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The Structural Stability of Maps with Heteroclinic Repellers

Yuanlong Chen

School of Financial Mathematics and Statistics, Guangdong University of Finance, Guangzhou 510521, P. R. China

E-mail Address: chernylong@163.com

E-mail Address: 26-022@gduf.edu.cn

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Liangliang Li

Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Guangzhou 510275, P. R. China

E-mail Address: liliangliang@mail.sysu.edu.cn

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Xiaoying Wu

http://orcid.org/0000-0003-0144-8440

School of Financial Mathematics and Statistics, Guangdong University of Finance, Guangzhou 510521, P. R. China

E-mail Address: wxyingyy@163.com

E-mail Address: 26-082@gduf.edu.cn

Corresponding author.

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, and

Feng Wang

School of Financial Mathematics and Statistics, Guangdong University of Finance, Guangzhou 510521, P. R. China

E-mail Address: 27-010@gduf.edu.cn

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https://doi.org/10.1142/S0218127420502077Cited by:6 (Source: Crossref)

Abstract

This note is concerned with the effect of small $C^{1}$ $C^{1}$ perturbations on a discrete dynamical system $(X, f)$ $(X, f)$ , which has heteroclinic repellers. The question to be addressed is whether such perturbed system $(X, g)$ $(X, g)$ has heteroclinic repellers. It will be shown that if $∥ f - g ∥_{C^{1}}^{1}$ $∥ f - g ∥_{C^{1}}$ is small enough, $(X, g)$ $(X, g)$ has heteroclinic repellers, which implies that it is chaotic in the sense of Devaney. In addition, if $X = R^{n}$ $X = R^{n}$ and $(X, f)$ $(X, f)$ has regular nondegenerate heteroclinic repellers, then $(X, g)$ $(X, g)$ has regular nondegenerate heteroclinic repellers, where $g$ $g$ is a small Lipschitz perturbation of $f$ $f$ . Three examples are presented to validate the theoretical conclusions.

Keywords:

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