Computation of center manifold and normal form and application

中心流形和范式的计算及应用

• 批准号: 183636-2010
• 负责人: Yu, Pei
• 金额: $ 1.46万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2012
• 资助国家: 加拿大
• 起止时间: 2012-01-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=508704
• 关键词: Computation; center; manifold; normal; form

An important problem encountered in studying nonlinear dynamical systems is to find the governing equation of a system, which can be used to analyze important dynamical behaviors such as instability and bifurcation. In practical applications, one often needs to find analytical form of the governing equations involving perturbation and/or control parameters, which can directly generate stability conditions and bifurcation diagrams. Center manifold theory and normal form theory are two of the useful and important tools in the study of bifurcation and stability of nonlinear systems. They are used to transform the original problem to a simpler form before trying to solve it. However, finding a normal form for a given system of differential equations is not a simple task. For applications, it usually requires computing the explicit expressions of center manifolds and normal forms in terms of the coefficients of the original nonlinear system, which is a very involved procedure and symbolic computation must be employed. In particular, to prove the existence of multiple limit cycles in high dimensional dynamical systems which usually contain a number of free parameters, center manifold and normal form computations are even more computationally demanding. Therefore, development of fundamental theory and efficient computational methods for center manifold and normal form are necessary in order to analyze higher dimensional, nonlinear problems. The objectives of this project include developing theory and efficient method for computing center manifold and normal form of general nonlinear dynamical systems, developing user-friendly automated Maple programs, and application to study bifurcation of multiple limit cycles in biological and engineering systems.

研究非线性动力系统遇到的一个重要问题是找到系统的控制方程，它可以用来分析重要的动力学行为，例如不稳定性和分岔。在实际应用中，常常需要找到涉及扰动和/或控制参数的控制方程的解析形式，从而可以直接生成稳定条件和分岔图。中心流形理论和范式理论是研究非线性系统分岔和稳定性的两个有用且重要的工具。它们用于在尝试解决问题之前将原始问题转换为更简单的形式。然而，找到给定微分方程组的范式并不是一件简单的任务。对于应用来说，通常需要根据原始非线性系统的系数计算中心流形和范式的显式表达式，这是一个非常复杂的过程，并且必须使用符号计算。特别是，为了证明通常包含许多自由参数的高维动力系统中多个极限环的存在，中心流形和范式计算的计算要求更高。因此，为了分析高维非线性问题，有必要发展中心流形和范式的基础理论和有效的计算方法。该项目的目标包括开发用于计算一般非线性动力系统的中心流形和范式的理论和有效方法，开发用户友好的自动化Maple程序，以及应用程序来研究生物和工程系统中多个极限环的分岔。