Bifurcations of Cubic Homoclinic Tangencies in Two-dimensional Symplectic Maps

¹ Departament de Mathemàtiques i Informàtica, Universitat de Barcelona, Spain
² N.I. Lobachevsky Nizhny Novgorod University, Russia
³ Fachbereich Mathematik und Informatik, Universität Bremen, Germany

^* Corresponding author. E-mail: gonchenko@ub.edu

Abstract

We study bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps. We distinguish two types of cubic homoclinic tangencies, and each type gives different first return maps derived to diverse conservative cubic Hénon maps with quite different bifurcation diagrams. In this way, we establish the structure of bifurcations of periodic orbits in two parameter general unfoldings generalizing to the conservative case the results previously obtained for the dissipative case. We also consider the problem of 1:4 resonance for the conservative cubic Hénon maps.