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Averaging principle for fully coupled dynamical systems and large deviations. (English) Zbl 1055.37025
For a long time it is known that in the study of systems which combine slow and fast motions, the averaging principle suggests that a good approximation of the slow motion on long time intervals can be obtained by averaging its parameters over the fast variables. When the slow and fast motions depend on each other (fully coupled dynamical systems), the averaging prescription cannot always be applied, and when it does work, this is usually only in some averaged (with respect to initial conditions) sense.
The paper gives necessary and sufficient conditions for the averaging principle (in the above sense) to hold. It is shown that these conditions can be verified in the case when the fast motions are Axiom A flows for each freezed slow variable. In this case, the Lebesgue measure of initial conditions with bad averaging approximation is proven to tend to zero exponentially fast as the parameter tends to zero.

MSC:
37C10 Dynamics induced by flows and semiflows
37H99 Random dynamical systems
34C29 Averaging method for ordinary differential equations
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
60F10 Large deviations
70K70 Systems with slow and fast motions for nonlinear problems in mechanics
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