Analytic and algebraic conditions for bifurcations of homoclinic orbits II: Reversible systems

TL;DR

This paper extends the analysis of homoclinic bifurcations in reversible systems, providing algebraic and analytic conditions, and demonstrates the theory with numerical examples in four-dimensional systems.

Contribution

It introduces new conditions for bifurcations of symmetric homoclinic orbits in reversible systems, extending previous work with algebraic, variational, and Melnikov methods.

Findings

Bifurcations occur when variational equations are integrable in differential Galois sense.

Extended Melnikov method yields conditions for saddle-node, transcritical, and pitchfork bifurcations.

Numerical examples confirm theoretical predictions in four-dimensional reversible systems.

Abstract

Following Part~I, we consider a class of reversible systems and study bifurcations of homoclinic orbits to hyperbolic saddle equilibria. Here we concentrate on the case in which homoclinic orbits are symmetric, so that only one control parameter is enough to treat their bifurcations, as in Hamiltonian systems. First, we modify and extend arguments of Part~I to show in a form applicable to general systems discussed there that if such bifurcations occur in four-dimensional systems, then variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory. We next extend the Melnikov method of Part~I to reversible systems and obtain theorems on saddle-node, transcritical and pitchfork bifurcations of symmetric homoclinic orbits. We illustrate our theory for a four-dimensional system, and demonstrate the theoretical results by numerical ones.