Stability of Non-Autonomous and Random Linear Systems

9972132 nerurkar1 .这个研究项目是研究一类随时间或随机系数的线性微分方程的稳定性和谱行为。通常，底层系数是时间的“循环”（特别是准周期）函数。这包括强迫振荡和（有限以及无限维）量子振荡与外部（时间相关）准周期和遍历强迫场的例子。最近对许多这样的例子进行了数值和理论上的研究。在这些例子的激励下，该项目旨在找到精确的分析结果。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”。相反，在这个项目中，我们将主要关注不稳定行为。我们通过与之相关的合适的“斜积流”的动力学性质来分析线性系统的稳定性。例如，强迫振荡的不稳定性被捕获为相关准能量算子的点谱的缺失，而这反过来又反映在给定线性系统产生的某些斜积流的遍历行为中。因此，我们的技术是拓扑动力学方法、遍历理论和控制理论思想的混合。将期望的扰动视为“具有特定约束的控制”，允许我们应用控制理论的方法来创建混沌行为。根据这一观点和迄今为止开发的技术，该项目计划检查混乱或不稳定行为的一般性和普遍性问题，零李雅普诺夫指数的缺失，物理和工程中出现的特定方程族的遍历性和邻近性。最后，将通过开发有关时间相关（或非平稳）线性化和范式的结果来检查所开发的理论的非线性含义。本项目研究长期演化，特别是某些动力系统的不稳定和混沌行为。受最近对随机或准周期强迫的强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这种系统的长期行为。我们考虑的系统模拟了几种实际和理论的机制，如(a)电子和其他“自旋1/2粒子”在外部随机磁场作用下的行为，(b)光纤中的声子传导和光的极化，以及(c)通过随机介质（如杂质不规则分布的（准）晶体）的传播。通常，大多数现有的知识体系都是关于两种极端情况，即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）。利用遍历理论、拓扑动力学和控制理论的分析工具，该项目将研究系统进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更精细的细节。该项目将试图找到并确定这些系统的潜在特征，这些特征决定了各种不稳定行为，并获得有关数量的定量信息，如李雅普诺夫指数，它是混乱程度的指标。我们研究的一些副产品将对诸如滤波理论和系统与控制工程等工程学科产生重要影响。

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.