Averaging and Hyperbolic Dynamics

许多自然现象涉及几个时间尺度。研究多尺度系统的最有力的工具之一是平均方法，它包括用它们的有效贡献代替快速变量。目前建议的目标是利用动力系统的最新进展，如参数消除方法，Sinai-Ruelle光滑度Bowen测度和部分双曲型系统的极限定理，为写出和严格证明一类系统的平均方程提供了有效的方法。这一建议的智力价值在于更好地理解微分方程的随机性。由于多尺度系统包含快速振荡的变量，它们通常会表现出对初始条件的敏感依赖性，从而提供了最自然的随机机制之一。平均法允许降低问题的维数，从而使其适合于数值研究。这项研究的结果应该是感兴趣的任何领域研究多尺度系统从分子动力学天文学。

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.