Topological horseshoes for perturbations of singular difference equations

Abstract

In this paper, we Study solutions of difference equations Phi(lambda)(y(n), y(n+1),...,y(n+m)) = 0, n is an element of Z, of order in with parameter lambda, and consider the case when Phi(lambda) has a singular limit depending on a single variable as lambda -> lambda(0), i.e. Phi(lambda 0)(y(0),...,y(m)) = phi(y(N)), where N is an integer with 0 <= N <= m and phi is a function. We prove that if phi has k simple zeros then for lambda close enough to lambda(0), the difference equation has a k-horseshoe among its solutions, that is, the dynamics is conjugate to the full shift with k symbols. Moreover, we show that these horseshoes change continuously in the uniform topology as lambda varies. As applications of these results, we establish the horseshoe structure in families of generalized Henon-like maps and of Arneodo-Coullet-Tresser maps near their anti-integrable limits as well as in steady states for certain lattice models.