Invariant tori for a class of singly thermostated hamiltonians

TL;DR

This paper establishes conditions under which invariant KAM tori exist in singly thermostated Hamiltonian systems, demonstrating their presence in key thermostats under certain analytic and temperature conditions, extending prior results.

Contribution

It provides new sufficient conditions for the existence of invariant tori in singly thermostated Hamiltonian systems, applying these to important thermostats and extending previous work.

Findings

Invariant tori exist under weak coupling and high temperature.

Results apply to four key thermostats in the literature.

Extends previous findings by Legoll, Luskin & Moeckel.

Abstract

This paper demonstrates sufficient conditions for the existence of a positive measure set of invariant KAM tori in a singly thermostated, 1 degree-of-freedom hamiltonian vector field. This result is applied to 4 important single thermostats in the literature and it is shown that in each case, if the hamiltonian is real-analytic and well-behaved, then the thermostated system always has a positive measure set of invariant KAM tori for sufficiently weak coupling and high temperature. This extends results of Legoll, Luskin & Moeckel.