Bifurcation of homoclinic orbits to a saddle-focus in reversible systems with SO(2)-symmetry

We study reversible, SO(2)-invariant vector fields in WS depending on a real parameter epsilon which possess for epsilon = 0 a primary family of homoclinic orbits TalphaHo, alpha is an element of S-1. Under a transversality condition with respect to epsilon the existence of homoclinic n-pulse solutions is demonstrated for a sequence of parameter values epsilon(k)((n)) --> 0 for k --> infinity. The existence of cascades of 2(l)3(m)-pulse solutions follows by showing their transversality and then using induction. The method relies on the construction of an SO(2)-equivariant Poincare 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. (C) 1999 Academic Press.