Averaging and Hyperbolic Dynamics

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Many natural phenomena involve several time scales. One of the most powerful tools in studying multiscale systems is the method of averaging which consists in replacing the fast variables by their effective contributions. The goal of current proposal is to use recent advances in dynamical system such as parameter elimination method, smoothness of Sinai-Ruelle-Bowen measures and limit theorems for partially hyperbolic systems for providing efficient ways of writing down and rigorously justifying the averaged equations for a broad class of systems.The intellectual merit of this proposal is in providing a better understanding of stochasticity in differential equations.Since multiscale systems contain rapidly oscillating variables they should often exhibit a sensitive dependence on initial conditions providing one of the most natural mechanisms of randomness.The broad impact of the proposed research is in development of the new tools for studying complicated equations. The averaging method allows to decrease the dimension of the problem and henceto make it amenable to numerical investigation. The results of this research should be of interest in any fields studying multiscale systems from molecular dynamics to astronomy.