Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps**Dedicated to the memory of Leonid Pavlovich Shilnikov, a Master whose works strongly influenced the mathematical theory of dynamical systems.

Abstract

We study the dynamics and bifurcations of two-dimensional reversible maps with non-transversal heteroclinic cycles containing symmetric saddle fixed points. We consider one-parameter families of reversible maps unfolding the initial heteroclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations and the birth of asymptotically stable, unstable and elliptic periodic orbits.