A Simple and Efficient Method for Computing Center Manifold and Normal Forms Associated with Semi-simple Cases

Abstract. In this paper we present a simple and efficient method for computing center manifold and normal forms of differential equations. The main attention is focused on general semi-simple singularities, that is, the Jacobian of a given system evaluated at an equilibrium contains semi-simple eigenvalues. The method combines the computation of center manifold and normal form in one unified procedure. The approach improves the existing general formulas and, in particular, leads to developing an efficient algorithm which directly yields the kth-order algebraic equations containing only the kth-order terms. The efficient method greatly saves computation time and reduces computer memory demanding. Symbolic programs using Maple have been developed, and applied to a physical oscillator model to show the efficiency of the new approach.