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.
Many natural phenomena involve several times scales. This project was devoted to study of slow-fast systems where the fast motion has sensetive dependence on initial conditions. Using the recent advances in the theory of chaotic systems the PI has derived effective equations for the evolution of slow variables. Main reseach findings of this project were the following: --the PI found universal behaviour for a wide class of models of Fermi acceleration mechanism. Namely, he developed a simple way to compute the correct scaling and effective behavior for the energy growth of particles moving in a disordered media. --It was found that transport coefficients for several systems with chaotic microscopic dynamics depend non-smoothly on the parameters involved. The PI developed methods to obtain the optimal regularity of transport coeficients in such problems. The project resulted in 13 publications one of which (Dmitry Dolgopyat, Bassam Fayad Unbounded orbits for semicircular outer billiard, Annales Henri Poincare vol 10 (2009) pp 357-375) won Annales Henri Poincare 2009 distinguished paper award. A discussion this award winning work suitable for general public can be found in the the AMS book What's happening in the Mathematical Sciences, Vol 8 (2010), artcile The ultimate billiard shut by Dana Mackenzie pp. 48-61. The author also mentored 3 graduate students and 2 postdocs. In the Spring semester of 2011 the PI coorganized a thematic program on Dynamics and transport in disordered media at the Fields Institute which among the other things introduced the participants to recent advances related to the current project.
