Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations
Mathematical analysis of a HIV model with quadratic logistic growth term
October 2012, 17(7): 2387-2412. doi: 10.3934/dcdsb.2012.17.2387
Construction of a finite-time Lyapunov function by meshless collocation
Peter Giesl 1,
Department of Mathematics, University of Sussex, Falmer BN1 9QH
Received July 2011 Revised February 2012 Published July 2012
We consider a nonautonomous ordinary differential equation of the form $\dot{x}=f(t,x)$, $x\in \mathbb{R}^n$ over a finite-time interval $t\in [T_1,T_2]$. The basin of attraction of an attracting solution can be determined using a finite-time Lyapunov function.
In this paper, such a finite-time Lyapunov function is constructed by Meshless Collocation, in particular Radial Basis Functions. Thereto, a finite-time Lyapunov function is characterised as the solution of a second-order linear partial differential equation with boundary values. This problem is approximately solved using Meshless Collocation, and it is shown that the approximate solution can be used to determine the basin of attraction. Error estimates are obtained and verified in examples.
Keywords: meshless collocation, basin of attraction, radial basis functions, error estimates., Nonautonomous ordinary differential equation, finite-time Lyapunov function.
Mathematics Subject Classification: Primary: 34D20, 65N35; Secondary: 37C60, 65N1.
Citation: Peter Giesl. Construction of a finite-time Lyapunov function by meshless collocation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2387-2412. doi: 10.3934/dcdsb.2012.17.2387
A. Berger, On finite-time hyperbolicity,, Comm. Pure Applied Anal., 10 (2011), 963. doi: 10.3934/cpaa.2011.10.963. Google Scholar
A. Berger, D. T. Son and S. Siegmund, Nonautonomous finite-time dynamics,, Discrete Cont. Dyn. Syst. Ser. B, 9 (2008), 463. Google Scholar
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM J. Numer. Anal., 36 (1999), 491. doi: 10.1137/S0036142996313002. Google Scholar
M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, in, (2002), 221. Google Scholar
T. S. Doan, D. Karrasch, N. T. Yet and S. Siegmund, A unified approach to finite-time hyperbolicity which extends finite-time Lyapunov exponents,, submitted., (). Google Scholar
T. S. Doan, K. Palmer and S. Siegmund, Transient spectral theory, stable and unstable cones and Gershgorin's theorem for finite-time differential equations,, J. Diff. Equations, 250 (2011), 4177. doi: 10.1016/j.jde.2011.01.013. Google Scholar
C. Franke and R. Schaback, Solving partial differential equations by collocation using radial basis functions,, Appl.\ Math.\ Comput., 93 (1998), 73. doi: 10.1016/S0096-3003(97)10104-7. Google Scholar
P. Giesl, "Construction of Global Lyapunov Functionsusing Radial Basis Functions,'', Lecture Notes in Mathematics, 1904 (2007). Google Scholar
P. Giesl and S. Hafstein, Local Lyapunov Functions for periodic and finite-time ODEs,, submitted., (). Google Scholar
P. Giesl and M. Rasmussen, Areas of attraction for nonautonomous differential equations on finite time intervals,, J. Math. Anal. Appl., 390 (2012), 27. doi: 10.1016/j.jmaa.2011.12.051. Google Scholar
P. Giesl and H. Wendland, Meshless Collocation: Error Estimates with Application to Dynamical Systems,, SIAM J. Numer. Anal., 45 (2007), 1723. doi: 10.1137/060658813. Google Scholar
P. Giesl and H. Wendland, Approximating the basin of attraction of time-periodic ODEs by meshless collocation,, Discrete Contin. Dyn. Syst., 25 (2009), 1249. doi: 10.3934/dcds.2009.25.1249. Google Scholar
P. Giesl and H. Wendland, Approximating the Basin of attraction of time-periodic ODEs by meshless collocation of a Cauchy problem,, Discrete Contin. Dyn. Syst., 2009 (): 259. Google Scholar
L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation,, Numer. Math., 75 (1997), 319. doi: 10.1007/s002110050241. Google Scholar
H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions,, Proc. Amer. Math. Soc., 136 (2008), 2793. doi: 10.1090/S0002-9939-08-09341-6. Google Scholar
S. Hafstein, A constructive converse Lyapunov Theoremon Exponential Stability,, Discrete Contin. Dyn. Syst., 10 (2004), 657. doi: 10.3934/dcds.2004.10.657. Google Scholar
G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields,, Chaos, 10 (2000), 99. doi: 10.1063/1.166479. Google Scholar
G. Haller, A variational theory of hyperbolic Lagrangian coherent structures,, Physica D, 240 (2011), 574. doi: 10.1016/j.physd.2010.11.010. Google Scholar
G. Haller and T. Sapsis, Lagrangian coherent structures and the smallest finite-time Lyapunov exponent,, Chaos, 21 (2011). Google Scholar
G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulance,, Physica D, 147 (2000), 352. doi: 10.1016/S0167-2789(00)00142-1. Google Scholar
C. Hsu, Global analysis by cell mapping,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992), 727. Google Scholar
B. Krauskopf and H. Osinga, Computing invariant manifolds via the continuation of orbit segments,, in, (2007), 117. Google Scholar
B. Krauskopf, H. Osinga, E. J. Doedel, M. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763. doi: 10.1142/S0218127405012533. Google Scholar
A. M. Lyapunov, Problème général de la stabilité du mouvement,, Ann. Fac. Sci. Toulouse, 9 (1907), 203. Google Scholar
F. J. Narcowich, J. D. Ward and H. Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting,, Mathematics of Computation, 74 (2005), 743. doi: 10.1090/S0025-5718-04-01708-9. Google Scholar
G. Osipenko, "Dynamical Systems, Graphs, and Algorithms,'', Lecture Notes in Mathematics, 1889 (1889). Google Scholar
M. Rasmussen, Finite-time attractivity and bifurcation for nonautonomous differential equation,, Differential Equations Dynam. Systems, 18 (2010), 57. doi: 10.1007/s12591-010-0009-7. Google Scholar
S. C. Shadden, F. Lekien and J. E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows,, Physica D, 212 (2005), 271. doi: 10.1016/j.physd.2005.10.007. Google Scholar
H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree,, J. Approx. Theory, 93 (1998), 258. doi: 10.1006/jath.1997.3137. Google Scholar
H. Wendland, "Scattered Data Approximation,'', Cambridge Monographs on Applied and Computational Mathematics, 17 (2005). Google Scholar
Z. Wu, Hermite-Birkhoff interpolation of scattered data by radial basis functions,, Approx. Theory Appl., 8 (1992), 1. Google Scholar
Peter Giesl, Holger Wendland. Approximating the basin of attraction of time-periodic ODEs by meshless collocation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1249-1274. doi: 10.3934/dcds.2009.25.1249
Peter Giesl, James McMichen. Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1835-1850. doi: 10.3934/dcdsb.2018094
Peter Giesl, Holger Wendland. Approximating the basin of attraction of time-periodic ODEs by meshless collocation of a Cauchy problem. Conference Publications, 2009, 2009 (Special) : 259-268. doi: 10.3934/proc.2009.2009.259
Peter Giesl, James McMichen. Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation. Journal of Computational Dynamics, 2016, 3 (2) : 191-210. doi: 10.3934/jcd.2016010
Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101
Hjörtur Björnsson, Sigurdur Hafstein, Peter Giesl, Enrico Scalas, Skuli Gudmundsson. Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4247-4269. doi: 10.3934/dcdsb.2019080
Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463
Najla Mohammed, Peter Giesl. Grid refinement in the construction of Lyapunov functions using radial basis functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2453-2476. doi: 10.3934/dcdsb.2015.20.2453
Imtiaz Ahmad, Siraj-ul-Islam, Mehnaz, Sakhi Zaman. Local meshless differential quadrature collocation method for time-fractional PDEs. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020223
Huijuan Li, Junxia Wang. Input-to-state stability of continuous-time systems via finite-time Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 841-857. doi: 10.3934/dcdsb.2019192
Peter Giesl, Najla Mohammed. Verification estimates for the construction of Lyapunov functions using meshfree collocation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4955-4981. doi: 10.3934/dcdsb.2019040
Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569
Khalid Addi, Samir Adly, Hassan Saoud. Finite-time Lyapunov stability analysis of evolution variational inequalities. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1023-1038. doi: 10.3934/dcds.2011.31.1023
Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75
Oliver Junge, Alex Schreiber. Dynamic programming using radial basis functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4439-4453. doi: 10.3934/dcds.2015.35.4439
Sohana Jahan, Hou-Duo Qi. Regularized multidimensional scaling with radial basis functions. Journal of Industrial & Management Optimization, 2016, 12 (2) : 543-563. doi: 10.3934/jimo.2016.12.543
Arno Berger. On finite-time hyperbolicity. Communications on Pure & Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963
Thierry Cazenave, Yvan Martel, Lifeng Zhao. Finite-time blowup for a Schrödinger equation with nonlinear source term. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1171-1183. doi: 10.3934/dcds.2019050
Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631
Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019041
Peter Giesl