Harmonic oscillators in the Nos\'e - Hoover thermostat

We study the dynamics of an ensemble of non-interacting harmonic oscillators in a nonlinear dissipative environment described by the Nos\'e - Hoover model. Using numerical simulation we find the histogram for total energy, which agrees with the analysis of the Nos\'e - Hoover equations effected with the method of averaging. The histogram does not correspond to Gibbs' canonical distribution. We have found oscillations at frequency proportional to $\sqrt{\alpha/m}$, $\alpha$ the dissipative parameter of thermostat and $m$ the characteristic mass of particle, about the stationary state corresponding to equilibrium. The oscillations could have an important bearing upon the analysis of simulating molecular dynamics in the Nos\'e - Hoover thermostat.Comment: 7 pages, 4 figure