  The Nose-Hoover thermostat is a deterministic dynamical system designed for
computing phase space integrals for the canonical Gibbs distribution. Newton's
equations are modified by coupling an additional reservoir variable to the
physical variables. The correct sampling of the phase space according to the
Gibbs measure is dependent on the Nose-Hoover dynamics being ergodic. Hoover
presented numerical experiments that show the Nose-Hoover dynamics to be
non-ergodic when applied to the harmonic oscillator. In this article, we prove
that the Nose-Hoover thermostat does not give an ergodic dynamics for the
one-dimensional harmonic oscillator when the ``mass'' of the reservoir is
large. Our proof of non-ergodicity uses KAM theory to demonstrate the existence
of invariant tori for the Nose-Hoover dynamical system that separate phase
space into invariant regions.
  We present numerical experiments motivated by our analysis that seem to show
that the dynamics is not ergodic even for a moderate thermostat mass. We also
give numerical experiments of the Nose-Hoover chain with two thermostats
applied to the one-dimensional harmonic oscillator. These experiments seem to
support the non-ergodicity of the dynamics if the masses of the reservoirs are
large enough and are consistent with ergodicity for more moderate masses.
