Parameter Dependence in Weakly Hyperbolic Dynamical Systems

摘要：DMS-0245359 PI: D DOLGOPYAT， U maryland光滑遍历理论研究了可由有限个数参数描述的自治系统的长时间渐近性。解析和数值结果表明，大多数系统属于以下一类：椭圆型（准周期）和双曲型（对初始条件表现出敏感的依赖性）。著名的Kolmogorov- Arnold- Moser理论描述了接近椭圆系统的系统。相比之下，双曲型系统的小摄动只在少数特殊情况下得到了分析。本文的目的是研究双曲系统的一般摄动理论。这包括对摄动动力学的描述，直到特定的时间，以及计算摄动系统的平衡特性。由于对初始条件有敏感的依赖性，处理单个轨道的摄动是没有希望的。相反，利用双曲系统通常具有强随机特性的事实，我将研究在相空间中访问给定域的频率的参数依赖性。我以前在双曲系统统计性质方面的工作对这个项目至关重要。近年来，人们已经认识到，我们遇到的许多系统都具有混沌行为。另一方面，有几种混沌系统模型可以精确求解，可以进行数学分析。然而，可以问的是，基于这些模型的预测可以信任多少，因为允许我们精确解决这些模型的特定特征可能在某种意义上使它们成为例外。特别是由于混沌系统对初始条件非常敏感，可以想象，系统参数的微小变化可能会极大地改变它们的行为。当前项目的目标是描述当这种剧烈变化没有发生时的情况。其主要思想是，大多数混沌系统都是如此复杂，以至于改变参数并不会导致新的行为模式的出现，而是会略微改变现有模式的概率。

Abstract: DMS-0245359 PI: D DOLGOPYAT, U MarylandSmooth ergodic theory studies the long-time asymptotics of those autonomous systemswhich can be described by a finite number of parameters. Both analytic andnumerical results suggest that most of systems fall in one of the followingclasses: elliptic (quasiperiodic) and hyperbolic (exhibiting sensitive dependence on initial conditions). Systems close to elliptic systemsare described by the famous Kolmogorov--Arnold--Moser theory. By contrastsmall perturbations of hyperbolic systems have been analyzed onlyin a few special cases. This proposal aims at the general perturbationtheory for hyperbolic systems. This includes both the description of theperturbed dynamics up to the certain specific times as well as computing equilibrium characteristics of the perturbedsystem. Because there is sensitive dependence on initial conditions, it is hopeless to deal with perturbations of individualorbits. Rather, using the fact that hyperbolic systems usually havestrong stochastic properties, I will study parameter dependence of frequenciesof visits to a given domain in the phase space. My previous work on statistical properties of hyperbolic systems is essential to this project.During the recent years it has been understood that many systemswe encounter have chaotic behavior. On the other hand thereare several models of chaotic systems which can be exactly solved, which canbe analyzed mathematically. However it can be asked how much onecould trust predictions made on the basis of these models since the specific features which allow us to precisely solve these models might make them in some sense exceptional. In particular since chaotic systems arevery sensitive to initial conditions it is conceivable that a slightchange in the parameters of the system could drastically change theirbehavior. The goal of the current project is to describe the conditionswhen this drastic change does not happen. The main idea is that mostchaotic systems are so complicated that changing parameters does notlead to an appearance of a new pattern of behavior but rather to changing slightly probabilities of already existing patterns.