FRG: Collaborative Research: Approximation of Lyapunov Exponents

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

This Focused Research Group project includes investigators from three different universities: Luca Dieci at Georgia Institute of Technology, Michal Jolly at Indiana University, and Erik Van Vleck at the University of Kansas. The project considers the approximation of Lyapunov exponents and other spectral information for dynamical systems. The main goal is to study and implement numerical techniques to approximate Lyapunov exponents of continuous dynamical systems, as defined by a system of time dependent differential equations. The investigators are implementing and comparing so-called continuous and discrete QR and SVD approaches. They distinguish between linear and nonlinear problems, the chief difference being that in the linear case no approximation of solution trajectory is attempted. The investigators study Lyapunov exponents for systems of large dimension, such as spatially discretized time dependent PDEs. In particular, they consider PDEs for which an inertial manifold is known to exist, and study the relative merits of techniques that compute the exponents after a prior inertial manifold reduction versus those that work with the full (spatially discretized) PDE. Methods that use the Jacobian and Jacobian-free methods are compared. The investigators also develop general purpose algorithms and software for approximation of Lyapunov exponents and aim to include the algorithms within standard software for differential equations. In many areas of science and engineering, physical and biological systems are modeled with differential equations. In a nutshell, a differential equation is a rule specifying how a given initial state of the system evolves into future states. In practice, we are given the differential equation and the initial state, and need to find the solution (i.e., the evolution of the initial condition). Realistic models depend on parameters and the solution of the differential equation will of course depend on the values of the parameters as well. The ultimate goal of this project is to provide scientists with quantitive means of assessing the dependency of solutions with respect to variations of the initial state or the parameters in the problem. Lyapunov exponents, and other related "spectral quantities," do exactly this. A chief effort of the investigators is the development of algorithms and computational software for approximation of Lyapunov exponents and other spectra.