Center manifold in continuous time systems and computation

Abstract

The objective in this paper is to give some results of bifurcation equations, is concerned with the bifurcation from an equilibrium point in the case when the linear approximation has eigenvalues with zero real parts As we know, there is an intimate relationship between changes of stability and bifurcation. We formulate the main theorems that allow one to reduce dimension of a given system near a local bifurcation. We treat only continuous case. Center manifold theory is a method which uses power series expansions in the neighborhood of an equilibrium point in order to reduce the dimension of a system of ordinary differential equation. We will discuss some aspects of the center manifold. In this paper we will be concerned with the question of how to reduce a system to its center manifold. The calculation of center manifolds involves the manipulation of truncated power series. Coefficients of the quadratic Taylor expansion representing the center manifold can be computed via a recursive procedure, each step of which involves solving a linear system of algebraic equations. We present programs by Maple to accomplish such computations.