Tokyo Journal of Mathematics

Hopf-homoclinic Bifurcations and Heterodimensional Cycles


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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 \emph{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.

Article information

Tokyo J. Math., Advance publication (2019), 21 pages.

First available in Project Euclid: 6 August 2018

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Primary: 37C05: Smooth mappings and diffeomorphisms
Secondary: 37C20: Generic properties, structural stability 37C29: Homoclinic and heteroclinic orbits


TOMIZAWA, Shuntaro. Hopf-homoclinic Bifurcations and Heterodimensional Cycles. Tokyo J. Math., advance publication, 6 August 2018.

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