Dynamics in Time Dependent Continuous and Discrete Equations and Applications

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NSF Award Abstract - DMS-0103381 Mathematical Sciences: Dynamics in Time Dependent Continuous and Discrete Equations and Applications AbstractDMS-0103381Wenxian ShenThis project investigates three aspects of the dynamics of time dependent (specifically time quasi-periodic and almost periodic) continuous and discrete equations: (i) traveling waves in time-dependent evolution equations; (ii) asymptotic behavior of time-dependent population models, and (iii) oscillatory dynamics for quasi- or almost periodic oscillators and wave equations. In the first area, problems related to traveling wave solutions in time dependent continuous and discrete equations of bistable and Kolmogorov-Petrovskii-Piskunov types are studied. The second research topic is focused primarily on uniform persistence, coexistence and convergence in time dependent multi-species competition models. The third area concerns a general study of oscillatory dynamics in quasi- or almost periodic oscillators and wave equations through investigation of the existence and structure of attractors in general quasi-periodically forced first-order "oscillators" as well as some second order oscillators. The results of the project will enhance understanding of dynamics in time-dependent continuous and discrete equations and will have application to numerous physical and biological problems.It is very important to understand the asymptotic behavior of solutions of nonautonomous differential equations, especially in situations where the nonautonomous part depends on time in a roughly, but not exactly, periodic way. Such equations are widely used as models for processes in biology, chemistry, physics, and engineering. A deep understanding of such almost-periodic or quasi-periodic equations will have a great impact on the development of theory as well as on applications. The objective of the proposed research is to investigate various aspects, of interest in applications, of the solutions to such equations. The results of the project will have significant impact for the analysis of a wide range of mathematical models that are based on these equations.