomes multi-constrained. Many control algorithms are designed for underactuated robots but they are specific mission and robot dependent. Motivations for the presented research come from the growing space exploration, which results in more space debris and requires sophisticated removal services. Service tasks and debris removal need to be performed by specialized robots. The debris interception scenario presented in the paper consists of estimation of target properties, a controller design to track and intercept the debris, and move it to the graveyard orbit. Simulation results of the theoretical control development for the robot intercepting a non-tumbling object are provided.
Implementation Assessment of a Wave Energy Converter, Based on Fully Enclosed Multi-axis Inertial Reaction Mechanisms
Ioannis A. Antoniadis1, Vasilis Georgoutsos1, Andreas Paradeisiotis1, Stratis A. Kanarachos2, Konstantinos Gryllias3
This paper examines the implementation of a standalone 1 MW Wave Energy Converter (WEC), based on a novel concept of a class of WECs, consisting in fully enclosed appropriate internal body configurations, which provide inertial reaction against the motion of an external vessel. Acting under the excitation of the waves, the external vessel is subjected to a simultaneous surge and pitch motion in all directions, ensuring maximum wave energy capture. The internal body is suspended from the external vessel body in such an appropriate geometrical configuration, that a symmetric four bar mechanism is essentially formed. The first advantage of this suspension geometry is that a linear trajectory results for the center of the mass of the suspended body with respect to the external vessel, enabling the introduction of a quite simple form of a Power Take-Off (PTO) design. The simplicity and symmetry of the suspension geometry and of the PTO, ensure a quite simple and robust technological implementation. Mass and inertia distribution of the internal body are optimized, ensuring maximal conversion and storage of wave energy. As a result, the internal body assembly is essentially, dynamically equivalent to a vertical physical pendulum. However, the resulting equivalent pendulum length and inertia can far exceed those that can be achieved by a simple horizontal or vertical pendulum, suspended or inverted, leading to a significant reduction of the suspended mass.
Multi-parametric Dependence of DeformationWork of Zona Pelucida in Fertilization Process Trough Quasi-static Continual Shell-like ZP Model
Andjelka Hedrih1†, Katica (Stevanovic) Hedrih1,2
<\it>Zona Pelucida (ZP) is highly specialized glycosylated and sulfated polymer gel that surrounds mammalian oocyte, exhibits elastic or viscoelastic properties and change of diameter in different maturation stages. In process of in vitro fertilization, sperm cells affect the external ZP surface and transfer certain amount of energy to the ZP structure that goes through the deformation work. ZP thickness and number of motile sperm cells are important factors that may influence the fertility. Using the quasi-static approximate ZP model in the form of hollow sphere the numerical analysis of how specific deformation work depends of different variables like: ZP thickness, specific point in ZP, external pressure that comes from sperm cells were done. According to the model, sperm that make pressure upon ZP surface are in the form of homogeny discrete continuum distribution in radial directions. For specific ZP point in the model, analytical expressions of component stresses and strains were obtained as well as deformations and volume dilatation in ZP. The results are discussed from biomechanical aspect of fertilization. Limitations of the model are also discussed.
Fatigue Cracks Detection in Rectangular Plates with Circular Hole with the Use of Elastic Waves
Marek Barski†, Adam Stawiarski, Piotr Pajak
The current work is devoted to the problem of fatigue cracks detection and evaluation in the case of isotropic rectangular plates with circular holes. The cutout is located in the geometrical center of the plate. The structure is subjected to the cyclic tension. It causes the formation of the fatigue cracks in the vicinity of the hole. This type of damages can be effectively detected by the analyzing of elastic waves propagation. In the presented work the propagation of the elastic waves in the vicinity of a hole is simulated with the use of the finite element method. It is assumed that the elastic waves are excited and picked up by piezoelectric sensors. The most effective position of the piezoelectric actuator is looked for while the position of the array of sensors is fixed. Four different possibilities of the location of the actuator are studied. Moreover, the advanced algorithm of crack detection and evaluation is also discussed.
Free Vibrations of Cantilever Bars with Linear and Nonlinear Variable Cross-Section
Jacek Jaworski, Olga Szlachetka
The topic of this study is the first mode of natural transverse vibrations of isotropic, homogeneous and elastic bars (columns or beams) with clamped bottom and free head. The columns of the first group are shaped as truncated solid cones or as tubes with linearly variable wall thickness and with different inclination of lateral faces, from cylinder to cone. The columns of the second group were shaped in similar way, but the generatrices of the solids of revolution were curvilinear ?in the shape of a parabola. The first frequency of free vibrations was determined using the Rayleigh method. The deflection line of the column axis during the vibration was assumed in form of the bending line of the column axis subjected to a uniform load. Resulting frequencies (or periods) were compared with these obtained with the use of FEM (ANSYS) and a good compliance of results was observed. As the expression for the energy of an elementary slice of material was integrated over the length of the rod, the formula for the frequency was obtained in form of an integral equation. In some cases an exact solution of integral equation was obtained, however in other cases only a numerical solution was possible.
Study of Negative Elements for Discrete Mechatronic Systems
Andrzej Buchacz†, DamianGałe¸ziowski
In the paper, the known problem of vibration control have been studied for elements which have defined configurations and connections of piezostack actuator with external electric network and can exhibits negative values. Depending on the phase of the synthesis process, used for the design of the mechatronic systems that have to comply with some given requirements, negative elements have been identified and described. Subsequently with the study on selection of corresponding negative stiffness and damping, presented in the graph form, optimal values for the systems can be compared. Following examples with selected calculations, the goal of this paper is to present limits and constrains that may support the physical realization, as well as applications of the considered systems.
Fibre Spring-damper Computational Models in a Laboratory Mechanical System and Validation with Experimental Measurement
Pavel Polach1†, Miroslav Byrtus1, ZbynekSika2, Michal Hajzman1
The paper deals with searching for the phenomenological model of a fibre. A multibody model of a simple weight-fibre-pulley-drive mechanical system is created. This mechanical system is focused on the investigation of a fibre behaviour and experimental laboratory measurements are performed on it. The carbon fibre, which is driven by one drive, is led over a pulley. On its other end there is a prism-shaped steel weight, which moves in a prismatic linkage on an inclined plane. Drive excitation signals can be of different shapes with the possibility of variation of a signal rate. Time histories of the weight position and of the force acting in the fibre are measured. At simulating with the multibody model of this system there is investigated the influence of dependences of the fibre spring-damper coefficients on the velocity of the weight motion in the computational model, on the coincidence of the simulation results and the experimental measurement results are evaluated. The aim of the simulations is to create a phenomenological model of the fibre, which will be utilizable in fibre modelling in the case of more complicated mechanical or mechatronic systems.
Existence and Uniqueness of Solutions for a Coupled System of Higher Order Fractional Differential Equations with Integral Boundary Conditions
P. Duraisamy, T. Nandha Gopal
In this article, we study the existence of solutions for a coupled system of higher order nonlinear fractional differential equations with non-local integral boundary condition by using Schaefer's fixed point theorem and the uniqueness result is proved by the contraction mapping principle. Finally, examples are provided to the applicability our main results.
Almost Periodicity in Chaos
pp. 15-29 | DOI: 10.5890/DNC.2018.03.002
Marat Akhmet, Mehmet Onur Fen
Periodicity plays a significant role in the chaos theory from the beginning since the skeleton of chaos can consist of infinitely many unstable periodic motions. This is true for chaos in the sense of Devaney[1],Li-Yorke[2]and the one obtained through period-doubling cascade [3]. Countable number of periodic orbits exist in any neighborhood of a structurally stable Poincare homoclinic orbit, which can be considered as a criterion for the presence of complex dynamics [4–6]. It was certified by Shilnikov [7] and Seifert [8] that it is possible to replace periodic solutions by Poisson stable or almost periodic motions in a chaotic attractor. Despite the fact that the idea of replacing periodic solutions by other types of regular motions is attractive, very few results have been obtained on the subject. The present study contributes to the chaos theory in that direction. In this paper, we take into account chaos both through a cascade of almost periodic solutions and in the sense of Li-Yorke such that the original Li-Yorke definition is modified by replacing infinitely many periodic motions with almost periodic ones, which are separated from the motions of the scrambled set. The theoretical results are valid for systems with arbitrary high dimensions. Formation of the chaos is exemplified by means of unidirectionally coupled Duffing oscillators. The controllability of the extended chaos is demonstrated numerically by means of the Ott-Grebogi-Yorke [9] control technique. In particular, the stabilization of tori is illustrated.
Backlund Transformation and Quasi-Integrable Deformation of Mixed Fermi-Pasta-Ulam and Frenkel-Kontorova Models
Kumar Abhinav, A Ghose Choudhury, Partha Guha
In this paper we study a non-linear partial differential equation (PDE), proposed by Kudryashov [arXiv:1611.06813v1[nlin.SI]], using continuum limit approximation of mixed Fermi-Pasta-Ulam and Frenkel-Kontorova Models. This generalized semi-discrete equation can be considered as a model for the description of non-linear dislocation waves in crystal lattice and the corresponding continuous system can be called mixed generalized potential KdV and sine-Gordon equation. We obtain the B¨acklund transformation of this equation in Riccati form in inverse method. We further study the quasi-integrable deformation of this model.
A New Comparison Theorem and Stability Analysis of Fractional Order Cohen-Grossberg Neural Networks
Xiaolei Liu, Jian Yuan, Gang Zhou, Wenfei Zhao
This paper proposes a new comparison theorem and stability analysis of fractional order Cohen-Grossberg neural networks. Firstly, a new comparison theorem for fractional order systems is proved. Secondly, the stability of a class of fractional order Cohen-Grossberg neural networks with Caputo derivative is investigated on the basis of the above comparison theorem. Thirdly, sufficient conditions of stability of the neural networks are obtained utilizing the property of Mittag-Leffler functions, the generalized Gronwall-Bellman inequality and the method of the integral transform. Furthermore, a numerical simulation example is presented to illustrate the effectiveness of these results.
Existence of Solutions of Stochastic Fractional Integrodifferential Equations
P. Umamaheswari, K. Balachandran, N. Annapoorani
In this paper, a general class of stochastic fractional integrodifferential equations is investigated. The Picard-Lindel¨of successive approximation scheme is used to establish the existence of solutions. The uniqueness of the solution is also studied under suitable conditions.
Evolution Towards the Steady State in a Hopf Bifurcation: A Scaling Investigation
Vin´ıcius B. da Silva, Edson D. Leonel
Some scaling properties describing the convergence for the steady state in a Hopf bifurcation are discussed. Two different procedures are considered in the investigation: (i) a phenomenological description obtained from time series coming from the numerical integration of the system, leading to a set of critical exponents and hence to scaling laws; (ii) a direct solution of the differential equations, which is possible only in the normal form. At the bifurcation, the convergence to the stationary state obeys a generalized and homogeneous function. For short time, the dynamics giving by the distance from the fixed point is mostly constant when acritical time is reached hence changing the dynamics to a convergence for the steady state given by a power law. Both the size of the constant plateau and the characteristic cross overtime depend on the initial distance from the fixed point. Near the bifurcation, the convergence is described by an exponential decay with a relaxation time given by a power law.
Integrability of a time dependent coupled harmonic oscillator in higher dimensions
Anand Malik, Ram Mehar Singh, S. B. Bhardwaj, Kushal Sharma, Richa Rani, Fakir Chand
Within the frame work of extended complex phase space characterized by x = x1 +ip4,y = x2 + ip5,z = x3 +ip6,px = p1 + ix4, py = p2 +ix5 and pz = p3 +ix6, we investigate the exact dynamical invariant for a coupled harmonic system in three dimensions. For this purpose Lie-algebraic method is employed and the invariant obtained in this work may play an important role in reducing the order of differential equations, solution of Cauchy system and to check the accuracy of a numerical simulation.
Approximation of random fixed point theorems
pp. 95-105 | DOI: 10.5890/DNC.2018.03.008
The aim of this paper is to establish and discuss the approximation of Caristi's random fixed point theorems. Our theorem is used to determine a large numbers of nonlinear stochastic problems.
Fibre Three-point multi-term fractional integral boundary value problems of fractional functional differential equations with delay
K. Sathiyanathan, V. Krishnaveni, M. Sivabalan
In this paper, we study fractional functional differential equations with three-point multi-term boundary conditions. Our method of analysis is based on the reduction of the given system to an equivalent system of integral equations. Existence and uniqueness results are obtained by using Schauder fixed point theorem and contraction principle. An illustrative example is also presented.
Dynamics and stability results of fractional integro-differential equations with complex order
D. Vivek, K. Kanagarajan, S. Harikrishnan
In this paper, we study the existence, uniqueness and Ulam stability of solutions of fractional integro-differential with complex order. Based on Krasnoselkii fixed point theorem and Banach contraction principle, we obtain existence and Ulam stability results.
Nonlinear Time Series Analysis of Complex Systems Using an e-Science Web Framework
Bruno B. F. Leonor, Walter A. dos Santos, Asiel Bomfin Jr., Reinaldo R. Rosa
The analysis of time series in the era of Big Data has become a major challenge for computational framework research. Furthemore, in the areas of space science which deals with a large variety of data, the practical consistence between workload, workflow and cloud computing is crucial. Here, such consistence is provided by an innovativee-Science framework project named Sentinel which is based on a NoSQL data base (MongoDB) and a containerization platform (Docker). This web framework supports researchers for time series analysis in a cloud environment where they can easily access, parameterize, initialize and monitor their applications. As a case study in the Brazilian Space Weather Program, we consider the intensive analysis of time series from a complex information system for solar activity monitoring and forecasting. As a prototype for implementing the framework, the DFA (detrended fluctuation analysis) techniquewas used as a nonlinear spectrum analyzer applied to the solar irradiance measurements from 1978 to 2012. Moreover, new applications can be added and managed by researchers on the portal easily to complement their data analysis purposes.
Simulating the Interaction of a Comet With the Solar Wind Using a Magnetohydrodynamic Mode
Edgard de F. D. Evangelista†, Margarete O. Domingues, Odim Mendes, Oswaldo D. Miranda
We present simulations of a comet interacting with the solar wind. Such simulations are treated in the framework of the ideal,2D magnetohydrodynamics (MHD), using the FLASH code in order to solve the equations of such a formalism. Besides, the cometis treated as a spherically symmetric source of ions in the equations of MHD. We generate results considering several scenarios, using different values for the physical parameters of the solar wind and of the cometin each case. Our aim is to study the influence of the solar wind on the characteristics of the comet and, given the nonlinear nature of the MHD, we search for the occurrence of phenomena which are typical of nonlinear systems such as instabilities and turbulence.
Using RQA for evaluating heart rate patterns under controlled breathing
Guênia Ladeira, Luís Cláudio Oliveira-Lopes, Gabriela V. Lima
In this work, the variation of time and amplitude of the electrical stimulation of the heart muscle is analyzed during respiratory control exercise, effect of the autonomic nervous system. The specifics are verified by recurrence plot, and recurrence quantification analysis. The analysis revealed distinguished behavior of the electrical stimulation of the heart. And, the cross recurrence between the variations of amplitude and time demonstrated the equilibrium relationship in the cardiac cycles, which are accentuatedin the exerciseof concentration.
Using Different Interval Extensions to Increase the Accuracy of the Exact Solution on Recursive Functions
H. M. Rodrigues Junior, M. L. C. Peixoto, M. E.G. Nepomuceno., S.A.M. Martins
The results obtained by numerical computation are not always precise. It happens because a computer has storage limitations and its set of numbers is finite. In this work, the interval analysis is used to give bounds around exact solution of the logistic map function. The connection between computer and interval mathematics makes possible to solve problems that can not be solved efficiently using floating point arithmetic. We use the intersection of different pseudo-orbits obtained by interval extensions to reduce the bounds of the exact solution. The method is applied using the Intlab toolbox. Without any substantial computational effort, we show a reduction of up to 26% in the width of interval by applying the method proposed in this paper.
Semantic networks of keywords from Brazilian dissertations on physics teaching
J. O. do Nascimento, R. L. S. Monteiro, D. M. Moreira, M. A. M. Simões Gonçalves, H. B. de Barros Pereira
Metrics of social and complex networks can be used in the study of emergent patterns in semantic networks. The goal of this article is to analyze the semantic network formed by the keywords of theses and dissertations on physics teaching that were defended in Brazil between the years 1996 and 2006. We describe the method used for constructing the network, including the keywords. We analyze the network's properties in local and global contexts. We study which keywords are presented as cut points, since their removal from the network result in a number of components greater than one. Our results suggest that the networkis small–world and scale–free. In addition, using the proposed method, we verified that most of the topics emphasized in the network are related to the education of physics instructors andnotto methods of teaching physics at different levels.
Exploiting the rounding mode of floating-point in the simulation of Chua's circuit
M. R. Silva, E. G. Nepomuceno, G. F. V. Amaral, S. A. M. Martins, L. G. Nardo
The Chua's circuit has been considered as one of the most important paradigms for nonlinear science studies. Its simulations is usually undertaken by means of numerical methods under the rules of IEEE 754-2008 floating-point arithmetic standard. Although, it is well known the propagation error issue, less attention has been given to its consequences on the simulation of Chua's circuit. In this paper we presented a simulation technique for the Chua's circuit, it exhibits qualitative differences in traditional approaches such as RK3, RK4 and RK5. By means of the positive largest Lyapunov exponent we show that for the same initial condition and same set of parameters, we produce a periodical and a chaotic solution.
Nonlinear integral inequalities with parameter and bounded solutions of dynamical systems
Taoufik Ghrissi, M. A. Hammami
We present in this paper some new nonlinear integral inequalities with parameter and we will use them to show the boundedness of solutions of perturbed nonlinear time-varying differential equations. Furthermore, some examples are given to illustrate the applicability of the obtained results.
Polar Amplification Projected by Energy Balance Model with Nonlinear Parametrization of Outgoing Longwave Radiation
Dmitry V. Kovalevsky, Genrikh V. Alekseev
A linear energy balance model (EBM) of polar amplification (PA) developed by Langen and Alexeev (2007) is modified by replacing the conventional linear parametrization of outgoing longwave radiation (OLR) with a more accurate nonlinear (quadratic) approximation for OLR.The nonlinear version of the model is studied analytically and numerically. The numerical results presented in the paper convincingly demonstrate the importance of adopting the nonlinear approximation for parametrizing the OLR in EBMs. The simulation results yielded by the nonlinear version of the model are visibly different from those provided by the linear model, especially for simulations of climate change in high latitudes and for quantification of PA with dedicated indices.
Global Stability Analysis of a General Scalar Difference Equation
Huseyin Merdan, Ozlem Ak Gumus, Gamzegul Karahisarli
We consider a general first-order scalar difference equation with and without Allee effect. The model without Allee effect represents asexual reproduction of a species while the model including Allee effect represents sexual reproduction. We analyze global stabilities of both models analytically and compare the results obtained. Numerical simulations are included to support the analytical results. We conclude that Allee effect has a destabilizing effect on the global stability of the model. This result is different fromthe local stability behaviour of the same fixed point of the model.
On the Well-posedness of the Magnetic, Semi-relativistic Schrodinger-Poisson System
We prove global existence and uniqueness of strong solutions for the Schr¨odinger-Poisson system in the repulsive Coulomb case with relativistic, magnetickinetic energy.
Asymptotic Stability of Caputo Fractional Singular Differential Systems with Multiple Delays
Sivaraj Priyadharsini, Venkatesan Govindaraj
In this work, Lyapunov functions combining Matrix inequalities or Matrix equations are developed to analyze the asymptotic stability of Caputo fractional singular systems with multiple time-varying delay. Integer-order derivatives of the Lyapunov functions are used to derive asymptotic stability criteria. The main advantage of applying stability criteriais that noneed of solving roots of transcendental equations. Some examples are provided to explain the effectiveness of the proposed criteria.
Stability of Fractional Differential Equations without Singular Kernal
S. Harikrishnan, K. Kanagarajan, D. Vivek
In this paper,we establish four types of Ulam stability,namely Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias and generalized Ulam-Hyers-Rassias stability for differential equation of fractional order.
Van der Pol Oscillators Generated from Grazing Dynamics
M. U. Akhmet, A. Kıvılcım
In this paper, we take into account two coupled Van der Pol equations with impacts. The main novelty is that the degenerated system for the model admits an oscillation with zero impact velocity. To prove presence of oscillations, beside the perturbation method, the newly developed linearization for dynamics with grazing has been applied. As different from the theoretical results such as Nordmark mapping and Zero time discontinuity mapping, the grazing is examined through another method of discontinuous dynamics, which diminishes the role of mappingsin the analysis. The rich diversity of changes in the dynamics is observed under regular perturbations, since of the grazing discontinuity. By means of the simulation results, the analytical studies are visualized.
A novel controllability analysis of impulsive fractional linear time invariant systems with state delay and distributed delays in control
B.Sundara Vadivoo, R.Raja, R.P.Agarwal, G.Rajchakit
In this paper, we investigate the controllability of impulsive fractional linear time invariant systems with state delay and distributed delays in control. By using the controllability Grammian matrix which is defined by the Mittag Leffler matrix function , a new set of sufficient conditions are obtained for the considered system to be controllable. Finally, two numerical examples are given to demonstrate the validity of the obtained the oretical results.
Tikhonov theorem for differential equations with singular impulses
M. Akhmet, S. Cag
The paper considers impulsive systems with singularities. The main novelty of the present research is that impulses (impulsive functions) are singular. This is beside singularity of differential equations. The most general Tikhonov theorem for the impulsive systems is obtained. Illustrative examples with simulations are given to support the theoretical results.
Approximate Controllability Results for Impulsive Partial Functional Nonlocal Integro-differential Evolution Systems through Resolvent Operators
Mahalingam Nagaraj, Selvaraj Suganya, Dumitru Baleanu, Mani Mallika Arjunan
This paper investigates the existence and approximate controllability results for a class of impulsive functional integro-differential evolution systems with nonlocal conditions via resolvent operators in Banach spaces. By making utilization of Banach contraction principle and Schaefer's fixed point theorem along with resolvent operators and semigroup theory, we build up the desired results. As an application, we also consider an impulsive partial functional integro-differential equations.
Nonlinear Modifications of Perturbation Theory with Some Examples of Applications to Complex System Analysis
V.V. Uchaikin, V.A. Litvinov, E.V.Kozhemjakina
The article reviews some nonlinear generalizations of the perturbation theory performed by various authors (including some co-authors of this article) in various years, and can be useful in analyzing complex systems and inverse problems. There is considered the problem of widening the range of perturbation theory applicability with simultaneous reducing the complexity of computational algorithm. The consideration is based on the duality principle, including adjoint (in the Lagrange sense) operators and importance functions. The article contains an exposition of perturbation theory of higher orders, variational methods of higher orders, symmetry transformation and some other question. Special attention is paid to such a promising approachas multi-reference methods. It is shown that this approach extends the range of applicability of perturbation theory and saves calculationstime. To demonstrate details of the method,some examples of transport problems from reactor engineering and astrophysics are considered.
On I-lacunary summability methods of order a in intuitionistic fuzzy n-normed spaces
E. Savas
In this paper we introduce and study the notion I -statistical convergence of order α , and I -lacunary statistical convergence of order α with respect to the intuitionistic fuzzy n-normed space and also we investigate their relationship and some inclusion theorems are proved.
Modularity, redundancy and the problem of "sex"
Christopher R. Stephens
Homologous recombination, and the associated notion of sexual reproduction, are ubiquitous in the evolution of biological organisms. However, there is still no clear, generally accepted understanding of why it exists and under what circumstances it is useful. Here, we argue that its utility is strongly linked to specific types of fitness landscape. In particular, landscapes that are quasi-modular (weak additive epistasis) or redundant (negative epistasis), two properties that are also ubiquitous in biological systems. We further argue that recombination and modularity together are responsible for the formation of "building block hierarchies" where, to make compatible the two different types of building block, landscape blocks associated with modularfitness landscapes and Building Block schemata defined by the recombination distribution, a meta-evolution is necessary, wherein the recombination distribution itself evolves, leading to recombination hot spots at the boundaries of landscape blocks. We finally argue that recombinative dynamics and modular landscapes are necessary conditions for the formation of building block hierarchies and, ultimately, life itself.
Complex Inference Networks: A new tool for spatial modelling
Christopher R. Stephens, Ra´ul Sierra Alcocer, Constantino Gonz´alez Salazar
All systems - physical, biological, ecological and social - are composed of hierarchies of building blocks- atoms, molecules, cells, tissues, individuals, species etc. - with corresponding interactions, wherein the presence of interactions- attractive and repulsive- affects the relative spatio-temporal distribution of the building blocks. In physical systems, in particular, the structure of the building blocks and the nature of their interactions has been deduced via systematic observations of their positions in space and time. Unfortunately, Complex Adaptive Systems are highly multi-factorial, so that, unlike many physical systems, it is impossible to systematically observe and characterize each and every interaction that exists in such systems. In this paper we discuss a general framework - Complex Inference Networks - wherein interactions, particularly in Complex Adaptive Systems, may be studied and characterized using position data about their building blocks. We compare and contrast physical versus Complex Adaptive Systems and give as an explicit example the identification of disease hosts in the ecology of emerging and neglected diseases, where it has been possible to discover previously unknown ecological interactions from species co-occurrence data.
MS-Stability Analysis of Predictor-Corrector Schemes for Stochastic Differential Equations
R. Zeghdane, A. Tocino
Deterministic predictor-corrector schemes are used mainly because of their numerical stability which they inherit from implicit counterparts of their corrector schemes. In principle these advantages carry over to the stochastic case. In this paper a complete study for the linear MS-stability of the one parameter family of weak order 1.0 predictor-corrector Taylor schemes for scalar stochastic differential equations is given. Figures of the MS-stability regions that confirm the theoretical results are shown. It is also shown that mean-square A-stability is recovered if the parameter is increased.
Investigation into the Regular and Chaotic States of Twitter
Victor Dmitriev, Andrey Dmitriev
The present paper is devoted to the investigation into the nonlinear dynamics of Twitter. A new model of Twitter as a thermodynamic non-equilibrium system is suggested. Dynamic variables of such system are represented by the variations of tweet/retweet number and instantaneous diversity between the densities of population on different levels around the equilibrium values. Regular and chaotic states of networks are described. It is pointed out, that the system is in a condition of an asymptotically stable equilibrium when the intensity values of an external information are small (the number of tweets eventually tends to its equilibrium value). If the intensity values of external information exceed the critical value, then the chaotic oscillations of tweets are to be observed. We have made the calculations of the correlation dimension and embedding dimension for the dynamics of the 10 most popular @ (TOP 100 by data of Twitter Counter). The results show, that all observed time series have clearly defined chaotic dynamical nature.
The Effect of non-Selective Harvesting in Predator-Prey ModelWith Modified Leslie-Gower and Holling Type II Shemes
I. El Harraki, R. Yafia, A. Boutoulout, M. A. Aziz-Alaoui
In this paper, we study the effect of harvesting on the qualitative properties of predator-prey model with modified Leslie-Gower and Holling Type II functional responses. The model is given by a system of two ordinary differential equations with non-selective constants harvesting terms. We investigate the impact of harvesting terms on the boundedness of solutions, on the existence of the attraction set, on the stability of different equilibrium points. A Lyapunov function is used to prove the global stability of the interior equilibrium. We also, discussed the policy of optimal harvest and we got the solution for the interior equilibrium by the Pontryagin maximum criterion. Finally, our theoretical results are illustrated by a numerical simulations.
Ant Colony Optimization Algorithm for Lesion Border Detection in Dermoscopic Images
Asmae Ennaji, Abdellah Aarab
Medical image segmentation plays a crucial role in computer aided diagnosis system that have a significant potential for early detection of skin cancer. The aim of segmentation process in this field is to facilitate the characterization and the visualization of the lesion in dermoscopic images. This paper proposes a new method for improving the lesion border detection in dermoscopic images, based on the ant colony optimization algorithm. Our experiments show that the proposed method achieved a significant improvement in image segmentation when compared to the deterministic canny procedure.
Bifurcation diagrams and Fomenko's Surgery on Liouville tori of a rigid body in the Goryachev-Chaplygin case on sokolov terms
Jaouad Kharbach, Mohammed Benkhali, Walid Chatar, Ahmed Sali, Abdellah Rezzouk, Mohammed Ouazzani-Jamil
In this paper, by taking advantage of the richness of Fomenko's theory of surgery on (bifurcations of) Liouville tori, we give a complete description of the topology and bifurcations of the invariant level sets of a heavy rigid body around a fixed point corresponding to the Goryachev-Chaplygin case on sokolov terms. In particular, for all values of the parameters of the system, the bifurcation diagrams of the momentum mapping are constructed, bifurcations of the common level sets of the first integrals are described, explicit periodic solutions were determined, the topology of the invariant manifolds and all generic bifurcations are illustrated numerically.
Approximate Controllability of Impulsive Neutral Functional Integrodifferential Systems with Nonlocal Conditions
A. Yasotha, K.Kanagarajan
In this paper, we study the approximate controllability of impulsive neutral functional integrodifferential systems with finite delay. The fractional power theory and α -norm are used to discuss the problem so that the obtained results can apply to the systems involving derivatives of spatial variables. By methods of functional analysis and semigroup theory, sufficient conditions of approximate controllability of the impulsive integrodifferential equation are formulated and proved. An example is provided to illustrate the theory.
Stable and Unstable Behaviors for Brushless Motor with Harmonic Disturbance via Discrete Implicit Maps
Jianzhe Huang
Dynamic model for brushless motor in the rotating frame is studied in this paper, where a periodic disturbance is introduced in the quadrature-axis voltage. The methodology of discrete implicit maps is adopted to investigate the dynamical response of such brushless motor system. Multiple closed semi-analytic bifurcation routes for period-1 motions are found by varying frequency of disturbance of the quadrature-axisvoltage. The stability conditions for periodic motions will be given through eigenvalue analysis. From the semi-analytical prediction of period-1 motions, the corresponding frequency-amplitude characteristics are obtained. Finally, the stable and unstable period-1 motions will be presented numerically. With such discrete implicit maps method, the unstable motion of such brushless motor with strong nonlinearity can be obtained. Compared to the method of generalized harmonic balance, the dimension of Jacobian matrix can be dramatically decreased and the numerical error can be avoided without using QR algorithm.
Existence, Uniqueness and Stability Results for Impulsive Neutral Stochastic Functional Differential Equations with Infinite Delay and Poisson Jumps
pp. 1-11 | DOI: 10.5890/DNC.2019.03.001
A. Anguraj, K. Banupriya
In this paper, we study the existence and uniqueness of mild solutions of impulsive neutral stochastic functional differential equations with infinite delay and Poisson jumps under non-Lipschitz condition with Lipschitz condition being considered as aspecial case by means of the successive approximation. Further, We study the continuous dependence of solutions on the initialvalue by means of a corollary of the Bihari inequality.
Synchronization of Time-delay Chaotic Systems with Uncertainties and External Disturbances
Ajit K. Singh, Vijay K. Yadav, Subir Das
In this article, the authors have studied the problem of synchronization of time-delay chaotic systems with uncertainties and external disturbances. The effectiveness of the problem statement is visualized through synchronization of time-delay advanced Lorenz system and double time-delay R¨ossler system with parametric uncertainties and disturbances using active control method. Numerical simulations are carried out using Runge-Kutta algorithm for delay differential equations (DDEs) and the results are depicted through graphs. The physical meaning of time-delay system is that a signal is transmitted and receive data latertime, which is found to occurin active sensing problems. The salient feature of the article is the demonstration of the efficiency of the considered method during synchronization of time-delay chaotic systems even in presence of uncertainties and external disturbances.
Spatiotemporal Dynamics of Meta Population Models with Application in Law
R.O. Walo, A.M. Ndondo, S. B. Mushi
The paper deals with a spatiotemporal dynamics study of criminality cityby-city in attempt to model mechanisms which give rise to synchronisation outbreak of crimes in different cities and villages. The study gives sharp relations between social and penal pressions.
Dynamics and Stability Results for Nonlinear Neutral Pantograph Equations via Hilfer-Hadamard Fractional Derivative
A study on the existence and uniqueness of a solution for nonlinear neutralpanto graphe quations with Hilfer-Hadamard fractional derivative is the main aim of this paper. The existence result is obtained by using Schauder fixed point theorem. Also, we discuss Ulam stability for these equations basedon Banach contraction principle.
Superdiffusive Searching Skill in Animal Foraging
Kishore Dutta
Random walk related to diffusion driven transport is widely used to describe phenomenain the context of physics, biology, statistics, and econometrics. However, contrary to the normal (Gaussian) diffusion process, anomalous diffusion takes place in many natural phenomena ranging from turbulence to animal foraging. In situations where spreading process is more rapid than the normal diffusion, the distribution of steps comprises many short movements broken episodically by less frequent long excursions,commonly known as L´evyflight. Recent quantification of high resolution foraging data of many free-ranging animals suggest that they display L´evyflight behaviorin environmentswhere prey is scarce. This article describes the fundamental properties of L´evy flight and how it encodes the statistical characteristicsof theanimal foraging patterns.
Existence of Solutions for Boundary Value Problem of Non-linear Integro-differential Equations of Fractional Order
J. Kavitha, V. Sadhasivam
In this article, we cultivate the existence theory for the following boundary value problem of fractional integro-differential equations togetherwith fractional integro-differential boundary conditions. By using the coincidence degree theory, we will obtain a new criteria for the existence of the solutions of the given boundary value problems. We presentan exampleto illustrate ourmain results.
Approximate Controllability of Sobolev-Type Fractional Neutral Evolution Inclusions
M. Tamil selvan, R. Murugesu
In this work, we establish a set of sufficient conditions for the approximate controllability for a class of Sobolev-type fractional neutral evolution inclusions in Banach spaces. We use Bohnenblust-Karlin'sfixed point theorem to prove our main results. Further, we extend our result to study the approximate controllability for nonlinear fractional control system with nonlocal conditions. An exampleis also givento illustrate our mainresults.
Controllability of Nonlinear Fractional Langevin Systems
P. Suresh Kumar, V. Govindaraj, K. Balachandran, N. Annapoorani
In this paper, we first derive the solution representation of fractional Langevin differential equation represented by the fractional differential coefficient in the sense of Caputo fractional derivative in terms of Mittag Leffler function. Based on this solution representation, controllability of linear fractional Langevin dynamical systems is studied by using Grammian matrix. Sufficient conditions for the controllability of the nonlinear system are established by using the Schauder's fixed point theorem. An exampleis givento verifythe results.
A New Local Fractional Derivative of q−uniform Type
Juan E. N´apoles Valdes†, Jorge A. Castillo Medina , Paulo M. Guzm´an , Luciano Miguel Lugo
In this work we present a new derivative of q-uniform type, which contains several definitions of known q-derivatives. Some examples and properties similar to those of the ordinary calculus are also presented.
Dynamics of Modified Leslie-Gower Harvested Predator-Prey Model with Beddington-DeAngelis Functional Response
R. Sivasamy, M. Sivakumar, K. Sathiyanathan, K. Balachandran
This paper considers a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and nonlinear prey harvesting strategy subject to the zero-flux boundary conditions. To understand the dynamics of the considered system, we derive sufficient conditions for permanence analysis, local stability, global stability and Hopf bifurcation of interior equilibrium point. Further we also investigate the existence and non-existenceof non-constant positive steady state solutions.
Stability and Bifurcation Analysis of Three Species Predator-Prey Model with Holling Type II
M. Sambath, C. Gokila
This paper considers a prey predator model with omnivore population and Holling type II response. First, we have studied the boundedness of the model system. The local and global stability of the equilibrium is investigated by analyzing the eigenvalues and constructing the appropriate Lyapunov functions respectively. The persistence of positive equilibrium is also discussed. The existence of Hopf bifurcation is investigated by analyzing the distribution of eigenvalues at the positive equilibrium point. By using the normal form theory and explicit formulae which determine the direction of bifurcating periodic solutions are derived. Some numerical simulationsare carryingout,to check our theoretical results.
Dynamical Behaviours of a Lotka Volterra Model with Katugampola Fractional Derivative
L. Vignesh, K. Kanagarajan, D. Vivek
In this paper, we study the dynamical behaviours of a very special type of delay differential equation known as Lotka Volterra model (predator-prey model) via Katugampola fractional derivative. The existence and uniqueness results are obtained using the Krasnoselskii's fixed point theorem. Moreover,we derived fractional Euler's method for delay differential equations. Illustrative examples are solved to show the applicability of theory results.
Permanence of Stochastic Biological Systems
Area of research related to prey-predator systems is an important topic. The concept of permanence is an important issues related to biological systems. In general permanence is considered as a combination of persistence and boundedness. Following this, this paper reviews few existing definitions of stochastic permanence. Following the existing definition, in this paper a new definition of permanence for stochastic biological systems has been proposed, which modifies the existing ones. The permanence of a general Lotka-Volterra stochastic two species cooperation, competition and predation systemsis provedby new definition.
Exact Solutions and Stability Analysis of a Nonlinear Model of Open-Ocean Deep Convection that Allows Multiple Steady States
Igor L. Bashmachnikov, DmitryV. Kovalevsky
We present analytical solutions of the two-basin model of open-ocean deep convection. Originally suggested by Whitehead [Whitehead, J.A. (2000), Stratified convection with multiple states, Ocean Modelling, 2(3-4), 109121], the model allows regimes with multiple steady states (multiple equilibria). We provide the full analytical description of the steady states for the particular case of constant surface heat flux from the ocean to the atmosphere, and explore analytically stability of the equilibria within the Lyapunov theory. The results show that, for this particular case, the steady state is unique and stable for all dynamic flow regimes. We also present analytical expressions for dependence of critical values of sea-surface heat flux,at which transitions between the dynamic regimes occur,on the model parameters.
A Fundamental Approach on Non-integer Order Differential Equation Using Nonlocal Fractional Sub-Strips Boundary Conditions
M. Subramanian, A.R. VidhyaKumar, T. Nandha Gopal
The intention of this article is to exhibit the validity and applicability of a fractional sub-strips integral boundary condition on a nonlinear differential equation of fractional order. We propose convergent and stable methods to validate the existence and uniqueness of the stated problem. The scheme is also provided with an illustrative examples.
On the Way to Generalized Kinetics for Multi-scale Complex System
V. V. Uchaikin
Development of nano-technology throws a new challenge to statistical physics. The classical statistical physics is underlaid by classical or quantum mechanics of closed systems, classical version of the Central Limit Theorem, the Markov model of Brownian motion, the linear approximation of the perturbation theory as a main mathematical tool. The nanotechnology has caused constructing new models: nonlocal in space and time dynamics, generalized Limit Theorem for the Levy-stable family as limit distributions, non-Markovian (subordinated) Levy-motion and instead of the Brownian one,self-similar (fractal) and non-local (integral and fractional differential) analysis. The article gives a short review of these concepts. The multiscale complex system is considered to exacute a self similar motion obeying the Levy stable statistics, which generalized then to fractionally stable kind. The intermediate asymptotics is introduced on the base of truncated L´evy flight. This leads to the correspondence principle connecting classical and nano-statistics. Some numerical results are presented.
Regional Observability with Constraints on the State of Semilinear Parabolic Systems
Hayat Zouiten, Ali Boutoulout, Fatima-Zahrae El Alaoui
The paper is devoted to the investigation of regional observability with constraints on the state of semilinear parabolic systems. The purpose is to reconstruct the initial state between two prescribed functions only on an internal subregion ω of the system evolution domain Ω. The proofs use two approaches, the subdifferential and HUM approach. Finally, a numerical exampleis provided to verify the effectiveness of our theory results.
Controllability Criteria for Nonlinear Higher Order Fractional Delay Dynamical Systems with Impulses
M. Sivabalan, K. Sathiyanathan
This paper is concerned with the controllability of nonlinear higher order fractional delay with impulses, which involved Caputo derivatives of any different orders. A necessary and sufficient condition for the controllability of linear fractional delay dynamical systems with impulses has proved,and a sufficient condition for the corresponding nonlinear systems has obtained. Examples has given to verify the results.
Relation Between Autocorrelation Sequence and Average Shortest-Path Length in a Time Serie to Network Mapping
Amanda Leitede Camargo, Marcio Eisencraft
An invertibl emapping between time series and networks was recently proposed. It can be used as a tool to figure out properties of the mapped time series. In the present work we use controlled artificial signals to numerically investigate how correlation properties of time series are mapped in the topological measures of associated networks. More specifically, we employ filtered uniform white noise and analyse how the auto correlation sequence influences the average shortest-path length.
Approximate Controllability of Second-order Neutral Stochastic Non-autonomous Integrodifferential Inclusionsby Resolvent Operators
R. Nirmalkumar, R. Murugesu
In this paper, we formulate a set of sufficient conditions for the approximate controllability for a class of second-order neutral stochastic nonautonomous integrodifferential inclusions in Hilbert space. We establish the results with the help of resolvent operators and Bohnenblust-Karlin's fixed point theorem is to prove the main result. An application is given to illustrate the main result.
A Study on Langevin Equations with ψ-Hilfer Fractional Derivative
In this paper, we discuss the existence, uniquenessand four types of Ulam stability results for a general class of Langevin equations. An illustrate exampleis given to check the applicable of this results.
Fuzzy ( η ,ϕ )-mixed Vector Equilibrium Problems
In this research document, we study the fuzzy ( η ,ϕ )-mixed vector equilibrium problems. By using the concepts of KKM-mapping, we prove the existence theorem of fuzzy ( η , ϕ )-mixed vector equilibrium problems.
Quadratic Operators Defined on a Finite-dimensional Simplex of Idempotent Measures
I. T. Juraev, M. M. Karimov
We describe some quadratic operators which map the (n−1)-dimensional simplex of idempotent measures to itself. Such operators are divided to two classes: the first class contains all n×n×n- cubic matrices with nonpositive entries which in each n×n dimensional k-th matrix contains exactly one non-zerorow and exactly one non-zero column; the second class contains all n×n×n-cubic matrices with non-positive entries whichhasat leastonequadraticzero-matrix. These matrices play a role of the stochastic matrices in the case of idempotent measures. For both classes of quadratic mapswe find fixed points.
A Variational Problem on the Deformation Energy of an Elastic Medium
Panayotis Vyridis
The variational problem under consideration is a generalization to higher dimension (n > 2) of a free and constrained deformation of an elastic medium. In the case of the free body the existence of the minimizers of the corresponding energy functional is studied, using standard direct methods. When the elastic medium is subjected to a one parametric force field of the shell, the problem turns to the existence of bifurcation points, where the necessary conditions are also sufficient.
Breakup of Closed Curve - Quasiperiodic Route to Chaos in Vibroimpact System
V. A. Bazhenov, O. S. Pogorelova, T. G. Postnikova
At present chaotic vibrations are the one of the most interesting and explored subjects in nonlinear dynamics. Particularly the routes to chaos in non-smooth dynamical systems are of the special scientists' interest. Quasiperiodic route to chaos in nonlinear non-smooth discontinuous 2DOF vibroimpact system is studied in this paper. In narrow frequency range different oscillatory regimes have succeeded each other many times under very small control parameter varying. There were periodic subharmonic regimes - chatters, quasiperiodic, and chaotic regimes. There were the zones of transition from one regime to another, the zones of prechaotic and post chaotic motion. The hysteresis effects (jump phenomena) occurred for increasing and decreasing frequencies. The chaoticity of obtained regime has been confirmed by typical views of Poincar´e map and Fourier spectrum, by the positive value of the largest Lyapunov exponent, and by the fractal structureof Poincar´e map.
Stability Analysis of a Stochastic Viral Infection Model with General Infection Rate and General Perturbation Terms
Marouane Mahrouf, Khalid Hattaf, Noura Yousfi
In this paper, we propose a stochastic viral infection model with general incidence rate. In the proposed model, the stochastic perturbations are modeled by general functions. Further, the global existence and positivity of solutionsare investigated. In addition, the stochastic stability of the model is established by using the direct Lyapunov method. Finally, anapplication of the hepatitis B virus(HBV) is given to validate our theoretical results.
Dynamical Behavior of a Delayed Predator-prey Model in Periodically Fluctuating Environments
A. Moussaoui, M. A. Aziz Alaoui
In this paper we develop a non-autonomous predator-prey system with time delay to study the influence of water level fluctuations on the interactions between fish species living in an artificial lake. We derive persistence and extinction conditions of the species. Using coincidence degree theory, we determine conditions for which the system has at least one periodic solution. Numerical simulations are presented to illustrate theoretical results.
Analytic Solution of Time Fractional Boussinesq Equation for Groundwater Flow in Unconfined Aquifer
Ritu Agarwal, MahaveerPrasad Yadav1, Ravi P. Agarwal
Anapproximate analytical solution to the nonlinear time fractional Boussinesq equation is presented here. Derivative with respect to time variable is replaced with Caputo fractional derivative. The Natural transform method and Adomian decomposition method are employed to obtain the solution. Some test problems are solved to show the accuracy of the proposed method. Behavior of water table head is depicted graphically for various time values.
Regional Weak and Strong Stabilization of Semilinear Systems with Decay Estimate
A. El Alami, A. Boutoulout
The aim of this paper is to develop the question of the regional stabilization for infinite-semilinear parabolic systems on a spatial domain Ω. precisely the study of such system on a subregion ω of Ω. We consider a decomposition of the state space via the spectral properties of the system. Then we apply this approach to regional strong and weak stabilization problem using bounded feedback. Some applications and simulations are presented.
Monotone Dynamical Systems Theory for Epidemiological Models with Delay: A New Approach with Applications
Jaafar El Karkri†, Khadija Niri
Monotone dynamical systems theory is an efficient and powerful tool for the study of dynamical systems asymptotic behaviour. However it is rarely used in mathematical epidemiology. In this paper we present a comparison between two different approaches of convergence and stability for dynamical systems. We prove that the convergence in the sense of the monotone dynamical systems theory is equivalent to the uniform convergence in the classical Lyapunov theory. Then we provide a stability analysis of an SIS epidemiological model based on the monotone approach. Numerical simulations illustrate our theoretical results.
Stability and Hopf Bifurcation of an Epidemic Model With Logistic Growth and Delay
El Mehdi Lotfi, Khalid Hattaf, Noura Yousfi
In this work, we propose and analyze a delayed epidemic model with logistic growth, in which the growth of susceptible individuals is governed by the logistic equation and the delay represents the latent period of the disease. Firstly, we prove that our model is mathematically and biologically well posed. In addition, the stability of equilibria and the existence of Hopf bifurcation are established. Moreover, several epidemic models existing in the previous studies are extended and generalized. Finally, some numerical simulations are given to illustrate our main results.
Stability and Bifurcation Analysis in a Two-Dimensional Neutral Differential Equation
A. Moussaid, Talibi Alaoui Hamad
This paper discusses asymptotic stability and Hopf bifurcations occurs at the origin in certain two-dimensional neutral delay differential equations. We give necessary and sufficient conditions on the parameters to obtain the asymptotic stability and bifurcations. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. Finally, some numerical simulations are carried out to support the analytic results. Our results are a generalization of M. Liu and X. Xu [1].
Interval Criteria for Oscillation of Damped Second-order Nonlinear Impulsive Differential Equation with Variable Delay
V. Muthulakshmi, R. Manjuram
This paper is devoted to the study of the oscillatory behavior of damped second order nonlinear impulsive differential equations with variable delay. The results obtained here extend and complement to some known results in the literature.
Boundary Controllability of Fractional Order Nonlocal Semi-linear Neutral Evolution Systems with Impulsive Condition
Kamalendra Kumar, Rakesh Kumar
Sufficient conditions of boundary controllability of fractional nonlocal semi-linear neutral evolution equations with impulsive conditions are demonstrated. To get the result, we employ fixed point theorem and strongly continuous semi-group theory. An application is discussed to explain the theory.
Influence of Coupled Nonlocal Slit-Strip Conditions Involving Caputo Derivative in Fractional Boundary Value Problem
M. Subramanian†, A.R. Vidhya Kumar, T. Nandha Gopal
By employing the coupled nonlocal slit-strips integral boundary conditions involving Caputo derivative, we investigate the existence and uniqueness of a boundary value problem of fractional differential equations. The main result is illustrated with examples.
Using the Parameter Optimization Method for Solving Differential Equations with Contour Conditions: The nonlinear Euler-Bernoulli Beam
Ad´elcio C. Oliveira
The Parameter OptimizationMethod was presented with analytical and numerical examples. It was shown that it is a useful tool for solving differential equations with contour conditions. The numerical procedure is based only on Runge-Kutta integration routine and on optimization techniques, both frequently used and with many developed routines, thus it was shown that this approach is accessible and practical. The method was used to solve a one-dimensional Nonlinear Schr¨odinger Equation and a nonlinear Euler- Bernoulli beam.
Delimitation of Hyperchaotic Regions in Parameter Planes of a Four-Dimensional Dynamical System
Juliane C. Ramos, Paulo C. Rech
We report results of a numerical investigation on parameter planes of a set of four autonomous first-order nonlinear ordinary differential equations. More specifically, here are reported ten numerically computed parameter plane diagrams for a five-parameter four-dimensional system, where the dynamical behavior of each point is characterized by using the related Lyapunov exponents spectrum. Each of these parameter plane diagrams indicates parameter values for which hyperchaos, chaos, quasiperiodicity, and periodicity may be found, i.e., each of these diagrams shows delimited regions for each of these dynamical behaviors.
Study on Stochastic Quasi-Linear Partial Differential Equations of Evolution
A. Anguraj, K. Ramkumar
In this article, the existence and uniqueness of local mild solution of a stochastic counterpart of TosioKato's Quasi-linear partial differential equation with additive cylindrical wiener process in a separable Hilbert space is established using contraction mapping principle.
Bernoulli Mapping with Holeanda Saddle-Node Scenario of the Birth of Hyperbolic Smale–Williams Attractor
OlgaB. Isaeva, Igor R. Sataev1
One-dimensional Bernoulli mapping with hole is suggested to describe the regularities of the appearance of a chaotic set under the saddle-node scenario of the birth of the Smale–Williams hyperbolic attractor. In such a mapping, a non-trivial chaotic set (with non-zero Hausdorff dimension) arises in the general case as a result of a cascade of period-adding bifurcations characterized by geometric scaling both in the phase space and in the parameter space. Numerical analysis of the behavior of models demonstrating the saddle-node scenario of birth of a hyperbolic chaotic Smale– Williams attractor shows that these regularities are preserved in the case of multi dimensional systems. Limits of applicability of the approximate 1D model are discussed.
The Thomas Attractor with and without Delay: Complex Dynamics to Amplitude Death
Brayden McDonald, S. Roy Choudhury
Bifurcations in the Thomas cyclic system leading from simple dynamics into chaotic regimes are considered. In particular, the existence of only one non-trivial fixed point of the system in parameter space implies that this point attractor may only be destabilized via a Hopf bifurcation as the single system parameter is varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The periodic orbit immediately following the Hopf bifurcation is constructed analytically by the method of multiple scales, and its stability is determined from the resulting normal form and verified by numerical simulations. The dynamically rich range of parameters past the Hopf bifurcation is next systematically explored. In particular, the period-doubling sequences there are found to be more complex than noted previously, and include period-three-like windows for instance. As the system parameter is decreased below these period-incrementing bifurcations, various additional features of the subsequent crises are also carefully tracked. Finally, we consider the effect of delay on the system,leading to the suppression of both the Hopf bifurcation as well as all of the subsequent complex dynamics. In modern terminology, this is an example of Amplitude Death,rather than Oscillation Death, as the complex system dynamics is quenched, with all of the variables settling to a fixed pointof the original system.
Existence of Nonoscillatory Solutions for Mixed Neutral Fractional Differential Equation
Velu Muthulakshmi, Subramani Pavithra
In this paper, we establish some sufficient conditions for the existence of nonoscillatory solution for a class of mixed neutral fractional differential equations with Liouville fractional derivative of order α ≥0 on the halfaxis. Our results generalize some of the existing results in the literature. Some examples are given to illustrate our results.
Universal Behavior of the Convergence to the Stationary State for a Tangent Bifurcation in the Logistic Map
Joelson D. V. Hermes, Fl´avio Heleno Graciano, Edson D. Leonel
The scaling formalism is applied to understand and describe the evolution towards the equilibrium at and near at a tangent bifurcation in the logistic map. At the bifurcation the convergence to the steady state is described by a homogeneous function leading to a set of critical exponents. Near the bifurcation the convergence is rather exponential whose relaxation time is given by a power law. We use two different approaches to obtain the critical exponents: (1) a phenomenological investigation based on three scaling hypotheses leading to a scaling law relating three critical exponents and; (2) an approximation that transforms the recurrence equations in a differential equation which is solved under appropriate conditions given analytically the scaling exponents. The numerical results give support for the theoretical approach.
Steady-State and Dynamic Characteristics of Water-lubricated Rubber Bearings under Two Sets of Reynolds Boundary Conditions
Gang Liu, Ming Li
The lubrication characteristics of bearings are significantly influenced by boundary conditions. The double Reynolds boundary conditions (DRBCs) are introduced to analyze the hydrodynamic lubrication characteristics of water-lubricated rubber bearings for turbulent flows, considering the elasticity of rubber liner. Differences in its steady state and dynamic characteristics arising from the DRBCs and the Reynolds boundary conditions (RBCs) are discussed based on the finite difference method. The results show that the water-film reformation boundary is significantly different between the two sets of conditions, however,the load capacity of the bearing is only slightly different. The attitude angle and friction are greater for DRBCs than for RBCs. In the horizontal direction, direct stiffness and direct damping coefficients are larger for DRBCs than for RBCs, but in the vertical direction the opposite result holds.
Effect of Fractional-Order on the Dynamic of two Mutually Coupled van der Pol Oscillators: Hubs, Multistability and its Control
Ngo Mouelas Ad`ele1, Kammogne Soup Tewa Alain, Kengne Romanic, Fotsin Hilaire Bertrand1, EssimbiZobo Bernard
This paper presents a novel approach to analyze the dynamic effect of the fractional-order derivative of the two mutually coupled van der Pol oscillators. The stability analysis is presented by two complementary phase diagrams: the isospike diagrams and the two Lyapunov exponent spectra. These diagrams reveal precisely the Hubs, spirals bifurcation and chaos when the derivative order is fixed at q = 0.95. In addition, when the fractional-order is set as a control parameter, various methods for detecting chaos including bifurcation diagrams, spectrum of largest Lyapunov exponentare exploited to establish the connection between the system parameters and various complicated dynamics. A transition was also observed between a desynchronized state and a multistability situation. These diagrams displayed the coexistence of four disconnected attractors (two symmetric). We study the basins of attraction of the system in the multi stability regime which thereby reveal the coexistence of attractors in the systems when the fractional-order derivative is taken as a function of initials conditions. Based on the parametric control, we have controlled this striking phenomenon in the system. Finally, the hardware circuit is implemented and the results are found to be in good agreement with the numerical investigations.
Complex Dynamicsofan Exploited Prey-PredatorModel with NonlinearPrey Refuge
Md. Manarul Haque, Sahabuddin Sarwardi
In this paper we study the renewable resources of a prey-predator system with prey refuge and nonlinear harvesting. Taxation is imposed to prevent over exploitations to maintain ecological balance. The steady state of the system are determined and various dynamical behavior are discussed in its steady states under certain parametric conditions. The boundedness, feasibility of interior equilibria, bionomic equilibrium have been studied. The main observation is that the taxation plays an important role in regulating the dynamics of the present system. Moreover the variation of the taxation change the system from periodic behaviors to chaos. Some numerical illustration are given in order to support of our analytical and theoretical findings.
The Dynamical Behaviorof a Two Degrees of Freedom Oblique Impact System
Xiaowei Tang, Xilin Fu, Xiaohui Sun
The oblique impact phenomena is quite common in practical engineering. In this paper, by using the theory of discontinuous dynamical systems, we studied the complex dynamics behaviors of a two degrees of freedom oblique impact system. We can see that the dynamics of oblique impact is more complex than that of the direct impact. The occurrence or disappearance conditions of sticking motion and grazing motion on the separation boundaries are given in Section 3. The conditions here are necessary and sufficient, which generate better results than those obtained with only sufficient conditions. The results appropriately interpret the physical phenomenon of this oblique impact system, hence validate our conclusions. As a supplement, we also give the analytic conditions of the existence of periodic motions. Numerical simulations for sticking motion and grazing motion are presented at last.
Unsteady Magneto hydrodynamic Boundary Layer Flow towards a Heated Porous Stretching Surface with Thermal Radiation and Heat Source/Sink Effects
Santosh Chaudhary, Susheela Chaudhary, Sawai Singh
Mathematical model of unsteady boundary layer flow and heat transfer is explored for analyzing the study of influence of thermal radiation on incompressible viscous electrically conducting fluid over continuous stretching surface embedded in a porous medium in the presence of heat source/sink. The scope of influencing parameters that describing phenomenon are determined and governing time dependent boundary layer equations are transformed to ordinary differential equations by using appropriate similarity transformation. Numerical computation of the problem was carried out by shooting iteration technique together with Runge-Kutta fourth order integration scheme. Effects of unsteadiness parameter, permeability parameter, magnetic parameter, thermal radiation parameter, Prandtl number and heat source/sink parameter on velocity and temperature profiles are computed and illustrated graphically, where as local skin friction coefficient and local Nusselt number are represented numerically through tables. In nonmagnetic flow condition the result is found in concordance with earlier investigations.
Fluid Flow and Solute Transfer in a Permeable Tube with Influence of Slip Velocity
M. Varunkumar, P. Muthu
In this paper,the influence of slip velocity on the fluid flow and solute transfer in a tube with permeable boundary is studied as a mathematical model for blood flow in glomerular capillaries. The viscous incompressible fluid flow across the permeable tube wall, as a result of differences in both hydrostatic and osmotic pressure, is considered. The solutions of differential equations governing the fluid flow and solute transfer are obtained using analyticaland Crank-Nicolsontype numericalmethods. It is observedthat the effect of slip on the hydrostatic and osmotic pressures, velocity profiles, concentration profile, solute mass flux and total solute clearance is significantand the results are presented graphically.
Noise-induced Intermittent Oscillation Death in a Synergetic Model
R.Jaimes-Re´ategui, D.A.Magall´on-Garc´ıa1, A.Gallegos, G.Huerta-Cuellar1, J.H.Garc´ıa-L´opez, A. N. Pisarchik
We study noise-induced intermittency in asynergetic model of two coupled oscillators with asymmetric nonlinear coupling. This model was previously used to simulate visual perception of ambiguous images. We show that additive noise induces preferencefor one of the coexisting unstable steady states. When the noise intensity exceeds a certain threshold value,the oscillations of one of the coupled subsystems are interrupted during some time intervals, resulting in intermittent oscillation death, while another subsystem exhibitsnoisy oscillationsin the vicinity of an unstable fixed point.
A Mathematical Model for Vineyard Replacement with Nonlinear Binary Control Optimization
An´ıbal Galindro, AdelaideCerveira, Delfim F. M. Torres, Jo˜aoMatias, Ana Marta-Costa
Vineyard replacement is a common practice in every wine-growing farm since the grapevine production decays over time and requires a new vine to ensure the business sustainability. In this paper, we formulate a simple discrete model that captures the vineyard's main dynamics such as production values and grape quality. Then, by applying binary non-linear programming methods to find the vineyard replacement trigger, we seek the optimal solution concerning different governmental subsidies to the target producer.
Preventing Computer Virus Prevalence using Epidemiological Modeling and Optimal Control
Joa˜o N.C. Gonc¸alves1, Helena Sofia Rodrigues, M. Teresa T. Monteiro
Computer viruses are a serious threat to the general society, due to their implications for private life and corporative systems. This paper begins to briefly illustrate the dynamics of computer viruses within a network system, by taking advantage of the EpiModel R package and using a SIR (Susceptible–Infected–Recovered) epidemic model. However, since devices are not constantly immune to cyberattacks, a SIRS model with an optimal control application is proposed to minimize the levels of infections caused by malicious objects. Additionally, real numerical data related to the number of reported cybercrimes in Japan from 2012 to 2017 are considered. The existence and uniqueness of an optimal control for the proposed control problem are proved. Under proper investment costs, numerical simulations in Matlab show the effectiveness of the proposed control strategy in increasing the rate of immunity and decreasing the chances of re–susceptibility to cyberattacks.
An Improved Approach for Image Segmentation and Three-Dimensional Reconstruction
K. Bellaj, S. Boujena, E. EL Guarmah
The main contribution of this paper consists of introducing a novel model of three-dimensional reconstruction from multiple two-dimensional images. Actually, our proposed model presents the following main advantages; first, we improve the original region fitting energy in the general region-based level set method by an anisotropic diffusion to evolve the contour. Second, we use the Dijkstra algorithm to improve and allow simple and free initializations. Third, we adopt the domain decomposition method to reduce the computational cost for high-resolution images. Finally, in order to prove the efficiency and accuracy of our proposed method, experiments were performed on synthetic and real images.
A Generalized Reaction-Diffusion Epidemic Model with Time Delay
El Mehdi Lotfi, Radoune Yafia, Noura Yousfi
In this paper, we propose a generalized reaction-diffusion epidemic model with time delay. The proposed model is governed by partial differential equations(PDEs). The global existence, positivity and boundedness of solutions are obtained. By analyzing the corresponding characteristic equation and constructing appropriate Lyapunov functionals, the local/global stability of homogeneous steady states are investigated. Finally, an application of our analytical results is given.
Study of a Predator-Prey System with Monod-Haldane Functional Response and Harvesting
N.H. Gazi, M.R. Mandal, S. Sarwardi
In the present work we study a predator-prey harvesting model with Monod-Haldane functional response. The mathematical features of the model have been analyzed in terms of stability, bifurcations and harvesting. Threshold values for some parameters indicating the feasibility and stability conditions of all equilibria are determined. The range of significant parameters under which the system admits different types of bifurcations are investigated. Optimal harvesting criteria and the net economic revenue are analyzed. Numerical illustrations are performed finally in order to confirm the analytical findings.
Existence and Stability Results of Impulsive Stochastic Partial Neutral Functional Differential Equations with Infinite Delays and Poisson Jumps
A. Anguraj, K. Ravikumar
In this paper, we are focused upon the results on existence, uniqueness and stability of mild solution of impulsive stochastic partial neutral functional differential equations with infinite delays and poisson jumps. The results are obtained by using the method of successive approximation and Bihari's inequality.
Characterizations of Non-Associative Ordered Semigroups by Their Intuitionistic Fuzzy Bi-Ideals
Nasreen Kausar, Meshari Alesemi, Salahuddin, Mohammad Munir
The aim of this paper is to investigate, the characterizations of different classes of non-associative and non-commutative ordered semigroups in terms of intuitionistic fuzzy left (right, bi-, generalized bi-, (1,2)-) ideals.
Existence Results of Fractional Neutral Integrodifferential Equations with Deviating Arguments
B. Kamalapriya, K. Balachandran, N. Annapoorani
In this paper we prove the existence of solutions of fractional neutral integrodifferential equations with deviating arguments by using the resolvent operators and fixed point theorem. Examples are discussed to illustrate the theory.
Mathematical Contribution of Analytical Solution in Biological Control using Asymptotic Approach
A. Eswari
The theoretical model of uncolonised by antagonist, colonised by the antagonist and infected by ther pathogen in the biological control has been discussed. In this article, we present the non-steady state solutions for biological control. The simple analytical expressions of the dimensionless concentrations are derived for non-linear boundary value problem using the variational iteration method. Numerical solutions are obtained and the results are discussed graphically. We also compared our analytical results with numerical simulation, it gives a satisfactory agreement is noted.
On Different Approaches for Integrals of Physical Dynamical Systems
Jasvinder Singh Virdi
For driven time dependent harmonic oscillator a detailed and systematic study of several methods for the building of Integrals is carried out. The central feature of the present discussion is the establishment of Integrals for the dynamical system. In spite of their different procedural details all the approaches lead to the same invariant for the given classical system. Advantage and limitations of different methods are briefly highlighted.
A Mathematical Study of a Two Species Eco-Epidemiological Model with Different Predation Principles
Aktar Saikh, Nurul Huda Gazi
This paper formulates and analyzes a predator-prey model with disease in the prey. Mathematical analysis of the model system concerns the existence, uniqueness and uniform boundedness of solutions in the positive octant. The threshold condition for epidemic and the conditions for persistence are obtained. Moreover, the system is analyzed for local stability, global stability around several equilibria. Hopf-bifurcation with its nature and the stability of the bifurcating limit cycle are studied around the disease free equilibrium point. Numerical simulations are performedto justify the analytical findings. Eco-epidemilogical significance and implications of the concluded results are discussed as well.
Existence, Uniqueness and Stability of Impulsive Stochastic Partial Neutral Functional Differential Equations with Infinite Delays Driven by a Fractional Brownian Motion
A. Anguraj, K. Ramkumar, E. M. Elsayed
This article presents the result on existence, uniqueness and stability of mild solution of impulsive stochastic partial neutral functional differential equations driven by a fractional Brownian motion. The results are obtained by using the method of successive approximation and Bihari's inequality.
Homoclinic Solutions in Bazykin's Predator-Prey Model
Bashir Al-Hdaibat
In this paper we derive an explicit second-order approximation of the homoclinic solutions in the Bazykin's predator-prey model. The analytic solutions are compared with those obtained by numerical continuation.
Attractiveness and Exponential p-Stability of Neutral Stochastic Functional Integrodifferential Equations Driven by Wiener Process and fBm with Impulses Effects
Mahamat Hassan Mahamat Hamit, Fulbert Kuessi Allognissode, Mohamed salem Mohamed, Louk-Man Issaka, Mamadou Abdoul Diop
In this work, we consider a class of neutral stochastic integro-differential equations driven by Wiener process and fractional Brownian motion with impulses effects. This paper deals with the global attractiveness and quasiinvariant sets for neutral stochastic integro-differential equations driven by Wiener process and fractional Brownian motion with impulses effects in Hilbert spaces. We use new integral inequalities combined with theories of resolvent operators to establish a set of sufficient conditions for the exponential p-stability of the mild solution of the considered equations. An example is presented to demonstrate the obtained theory.
Stability Approach of a Fractional-Delayed Duffing Oscillator
Yusry O. El-Dib
In this proposal, a formulation for the approximate-analytical solution of a fractional-delayed damping Duffing oscillator is developed. The fractional derivative is established using the Riemann-Liouville definition. In this scheme, the solution used a homotopy perturbation. In this proposal, a transcendental frequency equation is established. Finally, an analytic solution to the complicated algebraic frequency equation is obtained. Stability conditions are formulated to maintain the structure of the oscillatory solution. The case of un-delayed damping Duffing equation is investigated through the modified homotopy technique which is assumed to be the successor to obtain the solution.
Spatial Patterns of an SIS Epidemic Model with Diffusion
Muhammad A. Yau, Nuru Huda Gazi
We consider an SIS spatial epidemic model with nonlinear incidence rate and diffusion, that consists of susceptible S and infective I individual populations which interact randomly in their physical environment. We use pattern formation to explain the spread and control of the epidemic over a course of time. Turing instability conditions are established and analysed for the model to exhibit spatial patterns. We find the exact Turing space in the parameter regime for these conditions to hold. An implicit pseudo-spectral method is used to numerically approximate the system and the patterns form reveal that the susceptible and infected populations behave of the same way. In some examples these populations are in isolation from each other. This happens because the susceptible individuals diffuse or move away from the infected individuals to avoid contact and the possibility of getting infected with the disease. Rigorous numerical experimentation reveals that the model has rich dynamics. We find that whenever the transmission rate β is less than the treatment r, there is no outbreak but for β ≥ r there is a possibility of having an outbreak. The results obtained extend well the findings of pattern formation in epidemic models and may have direct implications for the study of disease spread and control and perhaps the mechanistic impact of public health interventions on epidemics.
Incorporating Prey Refuge in a Prey-Predator Model with Beddington-DeAngelis Type Functional Response: A Comparative Study on Intra-Specific Competition
Hafizul Molla, Md. Sabiar Rahman, Sahabuddin Sarwardi
The present study deals with a prey-predator system with prey refuge dependingon both species with the Beddington-DeAngelisresponse function. We propose a mathematical model for predator-prey interactions, allowing prey refuge in the absence of intra-specific competition and in the presence of intra-specific competition among the predators. We have analyzed the models in terms of boundedness, persistence, existence of equilibria and their stability and Hopf bifurcation. Existence of paradox of enrichments are examined well in both the cases. The analytical findings of this study are substantially validated by sufficient numerical simulations. The ecological implications of the obtained results are discussed as well.
Fractional Differential Equations Involving Hadamard Fractional Derivatives with Nonlocal Multi-point Boundary Conditions
Muthaiah Subramanian, Murugesan Manigandan, Thangaraj Nandha Gopal
In this paper, we investigate the existence and uniqueness of solutions for the Hadamard fractional boundary value problems with nonlocal multipoint boundary conditions. By using Leray-Schauder nonlinear alternative, Leray Schauder degree theory, Krasnoselskii fixed point theorem, Schaefer fixed point theorem, Banach fixed point theorem, Nonlinear Contractions, the existence and uniqueness of solutions are obtained. As an application, two examples are given to demonstrate our results.
Evolutionary Dynamics of a Single-Species Population Model with Multiple Delays in a Polluted Environment
Ashok Mondal, A. K. Pal, G. P. Samanta
In this work, evolutionary dynamical behaviour of a single-species population model in a polluted environment has been analyzed. This model system describes the effect of toxicant on a single-species population. Two discrete time delays have been incorporated for proper description. Important mathematical characteristics of the proposed model such as positivity, boundedness, stability and Hopf-bifurcation for all possible combinations of both the delays at the interior equilibrium point of the model system have been discussed. It is observed that increase amount of delay may lead to the change of stable behaviour of stationary points through the creation of limit cycles and higher periodic oscillations. Furthermore, it is reported that Hopf-bifurcations may also occur around stationary points for corresponding non-delayed system. Various numerical simulations are performed to validate analytical findings.
Qualitative Analysis of a Modified Leslie-Gower Model with Addictive Allee Effect and Gestation Delay
S. Vinoth, R. Sivasamy, K. Sathiyanathan
This paper explores on the qualitative analysis of a modified Leslie-Gower prey-predator model where the consumption rate of prey is according to Beddington-DeAngelis functional response and Allee effect on prey population. Moreover time-lag ( τ ) is established to exploit gestation period of predations. The permanence analysis of proposed system is investigated. Then we study the local stability of non-delayed model at all possible equilibrium points and it is demonstrated that the given model experiences Hopf bifurcation about interior equilibrium point with respect to delay τ . Thereafter the stability and direction of Hopf bifurcation are formulated through normaland center manifold theorems. The derived criteria are justified with the help of numerical simulations.
Modeling Response Time Distributions with Generalized Beta Prime
M. Dashti Moghaddam, Jiong Liu, John G. Holden, R. A. Serota
We use Generalized Beta Prime distribution, also known as GB2, for fitting response time distributions. This distribution, characterized by one scale and three shape parameters, is incredibly flexible in that it can mimic behavior of many other distributions. GB2 exhibits power-law behavior at both front and tail ends and is a steady-state distribution of a simple stochastic differential equation. We apply GB2 in contrast studies between two distinct groups – in this case children with dyslexia and a control group – and show that it provides superior fitting. We compare aggregate response time distributions of the two groups for scale and shape differences (including several scale-independent measures of variability, such as Hoover index), which may in turn reflect on cognitive dynamics differences. In this approach, response time distribution of an individual can be considered as a random variate of that individual's group distribution.
Uniqueness and Non-Uniqueness of Signed Measure-Valued Solutions to the Continuity Equation
Paolo Bonicatto
We consider the continuity equation tmt +div(bmt ) = 0, where {mt}t∈R is a measurable family of (possibily signed) Borel measures on Rd and b: R×Rd → Rd is a bounded Borel vector field (and the equation is understood in the sense of distributions). We discuss some uniqueness and non-uniqueness results for this equation: in particular, we report on some counterexamples in which uniqueness of the flow of the vector field holds but one can construct non-trivial signed measure-valued solutions to the continuity equation with zero initial data. This is based on a joint work with N.A. Gusev [1].
Integrability and Jacobi last multipliers of cubic Liénard differential equations with quadratic damping
Maria V. Demina
We solve completely the problem of Liouvillian integrability for cubic Li´enard differential equations with quadratic damping. Our main tool is the method of Puiseux series. We find necessary and sufficient conditions for equations under consideration to have Jacobi last multipliers of a special form. It turns out that some particular sub–families being Liouvillian non–integrable possess Jacobi last multipliers. The Jacobi last multipliers give rise to non–standard Lagrangians and it is an interesting property of these dynamical systems. In addition, we prove that cubic Li´enard differential equations with quadratic damping do not have algebraic limit cycles.
Lax Equation on the Uhlenbeck Manifold
Ya. Dymarskii
We give an analytic and topological description of the Uhlenbeck manifold, that is a manifold of triples (a symmetric operator, an eigenvector, an eigenvalue), for the finite-dimensional symmetric matrices and the family of stationary periodic Schrodinger operators. Then, we describe an uplifting of Lax vector fields to these manifolds.
Periodic Behavior of Maps Obtained by Small Perturbations of Smooth Skew Products
L.S. Efremova
We study C1-smooth maps obtained by small perturbations of C1-smooth skew products of maps of an interval with W-stable quotients and present results on the coexistence of periods of periodic orbits for maps under consideration. In particular, C1-smooth W-stable maps of an interval do not contain maps of type 2¥, i.e. maps that have the unbounded set of (the least) periods of periodic orbits t for t = {2i}i≥0. We prove here that analogously to C1-smooth skew products of maps of an interval with W-stable quotients there exist the maps under consideration with t = {2i}i≥0.
On the Extreme Points of the Unit Ball in the Space of Solenoidal Vector Measures on the Plane
Nikolay A. Gusev
We consider the space of finite divergence-free Borel vector measures on Rd, endowed with the total variation norm. For d = 2 we present a characterization of the extreme points of the unit ball in this space. This allows one to decompose (for d =2) any finite divergence-free vectormeasure into measures induced by closed Lipschitz curves. The results are based on a joint work with P. Bonicatto.
Dynamical System Model with the use of Liouville Equation for Empirical Distribution Function Densities
Alexey A. Kislitsyn, Yurii N. Orlov
The difference approximation of the one-dimensional Liouville equation for the sample distribution function density of the non-stationary time series estimated by the histogram is considered. The scheme with semi-group property conservation is constructed for evolutionmodel of this sample distribution function density. We investigate the problem of appropriate Liouville equation construction for given initial and final distributions. We prove the necessary and sufficient condition of such a representation, which is a strong positivity of the initial density distribution in the inner class intervals. The determination of the corresponding Liouville velocity algorithm is constructed and its mechanical-statistical meaning is shown. The dynamical system, associated with this Liouville equation, is considered. We interpret the Liouville statistical velocity as a corresponding velocity of dynamical system, according to representation of statistical mechanics. We show, that this interpretation leads to monotonic discrete dynamical system with stationary point, corresponding to equality of initial and final distribution functions.
Monotone Maps on Dendrites
E. N. Makhrova
Let X be a dendrite, f : X → X be a monotone map. In the article the relationship between a structure of a dendrite X and dynamical properties of f is studied. Namely the relation between a structure of X and a structure of the sets of periodic points of f , non-wandering points of f , w-limit sets of trajectories is established. The structure of dendrites on which there exist monotone pointwise chain recurrent maps is characterized. The structure of dendrites on which there exist monotone maps with homoclinic points is described.
Ekeland's Variational Principle for Functions Unbounded from below
R. Sengupta, S. Zhukovskiy
A modification of the Ekeland variational principle for functions unbounded from below is obtained. For a wide class of differentiable functions not necessarily bounded below, it is shown that there exists a minimizing sequence satisfying the first-order necessary conditions, up to any desired approximation.
Complex Geometry of Universal Teichm¨uller Space
Armen Sergeev
We discuss complex geometric properties of the universal Teichm¨uller space T . It is a complex Banach manifold which name is motivated by the fact that all classical Teichm¨uller spaces T(G), associated with compact Riemann surfaces, are contained in T as complex subvarieties. Another important subset of T is the space S of orientation-preserving diffeomorphisms of S1 considered modulo M¨obius transforms. It is a K¨ahler Frechet manifold. Our interest in T was initially motivated by its relation to string theory which we have studied earlier in a series of papers.
One-Particle Approximation as a Simple Playground for Irreversible Quantum Evolution
A.E. Teretenkov
Both quantum information features and irreversible quantum evolution of the models arising in physical systems in one-particle approximation are discussed. It is shown that the calculation of the reduced density matrix and entanglement analysis are considerably simplified in this case. The irreversible quantum evolution described by Gorini–Kossakowski–Sudarshan–Lindblad equations in the one-particle approximation could be defined by a solution of a Shroedinger equation with a dissipative generator. It simplifies the solution of the initial equation on the one side and gives a physical interpretation of such a Shroedinger equation with non-Hermitian Hamiltonian on the other side.
Non-Autonomous Dynamics and Product Formula Approximation of Solution Operator
Valentin A. Zagrebnov
The paper is devoted to non-autonomous dynamics, which is generated by positive self-adjoint operator A and a family of non-negative self-adjoint operators {B(t)}t≥0 defined in a separable Hilbert space. It is shown that solution operator {U(t, s)}0≤s≤t of the evolution equation can be approximated in the operator norm topology by a product formula that involves A and B(t). We also established the rate of convergence of the product formula to the solution operator. These results are proved using the evolution semigroup approach to non-autonomous dynamics.
Weak Compactness Problem for Sets of Bounded Radon Measures on Various Topological Spaces
Valeriy K. Zakharov, Timofey V. Rodionov
The paper presents some weak compactness criterion for a subset M of the set RMb(T,G ) of all positive bounded Radon measures on a Hausdorff topological space (T,G ) similar to the Prokhorov criterion for a complete separable metric space. Since for a general topological space the classical space Cb(T,G ) of all bounded continuous functions on T can be trivial and so does not separate points and closed sets, we consider instead of Cb(T,G )-weak compactness S(T,G )-weak compactness with respect to the new uniformly closed linear space S(T,G ) of all (symmetrizable) metasemicontinuous functions..
Evaluation of Chaotic Properties Of CBC Mode of Encryption Embedded with RC5 Block Cipher Algorithm
Abdessalem Abidi, Christophe Guyeux, Mohsen Machhout
It has been recently proven that, under ad hoc conditions, the Cipher Block Chaining (CBC) mode of operation can behave chaotically according to the mathematical definition of Devaney, on the infinite discrete product set of finite memory coupled with (finite) media of unbounded size. This occurs when the chosen block cipher function satisfies some properties related to a well defined associated graph. Rudimentary examples taken from so-called transposition cipher methods have formerly been proposed as illustrative examples. In this paper, the same canvas will be followed by regarding the conditions under which the CBC mode behaves chaotically. But the encryption function is now the Rivest Cipher 5 (RC5) one, a very famous symmetric key block cipher algorithm. Therefore, our goal is to prove the chaotic behavior of RC5-CBC encryption algorithm according to the reputed Devaney's definition. Then, this unpredictability was checked in hardware through such sensitivity tests which allowed us to validate that RC5-CBC exhibits a high degree of randomness, key and plain sensitivity, in addition to the so-called avalanche effect.
Vibration and Stability Analysis Comparison for Nanoshell and Piezoelectric Nanoshell Subjected to Electrostatic Excitation
Sayyid H. Hashemi Kachapi
In current study, vibration and stability analysis comparison of two nanostructures i.e. nanoshell (NS) and piezoelectric nanoshell (PENS) subjected to electrostatic excitation and Visco-Pasternak medium is investigated using the Gurtin–Murdoch surface/interface (S/I) theory. For this analysis, Hamilton's principles, the assumed mode method combined with Lagrange–Euler's and also Complex averagingmethod combinedwith Arclength continuation are used. It can be seen that by changing the surface/ interface densities and as a result, increasing or decreasing the system stiffness, the natural frequency can be less or greater than the state without taking into account the S/I effects. In both nanostructures (NS and PENS), considering the surface/interface effects increase the nonlinear behaviour compared with without S/I effects.
Iterative Method for Non-Stationary Mixed Variational Inequalities
We consider a non-stationary mixed variational inequality problem involving an integrable mapping and a convex function, where only approximation sequences are known instead of exact values of the cost mapping and function, and feasible set. We apply a descent method and partial penalization to prove the convergence is attained without concordance of penalty, accuracy, and approximation parameters under coercivity type conditions.
DNC Volume 10, Issue 1
Qualitative Analysis for a Phytoplankton-Zooplankton Model with Allee Effect and Holling Type II Response
M.S. Surendar, M. Sambath
This paper investigates a predator-prey system with Allee effect and Holling type-II functional response consisting of phytoplankton as prey and zooplankton as a predator. For the non-spatial system, we analyzed the stability, the existence of Hopf bifurcation at the coexistence equilibrium and stability of bifurcating periodic solutions are obtained. Moreover, the diffusion driven instability, Hopf bifurcation of the corresponding spatial system with zero flux boundary conditions and Turing instability region regarding parameters are established. Numerical simulations are provided to illustrate theoretical results.
Dynamics of an HIV pathogenesis Model with Absorption and Saturation Incidence
Vinoth Sivakumar, Jayakumar Thippan, Prasantha Bharathi Dhandapani
In this paper, the dynamics of an HIV infection model with absorption and saturation incidence are proposed and analyzed. Further, we introduce a time delay to the model, which describes a time between infected cells and excretion of the viral particles. This model is used to explain existence, characteristic equations, and stability of infected and disease free steady states. Numerical simulations are provided to illustrate the theoretical results.
Uniqueness and Decay of Weak Solutions to Phase-Lock Equations
Jishan Fan, Gen Nakamura, and Mei-Qin Zhan
In this paper, we prove the uniqueness of weak solutions $(f ,Q)$ to the phase-lock equations with $f_0$ ∈ $L^2$ and $Q_0$ ∈ $L^3$ when the space dimension $d = 3$. We also prove the uniqueness of weak solutions $(f ,a)$ to the Ginzburg-Landau equations with $(f_0,a_0)$ ∈ $L^p ×L^p$ and $1 < p < 2$ when $d = 1$. We will also present a result on the decay of $Q$ as time $t →\infty$.
Evolutionary Dynamics of Zero-sum Games with Degenerate Payoff Matrix and Bisexual Population
N.N. Ganikhodjaev, U.U. Jamilov, M. Ladra
In this paper we consider the quadratic stochastic operators describing evolution of a bisexual population. We establish correlation between such operators and evolutionary games, namely demonstrate that Volterra quadratic stochastic operator with degenerate payoff matrix is non-ergodic and corresponding evolutionary game is rock-paper-scissors game. To prove this statements we study the asymptotic behavior of trajectories of the Volterra quadratic stochastic operators with the non-hyperbolic fixed points.
Dynamics of a Stage-Structured-Prey and Predator Model with Linear Harvesting of Mature Prey and Predator
In this paper we have studied the dynamical behaviors of stage-structuredprey and predator model with linear harvesting of both mature prey and predator. Optimal harvesting policy, positivity, boundedness and Hopfbifurcation have been studied. Finally, some graphical and numerical simulations are given in order to validate our analytical and theoretical findings.
Numerical Solution of Hybrid Fuzzy Mixed Delay Differential Equation by Fourth Order Runge-Kutta Method
D. Prasantha Bharathi, T. Jayakumar, S. Vinoth
The hybrid systems are used to study many engineering problems over a period of time. Now, we are adding another parameter called delay to make the hybrid systems to model any biological problems in future. We are extending the system of hybrid fuzzy differential equation to hybrid fuzzy mixed delay differential equation in this research article. The Runge-Kutta method of order four has been used to solve the problem by converting the mixed delay differential equation to multiple retarded delay differential equation. Finally, the numerical example is presented and the results are analyzed.
Distributed Delay Effects on Coupled van der Pol Oscillators, and a Chaotic van der Pol-Rayleigh System
Ryan Roopnarain, S. Roy Choudhury
Distributed delays modeled by 'weak generic kernels' are introduced in two well-known coupled van der Pol systems, as well as a chaotic van der PolRayleigh system with parametric forcing. The systems are closed via the 'linear chain trick'. Linear stability analysis of the systems and conditions for Hopf bifurcation that initiates oscillations are investigated, including deriving the normal form at bifurcation, and deducing the stability of the resulting limit cycle attractor. The value of the delay parameter $a = a_{Hopf}$ at Hopf bifurcation picks out the onset of Amplitude Death(AD) in all three systems, with oscillations at larger values (corresponding to weaker delay). In both van der Pol systems, the Hopf-generated limit cycles for $a > a_{Hopf}$ turn out to be remarkably stable under very large variations of all other system parameters beyond the Hopf bifurcation point, and do not undergo further symmetry breaking, cyclic-fold, flip, transcritical or Neimark-Sacker bifurcations. Numerical simulations reveal strong distortion and rotation of the limit cycles in phase space as the parameters are pushed far into the post-Hopf regime, and also reveal other features, such as how the oscillation amplitudes and time periods of the physical variables on the limit cycle attractor change as the delay and other parameters are varied. For the chaotic system, very strong delays may still lead to the cessation of oscillations and the onset of AD. Varying of the other important system parameter, the parametric excitation, leads to a rich sequence of dynamical behaviors, with the bifurcations leading from one regime (or type of attractor) into the next being carefully tracked.
Watermarking of Electronic Patient Record in Parkinson Disease Affected Speech: A Robust and Secure Audio Hiding Technique for Smart e-healthcare Application
Aniruddha Kanhe, Aghila Gnanasekaran
A cloud-based framework for security of electronic patient record, in telediagnosis of Parkinson's disease (PD) speech signal has been proposed in this paper. The security and authenticity of patient's personal information is achieved using proposed Discrete Cosine Transform-Singular Value Decomposition (DCT-SVD) based audio watermarking technique. The automatic classification of PD affected speech signal from healthy person's speech signal requires high computational accuracy and in this work, it is addressed by extracting various time frequency features for the Support Vector Machine (SVM) classifier. In the proposed framework the speech signal of PD suspected person is recorded by a mobile application. The personal information is securely embedded in this speech signal using proposed DCT-SVD based audio watermarking technique, before transmitting to cloud for automatic classification. To ensure the security of electronic patient record (EPR), proposed watermarking technique is tested against various signal processing attacks and the performance of classifier has been evaluated by computing classification sensitivity, specificity, accuracy and area under the curve of receiver operating characteristics (ROC). The theoretical proofs and experimental results, show that proposed framework provides telediagnosis of PD affect patients without compromising the security and privacy of patient's records. The classification accuracy of 89% with high data payload of 6kbps is proposed leading to a smart and secure e-healthcare application.
Abstract Fractals
Marat Akhmet, Ejaily Milad Alejaily
We develop a new definition of fractals which can be considered as an abstraction of the fractals determined through self-similarity. The definition is formulated through imposing conditions which govern a relation between subsets of a metric space to build a porous self-similar structure. Examples are provided to confirm that the definition satisfies a large class of self-similar fractals. The new concepts create new frontiers for fractals and chaos investigations.
The Global Attractiveness of the Fixed Point of a Gonosomal Evolution Operator
Akmal T. Absalamov
In the paper we prove a conjecture of U.A. Rozikov and R. Varro about globally attractiveness of a unique nonhyperbolic fixed point of the normalized gonosomal evolution operator of a sex linked inheritance.
Holling-Tanner Predator-Prey Model with Type-IV Functional Response and Harvesting
Nurul Huda Gazi, Subrata Kumar Biswas
In this paper we modify Holling-Tanner predator-preymodel by using type-IV functional response in prey species in lieu of type-II functional response. Harvesting is used in predator as well as prey species. This model is compared with a special type of Kolmogorov model. In the case of quadratic harvesting, the fixed points are computed after nondimensionalization. For the non-existence of periodic orbits in the first quadrant we apply a condition of the general Kolmogorov model to exist a Dulac function. We show that this system does not have periodic orbits with the help of numerical simulation and graphical representation.
Bifurcation Analysis of Wilson-Cowan Model with Singular Impulses
Marat Akhmet, Sabahattin C¸ a˘g
The paper concernswithWilson-Cowan neuralmodel. Themain novelty of the study is that besides the traditional singularity of themodel, we consider singular impulses. A new technique of analysis of the phenomenon is suggested. This allows to consider the existence of solutions of the model and bifurcation in ultimate neural behavior observed through numerical simulations. The bifurcations are reasoned by impulses and singularity in the model and they concern the structure of attractors, which consist of newly introduced sets in the phase space such that medusas and rings.
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