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Motivated by applications to mathematical biology, we study the averagingproblem for slow-fast systems, {\em in the case in which the fast dynamics is astochastic process with multiple invariant measures}. We consider both the casein which the fast process is decoupled from the slow process and the case inwhich the two components are fully coupled. We work in the setting in which theslow process evolves according to an Ordinary Differential Equation (ODE) andthe fast process is a continuous time Markov Process with finite state spaceand show that, in this setting, the limiting (averaged) dynamics can bedescribed as a random ODE (that is, an ODE with random coefficients.)  Keywords. Multiscale methods, Processes with multiple equilibria, Averaging,Collective Navigation, Interacting Piecewise Deterministic Markov Processes.

On the study of slow-fast dynamics, when the fast process has multiple invariant measures