The C¹ persistence of heteroclinic repellers in ${\mathbb{R}}^n$

Abstract

In this paper, we show that if f is a $C^1$-map from ${\mathbb{R}}^n$ into itself and has heteroclinic repellers, then g also has heteroclinic repellers with ${\Vert f-g\Vert }_{C^1}$ being small enough and exhibits Devaney's chaos. The results demonstrate $C^1$ structural stability of heteroclinic repellers in Euclidean spaces. In the end, we give some examples to illustrate our theoretical results.

Introduction

Research on chaotic behaviors in dynamics has attracted more and more attentions due to their potential applications in various fields (see [1], [3], [5], [7], [8], [14], [17]). Although chaotic phenomenon had previously been observed in various fields, the term “chaos” has first been introduced by Li and Yorke in 1975, where they obtained that “period three implies chaos”. When the study of chaos theory was in its infancy, Marotto generalized the work of Li and Yorke to multidimensional discrete systems and presented that snap-back repellers implies chaos [15]. Further, Blanco demonstrated that a snap-back repeller implies positive topological entropy in [2]. Later, Devaney outlined an explicit definition of chaos in 1989 [9].

From the viewpoint of differential dynamical systems, it is natural to ask whether a dynamical system with some property P still has the property P under any small perturbations. In [16], Marotto showed that the discrete system with a snap-back repeller under delayed perturbations still has chaotic behaviors. In 2009, Li et al. showed the persistence of snap-back repellers for any small $C^1$ perturbations [10]. Chen et al. studyed the structural stability of snap-back repellers in Banach space [6]. In 2016, Chen and Li provides a sufficient condition for any small $C^1$ perturbation of a high-dimensional difference equation to have symbolic embedding [4].

In this paper, we consider the following differential dynamical systems:

$$x(n+1)=f(x(n)),n=1,2,\cdots ,$$

where f is a continuous map from ${\mathbb{R}}^n$ into itself. We show the persistence of the systems with heteroclinic repellers under any small $C^1$ perturbation. We obtain that the following main result:

Theorem 3.1

Let $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be a continuously differentiable map. Suppose that f has heteroclinic repellers $z_1(0),z_2(0),\cdots ,z_k(0)(k\ge 2)$. Then g also has heteroclinic repellers for any g with ${\Vert g-f\Vert }_{C^1}$ being small enough and there exist a positive integer n and a Cantor invariant set Λ such that $(g^n,\mathit{\Lambda })$ is topologically conjugate to the symbolic dynamical system $(\sigma ,{\mathit{\Sigma }}_2^{+})$. Therefore g is chaotic in the sense of Devaney.

This paper is organized as follows: In Section 2, we first recall the definitions of heteroclinic repellers. Some important lemmas in the sequel are given. The main results in the paper are presented in Section 3. At the end, we give some examples to illustrate the theoretical results.