FRG Collaborative Research: Approximation of Lyapunov exponents

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：乔治亚理工学院的卢卡·迪埃奇，印第安纳大学的迈克尔·乔利，堪萨斯大学的安德里克·范·弗莱克。该项目考虑了李雅普诺夫指数的近似和动力系统的其他谱信息。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，由时间相关微分方程系统定义。研究人员正在实施和比较所谓的连续和离散QRand 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。研究者研究了大维系统的李雅普诺夫指数，如空间离散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。比较了使用雅可比矩阵和不使用雅可比矩阵的方法。研究人员还开发了通用算法和李雅普诺夫指数近似软件，并旨在将算法包含在微分方程的标准软件中。在科学和工程的许多领域，物理和生物系统是用微分方程建模的。简而言之，微分方程是规定系统的给定初始状态如何演变成未来状态的规则。在实践中，我们给出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问题中初始状态或参数变化的依赖性。李亚普诺夫指数和其他相关的“谱量”正是这样做的。研究人员的一项主要工作是开发算法和计算软件，以逼近弗莱亚普诺夫指数和其他谱。

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.