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We consider the averaging principle for deterministic or stochastic systemswith a fast stochastic component (family of continuous-time Markov chainsdepending on the state of the system as a parameter). We show that, due tobifurcations in the simplex of invariant probability measures of the chains,the limiting system should be considered on a graph or on an open book withcertain gluing conditions in the vertices of the graph (or on the bifurcationsurface).

Averaging in the case of multiple invariant measures for the fast system