Positive topologicalentropy for multidimensional perturbations of topologically crossinghomoclinicity

Ming-Chia Li; Ming-Jiea Lyu

doi:10.3934/dcds.2011.30.243

Article Contents

2011, Volume 30, Issue 1: 243-252. Doi: 10.3934/dcds.2011.30.243

This issue Previous Article $C^{\alpha}$-Hölder classical solutions for non-autonomous neutral differential equations Next Article Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain

Positive topological entropy for multidimensional perturbations of topologically crossing homoclinicity

Ming-Chia Li^1, and
Ming-Jiea Lyu^1,

1.
Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300

Received: December 31, 2009

Revised: September 30, 2010

Published: December 2010

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Abstract

In this paper, we consider a one-parameter family $F_{\lambda }$ of continuous maps on $\mathbb{R}^{m}$ or $\mathbb{R}^{m}\times \mathbb{R}^{k}$ with the singular map $F_{0}$ having one of the forms (i) $F_{0}(x)=f(x),$ (ii) $F_{0}(x,y)=(f(x),g(x))$, where $g:\mathbb{R}^{m}\rightarrow \mathbb{R} ^{k}$ is continuous, and (iii) $F_{0}(x,y)=(f(x),g(x,y))$, where $g:\mathbb{R}^{m}\times \mathbb{R}^{k}\rightarrow \mathbb{R}^{k}$ is continuous and locally trapping along the second variable $y$. We show that if $f:\mathbb{R}^{m}\rightarrow \mathbb{R}^{m}$ is a $C^{1}$ diffeomorphism having a topologically crossing homoclinic point, then $F_{\lambda }$ has positive topological entropy for all $\lambda $ close enough to $0$.

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Mathematics Subject Classification: Primary: 37D45, 37B30; Secondary: 37B10, 37B40, 37J40.

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