On Two-Dimensional Diffeomorphisms with Homoclinic Orbits to Nonhyperbolic Fixed Points

Abstract

We consider two-dimensional diffeomorphisms with homoclinic orbits to nonhyperbolic fixed points. We assume that the point has arbitrary finite order degeneracy and is either of saddle-node or weak saddle type. We consider two cases when the homoclinic orbit is transversal and when a quadratic homoclinic tangency takes place. In the first case we give a complete description of orbits entirely lying in a small neighborhood of the homoclinic orbit. In the second case we study main bifurcations in one-parameter families that split generally the homoclinic tangency but retain the degeneracy type of the fixed point.