Geometric Singular Perturbations with Turning Points and Synchronization of Coupled Oscillators

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NSF Award Abstract - DMS-0071931Mathematical Sciences: Geometric Singular Perturbations with Turning Points and Synchronization of Coupled OscillatorsAbstract0071931 LiuThis project in the area of dynamical systems concerns two main areas of investigation. First, it investigates singularly perturbed systems with turning points, a class of dynamical systems that occur in applications involving multi-scale physical phenomena. The presence of turning points, where some eigenvalues of linearization along the slow variables change sign, indicates stability loss of the slow dynamic and complicates dynamical behavior. This project extends geometric singular perturbation theory to classes of problems to which current theory cannot be applied directly. Second, the project investigates the phenomenon of synchronization in systems of non-identical coupled oscillators. The main focus is on the effects of diffusive couplings and individual dissipations on synchronizations and their stabilities.The theory of dynamical systems has application to a wide array of natural systems that change in time, from the transmission of disease to the motion of planets and spacecraft. This project investigates two open problems in dynamical systems whose solution will advance the theory and as a result allow prediction and control of important natural systems. Results of the project will have significant impact on the understanding of multi-scale phenomena in biology, fluid dynamics, electrical circuit design, and other areas.