Averaging principle for fully coupled dynamical systems and large deviations.

*(English)*Zbl 1055.37025For 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.

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.

Reviewer: Eugene Ershov (St. Petersburg)

##### 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 |