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2000 Dissipative systems generating any structurally stable chaos
S. A. Vakulenko
Adv. Differential Equations 5(7-9): 1139-1178 (2000). DOI: 10.57262/ade/1356651296

Abstract

Reaction-diffusion and coupled oscillator systems are considered. They possess inertial manifolds and, moreover, in a sense, their inertial dynamics can be "controlled" by some system parameters $\mathcal P$: the inertial dynamics can be specified to within an arbitrarily small error by adjusting of $\mathcal P$. Due to the classical persistence hyperbolic set theorem, this property of their inertial forms yields that any hyperbolic local attractors and invariant sets can be embedded into the global attractors of these systems. The method of the proof is based on recent results from the neural network theory and the ideas connected with the dynamics of localized modes in singular perturbed systems. This approach can be considered as a development of the method of the realization of vector fields pioneering by P. Poláčik and has a physical interpretation. It is shown, in particular, that the fundamental models of neural network type (the Hopfield systems) can generate any structurally stable (persistent) large-time behaviour.

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S. A. Vakulenko. "Dissipative systems generating any structurally stable chaos." Adv. Differential Equations 5 (7-9) 1139 - 1178, 2000. https://doi.org/10.57262/ade/1356651296

Information

Published: 2000
First available in Project Euclid: 27 December 2012

zbMATH: 1192.37104
MathSciNet: MR1776350
Digital Object Identifier: 10.57262/ade/1356651296

Subjects:
Primary: 37L25
Secondary: 35B42 , 35K55 , 37L30

Rights: Copyright © 2000 Khayyam Publishing, Inc.

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Vol.5 • No. 7-9 • 2000
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