Analytic and algebraic conditions for bifurcations of homoclinic orbits I: Saddle equilibria

TL;DR

This paper investigates bifurcations of homoclinic orbits to saddle equilibria in four-dimensional systems, using Melnikov's method and differential Galois theory, with applications to Ginzburg-Landau PDEs.

Contribution

It provides new conditions for bifurcations of homoclinic orbits in both Hamiltonian and non-Hamiltonian systems, including integrability results via differential Galois theory.

Findings

Bifurcation conditions for homoclinic orbits are established.

The variational equations are shown to be integrable under certain conditions.

Numerical simulations confirm the theoretical bifurcation results.

Abstract

We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but two parameters are needed in general systems. We apply a version of Melnikov's method due to Gruendler to obtain saddle-node and pitchfork types of bifurcation results for homoclinic orbits. Furthermore we prove that if these bifurcations occur, then the variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds. We illustrate our theories with an example which arises as stationary states of coupled real Ginzburg-Landau partial differential equations, and demonstrate the theoretical results by numerical ones.