  The relative stability and ergodicity of deterministic time-reversible
thermostats, both singly and in coupled pairs, are assessed through their
Lyapunov spectra. Five types of thermostat are coupled to one another through a
single Hooke's-Law harmonic spring. The resulting dynamics shows that three
specific thermostat types, Hoover-Holian, Ju-Bulgac, and
Martyna-Klein-Tuckerman, have very similar Lyapunov spectra in their
equilibrium four-dimensional phase spaces and when coupled in equilibrium or
nonequilibrium pairs. All three of these oscillator-based thermostats are shown
to be ergodic, with smooth analytic Gaussian distributions in their extended
phase spaces ( coordinate, momentum, and two control variables ). Evidently
these three ergodic and time-reversible thermostat types are particularly
useful as statistical-mechanical thermometers and thermostats. Each of them
generates Gibbs' universal canonical distribution internally as well as for
systems to which they are coupled. Thus they obey the Zeroth Law of
Thermodynamics, as a good heat bath should. They also provide dissipative heat
flow with relatively small nonlinearity when two or more such bath temperatures
interact and provide useful deterministic replacements for the stochastic
Langevin equation.
