Time Dependent Stability and Applications
Professor Erik Van Vleck
Department of Mathematics
University of Kansas
Abstract: Dynamical systems of many different types are employed in science and engineering as models of physical and biological phenomena. These dynamical systems include linear and nonlinear mappings, ordinary and partial differential equations, delay equations and other non-local equations. Mathematically the first step in understanding the behavior of solutions is to determine the existence and uniqueness of solutions. This is often followed by analysis of the stability of solutions typically equilibrium or steady state solutions. Stability of solutions determine whether nearby solutions are attracted or repelled. If they repel nearby solutions then it is of interest to understand the types of perturbations associated with such repelling behavior. Spectral analysis of linear operators or matrices provides insight into the stability of time independent solutions. For time varying solutions such as periodic orbits, e.g., via Floquet theory, and for more general time dependent solutions stability analysis depends on understanding time dependent linear operators or matrix functions. Lyapunov exponents as first developed in the thesis of A.M. Lyapunov are among time dependent stability theories that provide information on the stability of time dependent solutions as well as other information (existence of chaotic behavior, dimension of attractors, entropy, etc.). In general stability spectra for time dependent solutions play the role that the real parts of eigenvalues play for linearization about time independent solutions. In this lecture we discuss the theoretical development and application of time dependent stability theory.
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