Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies

TL;DR

This paper investigates the complex bifurcation structures in 2D reversible maps with symmetric saddle points and homoclinic tangencies, revealing cascades of bifurcations leading to various periodic orbits.

Contribution

It introduces a detailed analysis of bifurcation cascades in reversible maps with symmetric homoclinic tangencies, highlighting the existence of infinitely many bifurcation sequences.

Findings

Existence of infinitely many bifurcation cascades.

Birth of stable, unstable, and elliptic periodic orbits.

Complex mixed dynamics near homoclinic tangencies.

Abstract

We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.