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Nonlinear Sciences
Submissions received from Thu 14 Jan 21 to Fri 15 Jan 21, announced Mon, 18 Jan 21
Cross-lists
[ total of 9 entries: 1-9 ]
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New submissions for Mon, 18 Jan 21
Title: Unveiling the role of plasticity rules in reservoir computing
Authors: Guillermo B. Morales, Claudio R. Mirasso, Miguel C. Soriano
Subjects: Adaptation and Self-Organizing Systems (nlin.AO); Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE)
Reservoir Computing (RC) is an appealing approach in Machine Learning that combines the high computational capabilities of Recurrent Neural Networks with a fast and easy training method. Likewise, successful implementation of neuro-inspired plasticity rules into RC artificial networks has boosted the performance of the original models. In this manuscript, we analyze the role that plasticity rules play on the changes that lead to a better performance of RC. To this end, we implement synaptic and non-synaptic plasticity rules in a paradigmatic example of RC model: the Echo State Network. Testing on nonlinear time series prediction tasks, we show evidence that improved performance in all plastic models are linked to a decrease of the pair-wise correlations in the reservoir, as well as a significant increase of individual neurons ability to separate similar inputs in their activity space. Here we provide new insights on this observed improvement through the study of different stages on the plastic learning. From the perspective of the reservoir dynamics, optimal performance is found to occur close to the so-called edge of instability. Our results also show that it is possible to combine different forms of plasticity (namely synaptic and non-synaptic rules) to further improve the performance on prediction tasks, obtaining better results than those achieved with single-plasticity models.
Title: Gauge Symmetry Origin of Bäcklund Transformations for Painlevé Equations
Authors: V.C.C. Alves, H. Aratyn, J.F. Gomes, A.H. Zimerman
Subjects: Exactly Solvable and Integrable Systems (nlin.SI)
We identify the self-similarity limit of the second flow of $sl(N)$ mKdV hierarchy with the periodic dressing chain thus establishing % a connection to $A^{(1)}_{N-1}$ invariant
Painlev\'e equations. The $A^{(1)}_{N-1}$ B\"acklund symmetries of dressing equations and Painlev\'e equations are obtained in the self-similarity limit of gauge transformations of the mKdV hierarchy realized as zero-curvature equations on the loop algebra $\widehat{sl}(N)$ endowed with a principal gradation.
Title: Discordant synchronization patterns on directed networks of identical phase oscillators with attractive and repulsive couplings
Authors: Thomas Peron
Subjects: Adaptation and Self-Organizing Systems (nlin.AO); Disordered Systems and Neural Networks (cond-mat.dis-nn); Statistical Mechanics (cond-mat.stat-mech)
We study the collective dynamics of identical phase oscillators on globally coupled networks whose interactions are asymmetric and mediated by positive and negative couplings. We split the set of oscillators into two interconnected subpopulations. In this setup, oscillators belonging to the same group interact via symmetric couplings while the interaction between subpopulations occurs in an asymmetric fashion. By employing the dimensional reduction scheme of the Ott-Antonsen (OA) theory, we verify the existence of traveling-wave and $\pi$-states, in addition to the classical fully synchronized and incoherent states. Bistability between all collective states is reported. Analytical results are generally in excellent agreement with simulations; for some parameters and initial conditions, however, we numerically detect chimera-like states which are not captured by the OA theory.
Cross-lists for Mon, 18 Jan 21
Title: Soliton resolution for the Hirota equation with weighted Sobolev initial data
Authors: Jin-Jie Yang, Shou-Fu Tian, Zhi-Qiang Li
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)
In this work, the $\overline{\partial}$ steepest descent method is employed to investigate the soliton resolution for the Hirota equation with the initial value belong to weighted Sobolev space $H^{1,1}(\mathbb{R})=\{f\in L^{2}(\mathbb{R}): f',xf\in L^{2}(\mathbb{R})\}$. The long-time asymptotic behavior of the solution $q(x,t)$ is derived in any fixed space-time cone $C(x_{1},x_{2},v_{1},v_{2})=\left\{(x,t)\in \mathbb{R}\times\mathbb{R}: x=x_{0}+vt ~\text{with}~ x_{0}\in[x_{1},x_{2}]\right\}$. We show that solution resolution conjecture of the Hirota equation is characterized by the leading order term $\mathcal {O}(t^{-1/2})$ in the continuous spectrum, $\mathcal {N}(\mathcal {I})$ soliton solutions in the discrete spectrum and error order $\mathcal {O}(t^{-3/4})$ from the $\overline{\partial}$ equation.
In this paper we extensively study the notion of Hamiltonian structure for nonabelian differential-difference systems, exploring the link between the different algebraic (in terms of double Poisson algebras and vertex algebras) and geometric (in terms of nonabelian Poisson bivectors) definitions. We introduce multiplicative double Poisson vertex algebras (PVAs) as the suitable noncommutative counterpart to multiplicative PVAs, used to describe Hamiltonian differential-difference equations in the commutative setting, and prove that these algebras are in one-to-one correspondence with the Poisson structures defined by difference operators, providing a sufficient condition for the fulfilment of the Jacobi identity. Moreover, we define nonabelian polyvector fields and their Schouten brackets, for both finitely generated noncommutative algebras and infinitely generated difference ones: this allows us to provide a unified characterisation of Poisson bivectors and double quasi-Poisson algebra structures. Finally, as an application we obtain some results towards the classification of local scalar Hamiltonian difference structures and construct the Hamiltonian structures for the nonabelian Kaup, Ablowitz-Ladik and Chen-Lee-Liu integrable lattices.
Replacements for Mon, 18 Jan 21
[6] arXiv:1909.06813 (replaced) [pdf, other]
Title: Scaling of causal neural avalanches in a neutral model
Authors: Sakib Matin, Thomas Tenzin, W. Klein
Subjects: Biological Physics (physics.bio-ph); Adaptation and Self-Organizing Systems (nlin.AO); Neurons and Cognition (q-bio.NC)
Title: A new class of integrable maps of the plane: Manin transformations with involution curves
Authors: Peter H van der Kamp
Subjects: Exactly Solvable and Integrable Systems (nlin.SI); Algebraic Geometry (math.AG)
Title: Directed long-range transport of a nearly pure component atom clusters by the electromigration of a binary surface alloy
Authors: Mikhail Khenner
Subjects: Materials Science (cond-mat.mtrl-sci); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Pattern Formation and Solitons (nlin.PS); Applied Physics (physics.app-ph); Computational Physics (physics.comp-ph)
[9] arXiv:2012.07573 (replaced) [pdf, ps, other]
Title: Intersection numbers on $\overline {\mathcal M}_{g,n}$ and BKP hierarchy
Authors: Alexander Alexandrov
Comments: 16 pages, minor corrections, references added
Subjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Algebraic Geometry (math.AG); Exactly Solvable and Integrable Systems (nlin.SI)
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