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Topological spatial chaos and homoclinic points of Zd-Actions in lattice dynamical systems

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Abstract

We study aZ d-action on a finite-dimensional subset of a Banach space. This subset represents a set of equilibrium solutions of a lattice dynamical system, i.e. a system with discrete spatial variables. Such a system can appear as a discrete version of a partial differential equation or an infinite lattice of dissipatively coupled oscillators, etc. Stochastic behavior of theZ d-action corresponds to the existence of spatial chaos; in other words, the existence of an infinite number of stable structures which are randomly situated along spatial coordinates. We generalize the notion of homoclinic points into the case of theZ d-action and prove that the existence of a homoclinic point of theZ d-action implies spatial chaos in a system under consideration.

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Authors and Affiliations

  1. Georgia Institute of Technology, Center for Dynamical Systems and Nonlinear Studies, 30332-0190, Atlanta, GA

    Valentine S. Afraimovich

  2. Georgia Institute of Technology, School of Mathematics, 30332-0160, Atlanta, GA

    Shui-Nee Chow

Authors
  1. Valentine S. Afraimovich
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  2. Shui-Nee Chow
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Additional information

Research supported in part by the Army Research Office Grant #DAAH04-93-G-0199 and the National Institute of Standards and Technology Grant #60NANB201276.

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Afraimovich, V.S., Chow, SN. Topological spatial chaos and homoclinic points of Zd-Actions in lattice dynamical systems. Japan J. Indust. Appl. Math. 12, 367–383 (1995). https://doi.org/10.1007/BF03167235

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  • DOI: https://doi.org/10.1007/BF03167235

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