Bifurcation of degenerate homoclinic orbits to saddle-center in reversible systems (Q1044778)

This paper deals with bifurcations from a homoclinic orbit \(\Gamma\) to a saddle-center in a two parameter family of reversible systems in \({\mathbb R}^{2n+2}\). The saddle-center has a two-dimensional center manifold filled with periodic orbits and \(n\)-dimensional stable and unstable manifolds in each case. Further it is assumed that the homoclinic orbit \(\Gamma\) is degenerate, meaning that along \(\Gamma\) the stable and unstable manifolds (of the equilibrium) have an additional common tangent (linearly independent into the direction of the vector field). Using a Liapunov-Schmidt reduction, the nearby 1-homoclinic orbits to the equilibrium are studied.