  We present the validity of stochastic averaging principle for non-autonomous
slow-fast stochastic differential equations (SDEs) whose fast motions admit
random periodic solutions. Our investigation is motivated by some problems
arising from multi-scale stochastic dynamical systems, where configurations are
time dependent due to nonlinearity of the underlying vector fields and the
onset of time dependent random invariant sets. Averaging principle with respect
to uniform ergodicity of the fast motion is no longer available in this
scenario. Lyapunov second method together with synchronous coupling and strong
Feller property of Markovian flows of SDEs are used to prove the ergodicity of
time periodic measures of the fast motion on certain minimal Poincare section
and consequently identify the averaging limit.
