Essential dynamics in chaotic attractors

TL;DR

This paper proves that complex heteroclinic knots in smooth flows on S^3 lead to infinitely many periodic orbits and introduces a topological framework for understanding chaos in three-dimensional flows, applicable to classical attractors.

Contribution

It establishes the existence of infinitely many periodic orbits from heteroclinic knots and introduces a topological method to analyze chaos in 3D flows, independent of hyperbolicity.

Findings

Heteroclinic knots imply infinite periodic orbits.

A topological template characterizes flow dynamics.

Criteria for chaos in R"ossler and Lorenz systems.

Abstract

We prove that if a smooth vector field $F$ of $S^3$ generates a sufficiently complicated heteroclinic knot, the flow also generates infinitely many periodic orbits, which persist under smooth perturbations which preserve the heteroclinic knot. Consequentially, we then associate a Template with the flow dynamics - regardless of whether $F$ satisfies any hyperbolicity condition or not. In addition, inspired by the Thurston-Nielsen Classification Theorem, we also conclude topological criteria for the existence of chaotic dynamics for three-dimensional flows - which we apply to study both the R\"ossler and Lorenz attractors.