Collision, explosion and collapse of homoclinic classes

TL;DR

This paper introduces a model demonstrating how two homoclinic classes in dynamical systems can collide, explode, and collapse, revealing complex interactions beyond the typical disjointness in generic cases.

Contribution

It constructs a one-parameter family of diffeomorphisms illustrating the transition from disjoint to intersecting homoclinic classes, including a saddle-node intersection.

Findings

Homoclinic classes can intersect in a saddle-node.

Disjoint classes can merge into a single class.

Transitions depend on a continuous parameter.

Abstract

Homoclinic classes of generic $C^1$-diffeomorphisms are maximal transitive sets and pairwise disjoint. We here present a model explaining how two different homoclinic classes may intersect, failing to be disjoint. For that we construct a one-parameter family of diffeomorphisms $(g_s)_{s\in [-1,1]}$ with hyperbolic points $P$ and $Q$ having nontrivial homoclinic classes, such that, for $s>0$, the classes of $P$ and $Q$ are disjoint, for $s<0$, they are equal, and, for $s=0$, their intersection is a saddle-node.