Two-heteroclinic orbits emerging in the reversible homoclinic pitchfork bifurcation

Wagenknecht, Thomas (2005) Two-heteroclinic orbits emerging in the reversible homoclinic pitchfork bifurcation. Nonlinearity, 18 (2). pp. 527-542. ISSN 0951-7715

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Abstract

We consider reversible and \Bbb{Z}_2 -symmetric systems of ordinary differential equations (ODEs) that possess a symmetric homoclinic orbit to a degenerate equilibrium. The equilibrium is supposed to undergo a reversible pitchfork bifurcation, controlled by the system's parameter. It has been shown in Wagenknecht (Nonlinearity 15 2097–119) that a multitude of homoclinic orbits emerges in this bifurcation. In particular, if a coefficient in the normal form of the local bifurcation has the correct sign such that this bifurcation is of eye-type, then globally a reversible homoclinic pitchfork bifurcation can be observed. This means, that similar to the local bifurcation in which two new equilibria emerge, two-homoclinic orbits to these equilibria bifurcate from the primary homoclinic orbit. In this paper, we investigate the emergence of two-homoclinic and two-heteroclinic orbits, that is, orbits making two windings in a neighbourhood of the primary orbit, in this bifurcation. Using a combination of geometrical and analytical techniques we prove the emergence of a family of two-homoclinic orbits to periodic orbits and of a two-heteroclinic cycle between equilibria. The general analysis is illustrated by numerical results for an example system of two second order ODEs.

Item Type:	Article
Uncontrolled Keywords:	homoclinic bifurcation, degenerate equilibrium, reversible system
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 34 Ordinary differential equations MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theory
Depositing User:	Thomas Wagenknecht
Date Deposited:	19 Oct 2006
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/631

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