Geometric Singular Perturbations with Turning Points and Synchronization of Coupled Oscillators

数学科学：具有拐点的几何奇异摄动和耦合振荡器的同步[Abstract] [liu]这个项目在动力系统领域涉及两个主要的研究领域。首先，它研究了具有转折点的奇摄动系统，这是一类发生在涉及多尺度物理现象的应用中的动力系统。在一些特征值沿慢变量变化符号线性化的拐点的存在，表明慢动力的稳定性损失和复杂的动力行为。本项目将几何奇异摄动理论扩展到当前理论不能直接应用的问题类别。其次，研究了非同构耦合振荡器系统中的同步现象。重点是扩散耦合和单个耗散对同步及其稳定性的影响。从疾病的传播到行星和宇宙飞船的运行，动力系统理论可以应用于一系列随时间变化的自然系统。该项目研究了动力系统中的两个开放问题，其解决方案将推进理论，并因此允许预测和控制重要的自然系统。该项目的成果将对生物学、流体动力学、电路设计等领域的多尺度现象的理解产生重大影响。

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.