Geometric Singular Perturbations with Turning Points and Synchronization of Coupled Oscillators

具有转折点的几何奇异扰动和耦合振荡器的同步

• 批准号: 0071931
• 负责人: Weishi Liu
• 金额: $ 7.4万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-15 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071931&HistoricalAwards=false
• 关键词: Geometric; Singular; Perturbations; Turning; Points

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.

NSF 奖项摘要 - DMS-0071931 数学科学：具有转折点的几何奇异扰动和耦合振荡器的同步摘要 0071931 刘这个动力系统领域的项目涉及两个主要研究领域。 首先，它研究具有转折点的奇扰动系统，这是在涉及多尺度物理现象的应用中出现的一类动力系统。转折点的存在，即沿慢变量线性化的一些特征值改变符号，表明慢动态的稳定性损失并使动态行为复杂化。 该项目将几何奇异摄动理论扩展到当前理论无法直接应用的问题类别。 其次，该项目研究非完全相同耦合振荡器系统中的同步现象。 主要重点是扩散耦合和个体耗散对同步及其稳定性的影响。动力系统理论广泛应用于随时间变化的自然系统，从疾病的传播到行星和航天器的运动。 该项目研究动力系统中的两个开放问题，其解决方案将推进理论发展，从而允许预测和控制重要的自然系统。 该项目的结果将对生物学、流体动力学、电路设计和其他领域的多尺度现象的理解产生重大影响。