Rigorous verification of chaotic behaviour of maps using validated shadowing

Abstract

In this paper discrete dynamical systems exhibiting \\complicated behaviour" are investigated. We present a computer assisted method to prove that the given system admits the shift map as a subsystem. The method is applied to the Henon map with the classical parameter values. 1 1. Introduction It is often rather easy to detect complicated behaviour of dynamical systems via numerical experiments. But it is usually very dicult to prove that a given system indeed behaves chaotically in a rigorous mathematical sense. In this paper we present a computer assisted method that allows us to prove that a discrete dynamical system admits the shift map as a subsystem. The classical way to achieve this goal is to use Smale's horseshoe construction (Smale [10], see also Kirchgraber, Stoer [4]): One shows that the stable and the unstable manifolds of a hyperbolic xed point intersect transversally. Then the horseshoe construction enables one to show that for some N the N{th iterate of the given m...