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Analysis of Slow Manifold of Dynamical Systems Using Several Methods

Rongrong Yu1;Yong Xie2 School of Aerospace;Xi'an Jiaotong University;Xi'an 710049;PR China

  As slow manifolds of dynamical systems contain information about the dynamics of the attractor structure of dynamical systems,it is useful to provide slow manifold equation using several methods in this work.Among these methods for the determination of slow manifold analytical equation,the classical one based on the singular perturbations theory is the so-called singular approximation method.In some special cases,however,it can not be used to determine the analytical equation of slow manifold.Thus,new alternative method of the determination of slow manifold equation,namely,flow curvature method,is proposed from the viewpoint of differential geometry.This new method provides three equivalent manners to determine the slow manifold analytical equation.Moreover,it can be seen that this method generalizes the tangent linear system approximation and includes the so-called geometric singular perturbation theory,and it is more useful to calculate the slow manifold for high-dimensional dynamical systems.In addition,Based on the specific use of acceleration,a new manifold called singular manifold will also be introduced to describe geometrical structure of the attractors.At last,various applications of several approaches to three-dimensional and high-dimensional models will be provided as tutorial examples.……