Mathematical Sciences: Oscillatory Behaviour in Asymmetric Systems with Loading

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This project will emphasize mathematical research on nonlinear differential equations motivated by studies of large loading on asymmetric systems which result in highly oscillatory behavior of solutions. By asymmetric systems one understands those systems in which there is a restoring force which is asymptotically linear along rays issuing from equilibrium to infinity, but not linear at infinity. The oscillatory behavior has been observed in partial differential equations of all types as well as with periodic solutions of ordinary differential equations and Hamiltonian systems. Recent work has led to new insights into nonlinear oscillations in suspension bridges, ships and elevators. One focus of the present work will be to develop a general theory of periodic solutions of systems of ordinary differential equations. Questions of primary interest concern the number of distinct solutions, the stability of solutions and the persistence of solutions under damping. Work is also expected to continue on computational approaches to Hamiltonian systems as well as on parabolic problems with nonlinearities which describe diffusion systems. Applications are expected to arise in areas already mentioned in addition to electronics, in the behavior of nonlinear capacitors. Methods from nonlinear functional analysis will be combined with computational tools and singularity theory in seeking broad general principles in the context of asymmetric systems.