FRG: Collaborative Research: Approximation of Lyapunov Exponents

这个聚焦研究小组的项目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·迪耶西，印第安纳州大学的迈克尔·乔利和堪萨斯大学的埃里克·货车·弗莱克。 该项目考虑了动力系统的李雅普诺夫指数和其他谱信息的近似。 主要目标是研究和实现数值技术来近似连续动力系统的李雅普诺夫指数，定义为一个系统的时间依赖微分方程。 研究人员正在实施和比较所谓的连续和离散QR和SVD方法。 他们区分线性和非线性问题，主要的区别是，在线性情况下没有近似的解决方案的轨迹是企图。 研究者们研究了高维系统的李雅普诺夫指数，如空间离散的时间相关偏微分方程。 特别是，他们认为PDE的惯性流形是已知存在的，并研究的技术，计算指数后，以前的惯性流形reductionversus那些工作与完整的（空间离散）PDE的相对优点。方法，使用雅可比和雅可比自由的方法进行了比较。 研究人员还开发通用算法和软件近似的李雅普诺夫指数和目标，包括标准软件的算法为微分方程。 在科学和工程的许多领域中，物理和生物系统都是用微分方程来建模的. 在一个外壳中，微分方程是一个规则，它规定了系统的给定初始状态如何演变成未来的状态。 在实际中，我们给出了微分方程和初始状态，并需要找到解（即，初始条件的演化）。 实际模型依赖于参数，微分方程的解当然也依赖于参数的值。 该项目的最终目标是为科学家提供定量的方法来评估解决方案对初始状态或问题参数的依赖性。 李雅普诺夫指数和其他相关的“光谱量”，正是这样做的。 研究人员的主要努力是发展近似李雅普诺夫指数和其他谱的算法和计算软件。

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.