FRG Collaborative Research: Approximation of Lyapunov exponents

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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.