Homoclinic finger-rings in RN

In this paper we investigate bifurcations of a degenerate homoclinic loop in RN. We prove that a homoclinic finger-ring, an invariant manifold of a definite dimension textured with homoclinic orbits, arises from the degenerate homoclinic orbit. The size of the homoclinic finger-ring is decided by not only its dimension of manifold but also its width. For the rise of homoclinic finger-rings of different dimensions we give conditions, which are proved to form bifurcation manifolds in the parameter space. We further estimate the width for the homoclinic finger-ring and give a method to compute the bifurcation manifolds approximately.