Global bifurcation of homoclinic trajectories of discrete dynamical systems

TL;DR

This paper proves the existence of an unbounded connected branch of nontrivial homoclinic trajectories in a family of discrete nonautonomous systems, using topological properties of stable bundles.

Contribution

It introduces a novel topological approach to establish bifurcation of homoclinic trajectories in discrete dynamical systems.

Findings

Existence of an unbounded connected branch of homoclinic trajectories

Application of topological properties of stable bundles

Results for nonautonomous asymptotically hyperbolic systems

Abstract

We prove the existence of an unbounded connected branch of nontrivial homoclinic trajectories of a family of discrete nonautonomous asymptotically hyperbolic systems parametrized by a circle under assumptions involving the topological properties of the asymptotic stable bundles.