Lattice Differential Equations and the Computation of Stability Spectra

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The investigator and his colleagues consider issues in the approximation of solutions of differential equations. Differential equations are commonly used to model physical and biological phenomena in many areas of science and engineering. A differential equation is a rule, a relationship between the solution and its derivatives, that determines how an initial configuration evolves into future configurations. The focus of this project is on the approximation of Lyapunov exponents and related quantities that provide information on stability, the tendency for nearby configurations to evolve nearby, and instability, the tendency for nearby configurations to move apart, and on the analysis and computation of lattice differential equations, i.e., differential equations that are discrete in space and continuous in time. Much of this work emphasizes the blending of rigorous analysis with practical implementation of algorithms. This research has as a central theme: the combination of dynamical systems and numerical analysis ideas together with the modeling and analysis of differential equations. Discrete models play a prominent role in the modeling of physical and biological systems. Of particular interest are traveling wave solutions of lattice differential equations. Sacker-Sell and Lyapunov spectral intervals are natural analogues of the real parts of the eigenvalues that provide stability information for time-varying differential equations. The investigator develops, analyzes, and justifies the use of numerical techniques for the approximation of these spectral intervals. A suite of computational modules for the computation of stability information and for functional traveling waves is being developed. It is backed by analysis of the numerical techniques in a form that should prove useful to working scientists and engineers.