Dynamics forced by homoclinic orbits

TL;DR

This paper introduces a pruning theory-based method to identify the core dynamics forced by homoclinic orbits in Smale diffeomorphisms, advancing understanding of complex dynamical systems.

Contribution

It presents a novel method for determining the forced dynamics of homoclinic orbits in 2D Smale diffeomorphisms, with potential generalization to arbitrary Smale maps.

Findings

Method successfully identifies the dynamical core of homoclinic orbits.

Application to infinite families of homoclinic horseshoe orbits demonstrated.

Proposed generalization to broader classes of Smale maps.

Abstract

The complexity of a dynamical system exhibiting a homoclinic orbit is given by the orbits that it forces. In this work we present a method, based in pruning theory, to determine the dynamical core of a homoclinic orbit of a Smale diffeomorphism on the 2-disk. Due to Cantwell and Conlon, this set is uniquely determined in the isotopy class of the orbit, up a topological conjugacy, so it contains the dynamics forced by the homoclinic orbit. Moreover we apply the method for finding the orbits forced by certain infinite families of homoclinic horseshoe orbits and propose its generalization to an arbitrary Smale map.