Removable dynamics in the Nose-Hoover and Moore-Spiegel Oscillators

TL;DR

This paper investigates the topological dynamics of the Nose-Hoover and Moore-Spiegel oscillators, showing they can be modeled as flows on a solid torus with finite attracting periodic trajectories, and all periodic trajectories form torus knots.

Contribution

It demonstrates that the dynamics of these oscillators can be reduced to a flow on a solid torus, revealing their periodic trajectories are torus knots, a novel topological insight.

Findings

Dynamics reduce to flow on a solid torus

Finite number of attracting periodic trajectories

All periodic trajectories are torus knots

Abstract

We study the dynamics of the Nose-Hoover and Moore-Spiegel Oscillators, and in particular, their topological dynamics. We prove the dynamics of both these systems can be reduced to a flow on a solid torus, with at most a finite number of attracting periodic trajectories. As a consequence, we obtain that every periodic trajectory for the Nose-Hoover and the Moore-Spiegel Oscillators is a Torus knot.