## Tokyo Journal of Mathematics

### Hopf-homoclinic Bifurcations and Heterodimensional Cycles

Shuntaro TOMIZAWA

#### 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 \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

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

Dates
First available in Project Euclid: 6 August 2018

https://projecteuclid.org/euclid.tjm/1533520824

#### Citation

TOMIZAWA, Shuntaro. Hopf-homoclinic Bifurcations and Heterodimensional Cycles. Tokyo J. Math., advance publication, 6 August 2018. https://projecteuclid.org/euclid.tjm/1533520824

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