Computation of center manifold and normal form and application

求解系统的控制方程是研究非线性动力系统的一个重要问题，它可以用来分析系统的不稳定性和分叉等重要的动力学行为。在实际应用中，经常需要找到包含摄动和/或控制参数的控制方程的解析形式，这样可以直接生成稳定性条件和分岔图。中心流形理论和规范形理论是研究非线性系统分叉和稳定性的两个有用而重要的工具。它们被用来在试图解决问题之前将原始问题转化为更简单的形式。然而，找到一个给定的微分方程组的范式并不是一项简单的任务。在实际应用中，通常需要根据原始非线性系统的系数来计算中心流形和标准形的显式表达式，这是一个非常复杂的过程，必须使用符号计算。特别是，为了证明通常包含大量自由参数的高维动力系统存在多个极限环，中心流形和规范型计算的计算量更大。因此，为了分析高维、非线性问题，发展中心流形和规范形的基本理论和有效的计算方法是必要的。该项目的目标包括发展计算一般非线性动力系统的中心流形和标准型的理论和有效方法，开发用户友好的自动化Maple程序，以及应用于研究生物和工程系统中的多个极限环的分叉。

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.