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Discrete chaos in Banach spaces

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Abstract

This paper is concerned with chaos in discrete dynamical systems governed by continuously Frechét differentiable maps in Banach spaces. A criterion of chaos induced by a regular nondegenerate homoclinic orbit is established. Chaos of discrete dynamical systems in then-dimensional real space is also discussed, with two criteria derived for chaos induced by nondegenerate snap-back repellers, one of which is a modified version of Marotto’s theorem. In particular, a necessary and sufficient condition is obtained for an expanding fixed point of a differentiable map in a general Banach space and in ann-dimensional real space, respectively. It completely solves a long-standing puzzle about the relationship between the expansion of a continuously differentiable map near a fixed point in ann-dimensional real space and the eigenvalues of the Jacobi matrix of the map at the fixed point.

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Authors and Affiliations

  1. Department of Mathematics, Shandong University, 250100, Jina, China

    Yuming Shi

  2. Department of Electronic Engineering, City University of Hong Kong, China

    Guanrong Chen

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  1. Yuming Shi
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  2. Guanrong Chen
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Correspondence to Yuming Shi or Guanrong Chen.

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Shi, Y., Chen, G. Discrete chaos in Banach spaces. Sci. China Ser. A-Math. 48, 222–238 (2005). https://doi.org/10.1360/03ys0183

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  • DOI: https://doi.org/10.1360/03ys0183

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