Lorenz equations and the figure eight knot

TL;DR

This paper explores the Lorenz equations at a specific parameter point, developing a geometric model that reveals infinitely many periodic orbits, all of which are positive, prime, and fibered knots, linking dynamical systems to knot theory.

Contribution

It introduces a geometric model for Lorenz dynamics at a T-point, demonstrating the existence of infinitely many knot types with specific properties.

Findings

Infinitely many periodic orbits exist at the second T-point.

All these orbits are positive, prime, and fibered knots.

The model links Lorenz dynamics with knot theory properties.

Abstract

Lorenz equations were first presented in 1963 by Edward Lorenz, they depend on three real positive parameters. For some of these parameters which are called T-points, there are two heteroclinic orbits connecting the three singular points in the equations. The heteroclinic connections can be extended into an invariant curve passing through infinity. We consider the system at the second T-point parameter, and develop a geometric model for the flow that simulates the Lorenz dynamics there. We show that the model contains infinitely many periodic orbits, and that as knots they are all positive, prime and fibered.