On Two-Dimensional Diffeomorphisms with Homoclinic Orbits to Nonhyperbolic Fixed Points
Received 10 October 2023; accepted 01 December 2023; published 20 December 2023
2024, Vol. 20, no. 1, pp. 151-165
Author(s): Gonchenko S. V., Gordeeva O. V.
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.
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