FRG Collaborative Research: Approximation of Lyapunov exponents

FRG 协作研究：Lyapunov 指数的近似

• 批准号: 0139874
• 负责人: Michael Jolly
• 金额: $ 15.47万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-15 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0139874&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Approximation; Lyapunov

This Focused Research Group project includes investigatorsfrom three different universities: Luca Dieci at GeorgiaInstitute of Technology, Michal Jolly at Indiana University, andErik Van Vleck at the University of Kansas. The projectconsiders the approximation of Lyapunov exponents and otherspectral information for dynamical systems. The main goal is tostudy and implement numerical techniques to approximate Lyapunovexponents of continuous dynamical systems, as defined by a systemof time dependent differential equations. The investigators areimplementing and comparing so-called continuous and discrete QRand SVD approaches. They distinguish between linear and nonlinearproblems, the chief difference being that in the linear case noapproximation of solution trajectory is attempted. Theinvestigators study Lyapunov exponents for systems of largedimension, such as spatially discretized time dependent PDEs. Inparticular, they consider PDEs for which an inertial manifold isknown to exist, and study the relative merits of techniques thatcompute the exponents after a prior inertial manifold reductionversus those that work with the full (spatially discretized) PDE.Methods that use the Jacobian and Jacobian-free methods arecompared. The investigators also develop general purposealgorithms and software for approximation of Lyapunov exponentsand aim to include the algorithms within standard software fordifferential equations. In many areas of science and engineering, physical andbiological systems are modeled with differential equations. In anutshell, a differential equation is a rule specifying how agiven initial state of the system evolves into future states. Inpractice, we are given the differential equation and the initialstate, and need to find the solution (i.e., the evolution of theinitial condition). Realistic models depend on parameters andthe solution of the differential equation will of course dependon the values of the parameters as well. The ultimate goal ofthis project is to provide scientists with quantitive means ofassessing the dependency of solutions with respect to variationsof the initial state or the parameters in the problem. Lyapunovexponents, and other related "spectral quantities," do exactlythis. A chief effort of the investigators is the development ofalgorithms and computational software for approximation ofLyapunov exponents and other spectra.

这个重点研究小组项目包括来自三所不同大学的研究人员：佐治亚理工学院的 Luca Dieci、印第安纳大学的 Michal Jolly 和堪萨斯大学的 Erik Van Vleck。 该项目考虑了动力系统的李亚普诺夫指数和其他谱信息的近似。 主要目标是研究和实施数值技术来近似连续动力系统的李亚普诺夫指数，如时间相关微分方程组所定义。 研究人员正在实施并比较所谓的连续和离散 QR 和 SVD 方法。 它们区分线性问题和非线性问题，主要区别在于在线性情况下不尝试近似解轨迹。 研究人员研究大维系统的李亚普诺夫指数，例如空间离散的时间相关偏微分方程。 特别是，他们考虑了已知存在惯性流形的偏微分方程，并研究了在先前的惯性流形约简后计算指数的技术与使用完整（空间离散）偏微分方程的技术的相对优点。比较了使用雅可比和无雅可比方法的方法。 研究人员还开发了用于近似李亚普诺夫指数的通用算法和软件，并旨在将这些算法纳入微分方程的标准软件中。 在科学和工程的许多领域，物理和生物系统都是用微分方程建模的。 简而言之，微分方程是指定系统给定初始状态如何演化为未来状态的规则。 在实践中，我们给出了微分方程和初始状态，需要找到解（即初始条件的演化）。 现实模型取决于参数，微分方程的解当然也取决于参数的值。 该项目的最终目标是为科学家提供定量方法来评估解决方案对问题中初始状态或参数变化的依赖性。 李雅普诺夫指数和其他相关的“谱量”正是这样做的。 研究人员的主要工作是开发用于近似李亚普诺夫指数和其他谱的算法和计算软件。