Bifurcation of degenerate homoclinic orbits in reversible and conservative systems

Abstract

We study the bifurcation of homoclinic orbits from a degenerate homoclinic orbitγ in autonomous systems which are either conservative or reversible. More precisely we consider degenerate homoclinic orbits where along the orbit the tangent spaces to the stable and unstable manifolds of the equilibrium have an intersection which is two-dimensional. In the reversible case we demand thatγ is symmetric and we distinguish betweenγ being elementary and nonelementary. We show that the existence of elementary degenerate homoclinic orbits is a codimension-two phenomenon, while nonelementary homoclinic orbits and degenerate homoclinic orbits in conservative systems appear generically in one-parameter families. We give a complete description of the set of nearby 1-homoclinic orbits.