The structural stability of maps with heteroclinic repellers. (English) Zbl 1455.37020
Summary: This note is concerned with the effect of small \(C^1\) perturbations on a discrete dynamical system \((X,f)\), which has heteroclinic repellers. The question to be addressed is whether such perturbed system \((X,g)\) has heteroclinic repellers. It will be shown that if \(\Vert f-g \Vert_{C^1}\) is small enough, \((X,g)\) has heteroclinic repellers, which implies that it is chaotic in the sense of Devaney. In addition, if \(X=R^n\) and \((X,f)\) has regular nondegenerate heteroclinic repellers, then \((X,g)\) has regular nondegenerate heteroclinic repellers, where \(g\) is a small Lipschitz perturbation of \(f\). Three examples are presented to validate the theoretical conclusions.
MSC:
| 37C20 | Generic properties, structural stability of dynamical systems |
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
| 37C29 | Homoclinic and heteroclinic orbits for dynamical systems |
| 37B40 | Topological entropy |
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