About a homoclinic pitchfork bifurcation in reversible systems with additional ℤ2-symmetry

Abstract

This paper studies bifurcations from a homoclinic orbit to a degenerate fixed point. We consider reversible ℤ2-symmetric systems of ordinary differential equations (ODEs) and assume the existence of a symmetric homoclinic orbit to a fixed point which itself undergoes a pitchfork bifurcation. We are interested in bifurcations from the primary homoclinic orbit in an unfolding of the degenerate situation.The studies are motivated by numerical investigations on a model-system of second-order ODEs. There one finds a similar behaviour in the local and the global bifurcation. While locally two new fixed points are created, numerical computations show that at the same time two homoclinic orbits to these fixed points bifurcate from the primary orbit. We call the global scenario a reversible homoclinic pitchfork bifurcation.An analysis of this homoclinic bifurcation is performed in a general frame. Depending on the sign of a higher order coefficient in the normal form we distinguish two cases of the local pitchfork bifurcation: the eye case (which is the one encountered in the model-system) and the figure-eight case. Adopting Lin's method to the non-hyperbolic situation the bifurcation of one-homoclinic orbits to the local centre manifold of the fixed point is investigated. Rigorous existence results for homoclinic orbits to fixed points and periodic orbits are derived. The global bifurcation picture is found to depend crucially on the local bifurcation.