On the topology of the Lorenz system

TL;DR

This paper introduces a novel topological approach to understanding the Lorenz system by using knot theory to analyze its invariant structures and phase transitions, revealing connections to Anosov flows and knot complements.

Contribution

It demonstrates how knot theory can be applied to remove singularities in the Lorenz system and identifies topological phase transitions related to different knot types.

Findings

Existence of a trefoil knot as an invariant curve in the Lorenz system.

Topological equivalence to Anosov flows on knot complements.

Parameter-dependent topological phase transitions with different knots.

Abstract

We present a new paradigm for three dimensional chaos, and specifically for the Lorenz equations. The main difficulty in these equations and for a generic flow in dimension three is the existence of singularities. We show how to use knot theory as a way to remove the singularities. Specifically, we claim: (1) For certain parameters, the Lorenz system has an invariant one dimensional curve, which is a trefoil knot. The knot is a union of invariant manifolds of the singular points. (2) The flow is topologically equivalent to an Anosov flow on the complement of this curve, and even to a geodesic flow. (3) When varying the parameters, the system exhibits topological phase transitions, i.e. for special parameter values, it will be topologically equivalent to an Anosov flow on a knot complement, and different knots appear for different parameter values. The steps of a mathematical proof of these statements are at different stages. Some have been proven, for some we present numerical evidence and some are still conjectural.