Topology and Homoclinic Trajectories of Discrete Dynamical Systems

TL;DR

This paper investigates how homoclinic trajectories in discrete nonautonomous systems bifurcate from stationary solutions, depending on the twisting of stable bundles at infinity.

Contribution

It establishes a bifurcation criterion for homoclinic trajectories based on the twisting of asymptotic stable bundles in discrete systems.

Findings

Homoclinic trajectories bifurcate when stable bundles are twisted differently.

Bifurcation depends on the topological twisting of stable bundles.

Results apply to a family of discrete, nonautonomous, asymptotically hyperbolic systems.

Abstract

We show that nontrivial homoclinic trajectories of a family of discrete, nonautonomous, asymptotically hyperbolic systems parametrized by a circle bifurcate from a stationary solution if the asymptotic stable bundles Es(+{\infty}) and Es(-{\infty}) of the linearization at the stationary branch are twisted in different ways.