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 rea l parameter epsilon which possess for epsilon = 0 a primary family of homoc linic orbits TalphaHo, alpha is an element of S-1. Under a transversality c ondition with respect to epsilon the existence of homoclinic n-pulse soluti ons 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 meth od relies on the construction of an SO(2)-equivariant Poincare map which, a fter factorization, is a composition of two involutions: A logarithmic twis t map and a smooth global map. Reversible periodic orbits of this map corre sponds to reversible periodic or homoclinic solutions of the original probl em. As an application we treat the steady complex Ginzburg-Landau equation for which a primary homoclinic solution is known explicitly. (C) 1999 Acade mic Press.