FRG: Collaborative Research: Approximation of Lyapunov Exponents

这个重点研究小组包括来自三所不同大学的研究人员：乔治亚理工学院的卢卡·迪奇、印第安纳大学的米哈尔·乔利和堪萨斯大学的埃里克·范·弗莱克。该方案考虑了动力系统的Lyapunov指数和其他谱信息的近似。主要目的是研究和实现数值技术来逼近由含时微分方程组定义的连续动力系统的李雅普诺夫指数。研究人员正在实施和比较所谓的连续和离散QR和SVD方法。它们区分了线性问题和非线性问题，主要的区别是在线性情况下不尝试对解的轨迹进行近似。研究人员研究了大型系统的Lyapunov指数，例如空间离散的依赖时间的偏微分方程组。特别是，他们考虑了已知存在惯性流形的偏微分方程组，并研究了在先前的惯性流形简化后计算指数的方法与使用全(空间离散的)偏微分方程组的方法的相对优点。研究人员还开发了近似Lyapunov指数的通用算法和软件，目标是将这些算法包括在微分方程组的标准软件中。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式来建模的。简而言之，微分方程是一种规则，规定了系统的初始状态如何演变为未来状态。在实际应用中，我们得到了微分方程和初始条件，需要求解(即初始条件的演化)。现实的模型取决于参数，而微分方程式的解当然也取决于参数值。这个项目的最终目标是为科学家提供定量的手段，以评估解决方案相对于问题中初始状态或参数的变化的依赖性。李雅普诺夫指数和其他相关的“谱量”就是这样做的。研究人员的主要工作是开发算法和计算软件来逼近李亚普诺夫指数和其他光谱。

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.