高维非线性动力学系统降维方法的若干进展

RECENT DEVELOPMENTS IN DIMENSION REDUCTION METHODS FOR HIGH-DIMENSION DYNAMICAL SYSTEMS

综述近年来非线性动力系统降维理论与方法的研究现状.主要介绍非线性动力系统现有降维方法的基本思想、特点与局限性;这些方法包括:基于中心流形理论的降维方法,Lyapunov-Schmidt(L-S)方法,非线性Galerkin方法和本征正交分解技术(proper orthogonal decomposition,POD)方法;并简单介绍了基于规范形理论和快慢流形动力系统的降维方法.最后提出关于高维非线性动力系统降维的一些新设想,并讨论了今后研究工作的方向.

The current achievements in dimension reduction of nonlinear dynamic systems are reviewed. The basic concepts, features and limitations are elucidated for the existing dimension reduction methods of nonlinear dynamic systems. In addition to the typical dimension reduction methods （such as the model reduction method based on center manifold theorem, the Lyapunov-Schmidt method, the Galerkin method and the method of proper orthogonal decomposition）, the methods in terms of normal form and slow-fast dynamics are briefly presented. Finally, new ideas on the dimension reduction of high-dimensional dynamical systems are proposed, and future research directions are discussed.