Advances in Differential Equations

Dissipative systems generating any structurally stable chaos

S. A. Vakulenko

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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.

Article information

Adv. Differential Equations, Volume 5, Number 7-9 (2000), 1139-1178.

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37L25: Inertial manifolds and other invariant attracting sets
Secondary: 35B42: Inertial manifolds 35K55: Nonlinear parabolic equations 37L30: Attractors and their dimensions, Lyapunov exponents


Vakulenko, S. A. Dissipative systems generating any structurally stable chaos. Adv. Differential Equations 5 (2000), no. 7-9, 1139--1178.

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