Invariant tori for multi-dimensional integrable hamiltonians coupled to a single thermostat

TL;DR

This paper establishes conditions under which invariant tori exist in multi-dimensional integrable Hamiltonian systems coupled to a single thermostat, extending previous results to a broader class of thermostats and demonstrating the persistence of these tori under weak coupling.

Contribution

It generalizes the existence of invariant tori in thermostated Hamiltonian systems to variable-mass thermostats of order 2, including Nosé and other models, extending prior theoretical results.

Findings

Invariant tori exist for weakly coupled thermostated Hamiltonians.

Extension of previous theorems to variable-mass thermostats of order 2.

Positive-measure set of invariant tori in typical thermostated systems.

Abstract

This paper demonstrates sufficient conditions for the existence of KAM tori in a singly thermostated, integrable hamiltonian system with $n$ degrees of freedom with a focus on the generalized, variable-mass thermostats of order 2--which include the Nos\'e thermostat, the logistic thermostat of Tapias, Bravetti and Sanders, and the Winkler thermostat. It extends Theorem 3.2 of Legoll, Luskin & Moeckel, (Non-ergodicity of Nos\'e-Hoover dynamics, Nonlinearity, 22 (2009), pp. 1673--1694) to prove that a "typical" singly thermostated, integrable, real-analytic hamiltonian possesses a positive-measure set of invariant tori when the thermostat is weakly coupled. It also demonstrates a class of integrable hamiltonians, which, for a full-measure set of couplings, satisfies the same conclusion.