The invariant tori of knot type and the interlinked invariant tori in the Nos\'e-Hoover system

TL;DR

This paper investigates the Nosé-Hoover system, revealing the existence of invariant tori of different knot types and interlinked tori, demonstrating complex dynamic behaviors in this quadratic system.

Contribution

It demonstrates the presence of invariant tori of trefoil and trivial knot types and characterizes interlinked tori with their interlinking numbers in the Nosé-Hoover system.

Findings

Existence of invariant tori of trefoil and trivial knot types.

Presence of interlinked invariant tori with various interlinking numbers.

Rich dynamic properties of the Nosé-Hoover system.

Abstract

We revisit the famous Nos\'e-Hoover system in this paper and show the existence of some averagely conservative regions which are filled with an infinite sequence of nested tori. Depending on initial conditions, some invariant tori are of trefoil knot type, while the others are of trivial knot type. Moreover, we present a variety of interlinked invariant tori whose initial conditions are chosen from different averagely conservative regions and give all the interlinking numbers of those interlinked tori, showing that this quadratic system possesses so rich dynamic properties.