Dynamics in Time Dependent Continuous and Discrete Equations and Applications

瞬态连续和离散方程的动力学及其应用

• 批准号: 0103381
• 负责人: Wenxian Shen
• 金额: $ 8万
• 依托单位: Auburn University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-15 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103381&HistoricalAwards=false
• 关键词: Dynamics; Time; Dependent; Continuous; Discrete

NSF Award Abstract - DMS-0103381 Mathematical Sciences: Dynamics in Time Dependent Continuous and Discrete Equations and Applications AbstractDMS-0103381Wenxian ShenThis project investigates three aspects of the dynamics of time dependent (specifically time quasi-periodic and almost periodic) continuous and discrete equations: (i) traveling waves in time-dependent evolution equations; (ii) asymptotic behavior of time-dependent population models, and (iii) oscillatory dynamics for quasi- or almost periodic oscillators and wave equations. In the first area, problems related to traveling wave solutions in time dependent continuous and discrete equations of bistable and Kolmogorov-Petrovskii-Piskunov types are studied. The second research topic is focused primarily on uniform persistence, coexistence and convergence in time dependent multi-species competition models. The third area concerns a general study of oscillatory dynamics in quasi- or almost periodic oscillators and wave equations through investigation of the existence and structure of attractors in general quasi-periodically forced first-order "oscillators" as well as some second order oscillators. The results of the project will enhance understanding of dynamics in time-dependent continuous and discrete equations and will have application to numerous physical and biological problems.It is very important to understand the asymptotic behavior of solutions of nonautonomous differential equations, especially in situations where the nonautonomous part depends on time in a roughly, but not exactly, periodic way. Such equations are widely used as models for processes in biology, chemistry, physics, and engineering. A deep understanding of such almost-periodic or quasi-periodic equations will have a great impact on the development of theory as well as on applications. The objective of the proposed research is to investigate various aspects, of interest in applications, of the solutions to such equations. The results of the project will have significant impact for the analysis of a wide range of mathematical models that are based on these equations.

美国国家科学基金会奖摘要- DMS-0103381数学科学：时变连续和离散方程的动力学及其应用摘要DMS-0103381沈文贤本项目研究时变动力学的三个方面（特别是时间准周期和几乎周期）连续和离散方程：（i）时间相关演化方程中的行波;（ii）含时种群模型的渐近行为;（iii）准周期或概周期振子和波动方程的振荡动力学。 在第一个领域中，我们研究了含时连续和离散的Kolmogorov-Petrovskii-Piskunov型方程的行波解问题。 第二个研究主题主要集中在时间依赖的多种群竞争模型的一致持续生存，共存和收敛。 第三个领域涉及的一般性研究的振荡动力学的准或几乎周期振荡器和波动方程通过调查的存在性和结构的吸引子一般准周期强迫一阶“振荡器”，以及一些二阶振荡器。 该项目的结果将提高对依赖时间的连续和离散方程的动力学的理解，并将应用于许多物理和生物问题。理解非自治微分方程解的渐近行为非常重要，特别是在非自治部分以粗略但不精确的周期方式依赖于时间的情况下。 这些方程被广泛用作生物学、化学、物理学和工程学过程的模型。 深入理解这类准周期或准周期方程，对理论的发展和应用都有重要的意义。 所提出的研究的目的是调查的各个方面，感兴趣的应用，这样的方程的解决方案。 该项目的结果将对基于这些方程的各种数学模型的分析产生重大影响。