Averaging and Hyperbolic Dynamics

许多自然现象涉及几个时间尺度。平均法是研究多尺度系统最有力的工具之一，它是用快速变量的有效贡献来代替快速变量。本文的目标是利用动力系统的最新进展，如参数消去方法、Sinai-Ruelle-Bowen测度的平滑性和部分双曲系统的极限定理，为广泛的系统提供有效的平均方程的书写和严格证明方法。这一建议的智力价值在于对微分方程中的随机性提供了更好的理解。由于多尺度系统包含快速振荡的变量，它们应该经常表现出对初始条件的敏感依赖，从而提供最自然的随机性机制之一。所提出的研究的广泛影响在于开发了研究复杂方程的新工具。平均法可以减少问题的维度，从而使其适合于数值研究。这项研究的结果对从分子动力学到天文学的任何研究多尺度系统的领域都有意义。

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.