Hopf-homoclinic Bifurcations and Heterodimensional Cycles

Abstract

We consider a $C^r$ diffeomorphism having a Hopf point with $r\ge 5$. If there exists a homoclinic orbit associated with the Hopf point, we say that the diffeomorphism has a Hopf-homoclinic cycle. In this paper we prove that every $C^r$ diffeomorphism having a Hopf-homoclinic cycle can be $C^r$ approximated by diffeomorphisms with heterodimensional cycles. Moreover, we study stabilizations of such heterodimensional cycles.