  We consider an open (Brownian) classical harmonic oscillator in contact with
a non-Markovian thermal bath and described by the generalized Langevin
equation. When the bath's spectrum has a finite upper cutoff frequency, the
oscillator may have ergodic and nonergodic configurations. In ergodic
configurations (when exist, they correspond to lower oscillator frequencies)
the oscillator demonstrates conventional relaxation to thermal equilibrium with
the bath. In nonergodic configurations (which correspond to higher oscillator
frequencies) the oscillator in general does not thermalize, but relaxes to
periodically correlated (cyclostationary) states whose statistics vary
periodically in time. For a specific dissipation kernel in the Langevin
equation, we evaluate explicitly relevant relaxation functions, which describe
the evolution of mean values and time correlations. When the oscillator
frequency is switched from a lower value to higher one, the oscillator may show
parametric ergodic to nonergodic transitions with equilibrium initial and
cyclostationary final states. These transitions are shown to resemble phase
transitions of the second kind.
