Coupled slow and fast motions generated by ordinary differential equations are examined. The qualitative limit behaviour of the trajectories as the small parameter tends to zero is sought after. Invariant measures of the parametrised fast flow are employed to describe the limit behaviour, rather than algebraic equations which are used in the standard reduced order approach. In the case of a unique invariant measure for each parameter, the limit of the slow motion is governed by a chattering type equation. Without the uniqueness, the limit of the slow motion solves a differential inclusion. The fast flow, in turn, converges in a statistical sense to the direct integral, respectively the set-valued direct integral, of the invariant measures.

We describe the limit behavior of admissible trajectories in a singularly perturbed control system as the small parameter tends to zero. A general case is considered where, in the limit, the fast motion may infinitely rapidly oscillate in time. Invariant measures of the parameterized fast flow are employed to describe the limit behavior and construct the limit control problem. The notion of relatively slow controls is introduced. Approximating properties of the limit problem within the families of relatively slow controls are verified. The results are illustrated by examples.

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