  A fast harmonic oscillator is linearly coupled with a system of Ising spins
that are in contact with a thermal bath, and evolve under a slow Glauber
dynamics at dimensionless temperature $\theta$. The spins have a coupling
constant proportional to the oscillator position. The oscillator-spin
interaction produces a second order phase transition at $\theta=1$ with the
oscillator position as its order parameter: the equilibrium position is zero
for $\theta>1$ and non-zero for $\theta< 1$. For $\theta<1$, the dynamics of
this system is quite different from relaxation to equilibrium. For most initial
conditions, the oscillator position performs modulated oscillations about one
of the stable equilibrium positions with a long relaxation time. For random
initial conditions and a sufficiently large spin system, the unstable zero
position of the oscillator is stabilized after a relaxation time proportional
to $\theta$. If the spin system is smaller, the situation is the same until the
oscillator position is close to zero, then it crosses over to a neighborhood of
a stable equilibrium position about which keeps oscillating for an
exponentially long relaxation time. These results of stochastic simulations are
predicted by modulation equations obtained from a multiple scale analysis of
macroscopic equations.
