  We consider a Brownian oscillator whose coupling to the environment may lead
to the formation of a localized normal mode. For lower values of the
oscillator's natural frequency, $\omega\le\omega_c$, the localized mode is
absent and the unperturbed oscillator reaches thermal equilibrium. For higher
values of $\omega>\omega_c$, when the localized mode is formed, the unperturbed
oscillator does not thermalize but rather evolves into a nonequilibrium
cyclostationary state. We consider the response of such an oscillator to an
external periodic force. Despite the coupling to the environment, the
oscillator shows the unbounded resonance (with the response linearly increasing
with time) when the frequency of the external force coincides with the
frequency of the localized mode. An unusual resonance (``quasi-resonance")
occurs for the oscillator with the critical value of the natural frequency
$\omega=\omega_c$, which separates thermalizing (ergodic) and non-thermalizing
(nonergodic) configurations. In that case the resonance response increases with
time sublinearly, which can be interpreted as a resonance between the external
force and the incipient localized mode.
