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Two-heteroclinic orbits emerging in the reversible homoclinic pitchfork bifurcation. (English) Zbl 1067.37023
Summary: 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 [the author, ibid. 15, No. 6, 2097--2119 (2002; Zbl 1021.37014)] 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.

37C29Homoclinic and heteroclinic orbits
37C80Symmetries, equivariant dynamical systems
37C10Vector fields, flows, ordinary differential equations
34C23Bifurcation (ODE)
34C37Homoclinic and heteroclinic solutions of ODE
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