Applications of Nonlinear Dynamics

数学科学：非线性动力学的应用数学科学：非线性动力学的应用该研究探索与非线性物理系统建模和实验数据分析相关的数学现象。我们考虑了科学家们常用的从实验数据中“重建”非线性物理系统动力学的方法，并研究了重建动力学与真实动力学之间的数学关系。我们还将开发方法来预测时空混沌系统的行为，如地球的天气，给定有限的测量系统的状态。进一步，我们将研究混沌流体流动中出现的分形测度及其在混合和磁场产生问题中的应用。混沌动力学已经在自然界的许多方面被观察到；在天气方面，它被称为“蝴蝶效应”。在某些情况下（限制天气的可预测性），混乱可能是一种障碍，而在其他情况下（如工业混合应用程序），混乱可能是一种资产。我们的研究涉及科学家用来模拟和预测混沌系统的方法背后的数学。我们的目的是检查那些看似有用但未经证实的方法的有效性，并开发新的和改进的方法。长期的好处可能包括更好的天气预报和更有效的化学品混合程序。

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.