Bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps

TL;DR

This paper investigates bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps, classifying two types and analyzing their effects on bifurcation diagrams and periodic orbit structures.

Contribution

It introduces a classification of cubic homoclinic tangencies in symplectic maps and derives associated first return maps, extending bifurcation theory to conservative systems.

Findings

Different types lead to distinct bifurcation diagrams

Established bifurcation structures for periodic orbits in two-parameter unfoldings

Analyzed the 1:4 resonance in conservative cubic Hénon maps

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 H\'enon 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 H\'enon maps.