Persistence of Heteroclinic Cycles Connecting Repellers in Banach Spaces

Abstract

This paper is concerned with persistence of heteroclinic cycles connecting repellers in Banach spaces. It is proved that if a map with a regular and nondegenerate heteroclinic cycle connecting repellers undergoes a small perturbation, then the perturbed map can still have a regular and nondegenerate heteroclinic cycle connecting repellers. The perturbation rang is given by an explicit positive constant according to the properties of the original map. Hence, the perturbed map and the original map are simultaneously chaotic in the sense of both Devaney and Li-Yorke. Especially, the persistence of heteroclinic cycles connecting repellers is also discussed in the Euclidean space, where the repellers can expand in different norms. Finally, three examples are provided to illustrate the validity of the theoretical results.