Two-heteroclinic orbits emerging in the reversible homoclinic pitchfork bifurcation

Abstract

We consider reversible and -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.