Bifurcation of Homoclinic Orbits to a Saddle-Focus in Reversible Systems with SO(2)-Symmetry

Abstract

We study reversible, SO(2)-invariant vector fields in R4 depending on a real parameter ε which possess for ε=0 a primary family of homoclinic orbits TαH0, α∈S1. Under a transversality condition with respect to ε the existence of homoclinic n-pulse solutions is demonstrated for a sequence of parameter values ε(n)k→0 for k→∞. The existence of cascades of 2l3m-pulse solutions follows by showing their transversality and then using induction. The method relies on the construction of an SO(2)-equivariant Poincaré map which, after factorization, is a composition of two involutions: A logarithmic twist map and a smooth global map. Reversible periodic orbits of this map corresponds to reversible periodic or homoclinic solutions of the original problem. As an application we treat the steady complex Ginzburg–Landau equation for which a primary homoclinic solution is known explicitly.