Homotopy Classes for Stable Connections between Hamiltonian Saddle-Focus Equilibria

Abstract

For a class of Hamiltonian systems in ℝ4 the set of homoclinic and heteroclinic orbits which connect saddle-focus equilibria is studied using a variational approach. The oscillatory properties of a saddle-focus equilibrium and the variational nature of the problem give rise to connections in many homotopy classes of the configuration plane punctured at the saddle-foci. This variational approach does not require any assumptions on the intersections of stable and unstable manifolds, such as transversality. Moreover, these connections are shown to be local minimizers of an associated action functional. This result has applications to spatial pattern formation in a class of fourth-order bistable evolution equations.