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. © 1999 Academic Press.