Parameter Dependence in Weakly Hyperbolic Dynamical Systems

摘要：DMS-0245359 主要研究者：D DOLGOPYAT，U Maryland Smooth遍历理论研究了可由有限个参数描述的自治系统的长时间渐近性。分析和数值结果表明，大多数系统属于下列类别之一：椭圆（准周期）和双曲（表现出敏感的依赖于初始条件）。接近椭圆系统的系统由著名的Kolmogorov-Arnold-Moser理论描述。相比之下，双曲型方程组的小扰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型方程组的一般摄动理论。这既包括描述扰动动力学的某些特定的时间，以及计算的平衡特性的扰动系统。由于对初始条件的敏感依赖性，无法处理单个轨道的摄动。相反，利用双曲系统通常具有强随机性质的事实，我将研究相空间中访问给定域的频率的参数依赖性。我以前关于双曲系统的统计性质的工作对这个项目是必不可少的。在最近几年里，我们已经了解到我们遇到的许多系统都有混沌行为。另一方面，混沌系统有几种模型是可以精确求解的，可以进行数学分析。然而，它可以问有多少一个可以信任的预测的基础上，这些模型，因为特定的功能，使我们能够精确地解决这些模型可能使他们在某种意义上例外.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彻底改变它们的行为.当前项目的目标是描述这种剧烈变化不发生的条件。其主要思想是，大多数混沌系统是如此复杂，改变参数并不会导致一个新的行为模式的出现，而是稍微改变已经存在的模式的概率。

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.