Topological horseshoes for perturbations of singular difference equations

Ming-Chia Li*, Mikhail Malkin

*Corresponding author for this work

Research output: Contribution to journalArticle

18 Scopus citations

Abstract

In this paper, we study solutions of difference equations Φλ(yn, yn+1, ..., yn+m) = 0, , of order m with parameter λ, and consider the case when Φλ has a singular limit depending on a single variable as λ → λ0, i.e. , where N is an integer with 0 ≤ N ≤ m and φ is a function. We prove that if φ has k simple zeros then for λ close enough to λ0, 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 λ varies. As applications of these results, we establish the horseshoe structure in families of generalized Hénon-like maps and of Arneodo-Coullet-Tresser maps near their anti-integrable limits as well as in steady states for certain lattice models.

Original languageEnglish
Pages (from-to)795-811
Number of pages17
JournalNonlinearity
Volume19
Issue number4
DOIs
StatePublished - 1 Apr 2006

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