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

We study reversible, SO(2)-invariant vector fields in R depending on a real parameter = which possess for ==0 a primary family of homoclinic orbits T:H0 , : # S. Under a transversality condition with respect to = the existence of homoclinic n-pulse solutions is demonstrated for a sequence of parameter values = (n) k 0 for k . The existence of cascades of 2 3-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. 1999 Academic Press