Applications of Nonlinear Dynamics

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NSF Award Abstract - DMS-0104087Mathematical Sciences: Applications of Nonlinear DynamicsAbstractDMS-0104087YorkeThis project concerns the theory of chaotic dynamical systems. The research explores mathematical phenomena that are relevant to the modeling of nonlinear physical systems and the analysis of experimental data. We consider the method commonly used by scientists to "reconstruct" the dynamics of a nonlinear physical system from experimental data, and we study the mathematical relationship between the reconstructed dynamics and the true dynamics. We will also develop methods for forecasting the behavior of spatiotemporally chaotic systems, such as the Earth's weather, given limited measurements of the state of the system. Further, we will study fractal measures that arise in chaotic fluid flow and their application to problems of mixing and magnetic field generation. Chaotic dynamics has been observed in many aspects of nature; in weather, it has been called the "butterfly effect." Chaos can be an obstacle in some cases (limiting the predictability of the weather) and an asset in others (such as industrial mixing applications). Our research concerns the mathematics behind the methods scientists use to model and forecast chaotic systems. Our intent is both to examine the validity of apparently useful but unproven methods, and to develop new and improved methods. Long-term benefits may range from better weather forecasts to more efficient procedures for mixing chemicals.