Horseshoes for Singly Thermostated Hamiltonians

Leo T Butler

doi:10.1137/20m1315063

Abstract

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 27 January 2020Accepted: 16 June 2020Published online: 13 October 2020Keywordsthermostats, Nosé--Hoover thermostat, Hamiltonian mechanics, transverse homoclinic points, horseshoesAMS Subject Headings37J30, 37A60, 37J46, 82B03Publication DataISSN (online): 1536-0040Publisher: Society for Industrial and Applied MathematicsCODEN: sjaday

Highlights

The development of the theory ofuniformly hyperbolic dynamical systems at roughly the same time showed there is a class of dynamical systems to which the statistical-mechanical formalism is applicable [44, 6]; the centrality of such systems is demonstrated by the so-called chaotic hypothesis which states, in essence, that a chaotic attractor can be regarded as a transitive hyperbolic set [12]
The results in the current paper indicate that the quantities thatdrift"" should be some components of the momenta of the mechanical system and, reciprocally, the energy in the thermostat
In [50], the same authors visit the variant of the Nos\e'--Hoover thermostated harmonic oscillator that thermostats total energy, demonstrating the existence of a horseshoe

Summary

Introduction

Nos\e' [40], based on earlier work of Andersen [2], created a dynamical model of the exchange of energy between heat bath and system. This consists of adding an extra degree of freedom s and rescaling momentum by s:. There are numerous extensions of the Nos\e'--Hoover thermostat that model the exchange of energy with the heat bath using a single, additional thermostat variable. In [50], the same authors visit the variant of the Nos\e'--Hoover thermostated harmonic oscillator that thermostats total energy, demonstrating (numerically) the existence of a horseshoe. Mahdi and Valls [34] show that the Nos\e'--Hoover thermostated harmonic oscillator is not integrable in the class of Darboux integrals, which implies nonintegrability in the class of polynomial integrals, but not necessarily in the class of real-analytic or smooth integrals

Results

Elementary thermostats of order 2

Conclusion