Mathematical Sciences: Multiparameter Bifurcations and Applications

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This project will research bifurcation of multiparameter families of nonlinear differential equations. The primary focus is the analysis of bifurcations to periodic solutions. In the autonomous case the bifurcation of limit cycles from centers and from one parameter families of periodic solutions is considered while in the autonomous case bifurcation of subharmonic solutions is studied. A major goal is to develop an analysis that goes beyond the usual first order methods by being able to treat the response to a nonlinear perturbation to all orders. This will allow for a complete understanding of the bifurcation diagram of the multiparameter system by detecting periodic solutions which are invisible to first order methods and by mixing a priori information on the minimum order to which the perturbation analysis must be carried so as to exhaust the possibility of finding additional perturbed periodic solutions. The methods employed include geometric analysis of nonlinear dynamics, perturbation theory, "Andronov-Melnikov" theory, and algebraic geometric ideal theory. There is also an important computational component especially integration of differential equations, computer graphics and computer algebra. Applications of the theory are found wherever physical systems are modeled by nonlinear ordinary differential equations depending on parameters and, in particular, when determining the possible response of a nonlinear system to external stimuli. A typical application is illustrated in the project by the analysis of a wind oscillation problem.