This project is to investigate the dynamics of nonautonomous and random differential equations arising from a variety of applied problems. In particular, the investigator will explore (1) wave front dynamics in diffusive random and inhomogeneous media arising in phase transitions, nerve pulse propagation, population genetics, and cellular neural networks; (2) spectra and Lyapunov exponents for nonautonomous and random linear parabolic equations, with applications to uniform persistence and coexistence in competitive systems; and (3) global dynamics in stochastically forced oscillators, including Josephson junctions in superconducting circuits. The project involves extension of classical notions in differential equations and dynamical systems as well as the development and implementation of new tools and techniques for the study of the problems in the project. The results of the project will enhance the understanding of the dynamics of the systems under investigation, will provide theoretical and methodological foundations for the further study of these systems and related ones, and will bring closer together three separate, but related, branches of mathematics: differential equations, topological dynamical systems, and metric dynamical systems, enriching each of them.
Realistic physical and biological systems are influenced by variations in the external environment, and are often situated in anisotropic or inhomogeneous media. For this reason, the study of such systems via models involving nonautonomous or random or stochastic differential equations has been gaining more and more attention. Due to a lack of general methodology and difficulties in generalizing classical concepts and notions, there is still little understanding of many of these equations. The investigator will extend relevant classical concepts and notions and develop new tools and techniques to study the dynamics of these equations. The results of the project will provide deep insight into the effects of random external influences and inhomogeneity of media on the dynamics in models of physical and biological problems including phase transitions, nerve pulse propagation, population dynamics, cellular neural networks, pattern formation, and superconducting circuits. The project will provide graduate students training in the area and will create opportunities for students to interact with scientists from other disciplines.
