Mathematical Sciences: Large Scale Oscillatory Behavior Arising from Nonlinear Systems

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This research involves the following principle: imposing large loading on asymmetric systems results in highly oscillatory behavior. The greater the asymmetry, the larger the number of oscillatory solutions. This research applies this principle in one form or another to nonlinear boundary value problems of elliptic, hyperbolic, and parabolic type as well as with ordinary differential equations and Hamiltonian systems. In the periodic case, there is a considerable body of information which suggests that some of these solutions are asymptotically stable and commonly occurring. This work has led to new insights into nonlinear periodic oscillations in suspended masses, rods and beams, and in floating beams. This research is in the general area of applied mathematics. The project involves a mixture of analytic theory, functional analysis, numerical and computational analysis, and an extension of these results to concrete engineering problems such as the structure of bridges and ships.