Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps

TL;DR

This paper investigates the complex bifurcation structures in two-dimensional reversible maps, revealing infinite cascades of stable, unstable, and elliptic periodic orbits arising from heteroclinic tangencies.

Contribution

It demonstrates the existence of infinite bifurcation cascades of various periodic orbits in reversible maps with heteroclinic cycles, expanding understanding of their dynamical complexity.

Findings

Infinite cascades of bifurcations of periodic orbits

Existence of stable, unstable, and elliptic orbits

Unfolding of heteroclinic tangencies in reversible maps

Abstract

We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the initial heteroclinic tangency and prove that there are infinitely sequences (cascades) of bifurcations of birth of asymptotically stable and unstable as well as elliptic periodic orbits.