Discrete chaos in Banach spaces

Abstract

This paper is concerned with chaos in discrete dynamical systems governed by continuously Frechet 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.