Stability of Non-Autonomous and Random Linear Systems

9972132 NerurkarThe拟议的研究项目涉及的稳定性和谱行为的研究一族线性微分方程的时间依赖或随机系数。典型的基本系数是时间的“经常性”（特别是准周期性）函数。这包括受迫振荡和（有限维和无限维）量子振荡器与外部（时间相关）准周期和遍历强迫场的例子。许多这样的例子已经数值和理论研究最近。受这些例子的启发，该项目旨在寻找精确的分析结果。证明稳定性结果的标准方法是通过完善的技术，如KAM技术或使用“双曲线”。相反，在这个项目中，我们将主要关注不稳定的行为。我们分析了线性系统的稳定性，通过适当的动力学性质的“斜积流”与之相关联。例如，强迫振荡的不稳定性被捕获在相关的准能量运营商的点谱的情况下，这反过来又反映在遍历行为的某些斜积流所产生的给定的线性系统。 因此，我们的技术是一个混合的拓扑动力学，遍历理论的方法，和控制理论的思想。将期望的扰动视为“具有某些约束的控制”，使我们能够应用控制理论的方法来创造混沌行为。有了这个观点和迄今为止开发的技术，该项目计划研究混沌或不稳定行为的普遍性和普遍性，零李雅普诺夫指数的缺乏，遍历性和物理和工程中产生的特定方程族的近似性等问题。最后，非线性的理论发展的影响，将通过发展有关的时间依赖（或非平稳）的线性化和normal forms.This project investigates long term time evolution，特别是某些类的动力系统的不稳定和混沌行为的结果进行检查。受最近对具有随机或准周期强迫的受迫量子振荡器的理论和数值研究的启发，该项目计划提供潜在的分析原因并精确预测此类系统的长期行为。我们考虑的系统模型的实际和理论兴趣的几种机制，如（a）电子和其他'自旋1/2粒子'受到外部随机磁场的行为，（B）声子传导和光纤中的光的偏振，和（c）通过随机介质，如（准）晶体与不规则分布的杂质的传播。通常，大多数现有的知识都是关于两种极端情况的，即当外部强迫场没有随机性（例如周期性）或尽可能随机（技术上这意味着过程是随机独立的）。该项目将利用遍历理论、拓扑动力学和控制理论等分析工具，研究系统演化的混沌性质与这些随机强迫过程的“相关函数”和“混合率”性质的精确数据之间的关系。该项目将试图发现和确定这些系统的基本特征，这些特征决定了各种类型的不稳定行为，并获得有关数量的定量信息，如作为混沌程度指标的李雅普诺夫指数。我们的调查的一些副产品将有重要的影响，工程学科，如过滤理论和系统与控制工程。

9972132NerurkarThe proposed research project deals with the study of stability and spectral behavior of a family of linear differential equations with time-dependent or random coefficients. Typically the underlying coefficients are `recurrent' (in particular quasi-periodic) functions of time. This includes examples of forced oscillations and (finite as well as infinite dimensional) quantum oscillators with external (time dependent) quasi-periodic and ergodic forcing fields. Many such examples have been numerically and theoretically investigated recently. Motivated by these examples, the project is geared towards finding precise analytical results. The standard method of proving stability results is via well established techniques like the KAM technique or the use of `hyperbolicity'. On the contrary, in this project we shall mainly focus on the unstable behavior. We analyze the stability properties of the linear system via the dynamical properties of suitable `skew-product flows' associated with it. For example, instability of forced oscillations is captured in terms of the absence of point spectrum of the associated quasi-energy operator and this in turn is reflected in the ergodic behavior of certain skew-product flows generated by the given linear system. Consequently our technique is a blend of methods of topological dynamics, ergodic theory, and ideas from control theory. Viewing the desired perturbations as `controls with certain constraints' allow us to apply methods of control theory to create chaotic behavior. With this viewpoint and with the techniques developed thus far, the project plans to examine questions of genericity and prevalence of chaotic or unstable behavior, absence of zero Lyapunov exponents, ergodicity and proximality for specific families of equations arising in physics and engineering. Finally the non-linear implications of the theory developed will be examined by developing results about time-dependent (or non-stationary) linearization and normal forms.This project investigates long term time evolution, particularly the unstable and chaotic behavior of certain classes of dynamical systems. Motivated by the recent theoretical and numerical study of forced quantum oscillators with random or quasi-periodic forcing, this project plans to provide the underlying analytical reasons and precisely predict the long term behavior of such systems. The systems we consider model several mechanisms of practical and theoretical interest such as (a) behavior of electrons and other `spin 1/2 particles' subjected to external random magnetic fields, (b) phonon conduction and polarization of light in optical fibers, and (c) propagation through random media such as (quasi)-crystals with irregularly distributed impurities. Typically most of the existing body of knowledge is about two extreme cases, namely either when the external forcing field has no randomness (e.g. is periodic) or is as random as can be (technically this means the process is stochastically independent). Using the analytical tools of ergodic theory, topological dynamics and control theory, the project will investigate finer details relating the chaotic nature of evolution of the system to the precise data about the nature of `correlation function' and `rate of mixing' of these random forcing processes. The project will attempt to find and identify the underlying features of these systems that determine various types of unstable behavior and obtain quantitative information about quantities such as the Lyapunov exponents which are indicators of the degree of chaos. Some by-products of our investigations will have important implications to engineering disciplines such as filtering theory and to systems and control engineering.