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

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 saddlenode 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.