Chaos for implicit difference equations with snap-back repellers

Abstract

This paper is a study of chaos for generalized dynamical systems derived from implicit difference equations. We define a snap-back repeller for an implicit difference equation and show that its existence implies chaotic dynamics for all small C-1-perturbed systems. By chaotic dynamics, we mean that the solution set of an implicit difference equation contains a compact subset on which the Bernoulli shift map is invariant and has positive topological entropy.