A recursive approach to compute normal forms

Normal forms are instrumental in the analysis of dynamical systems describe d by ordinary differential equations, particularly when singularities close to a bifurcation are to be characterized. However, the computation of a no rmal form up to an arbitrary order is numerically hard. This paper focuses on the computer programming of some recursive formulas developed earlier to compute higher order normal forms. A computer program to reduce the system to its normal form on a center manifold is developed using the Maple symbo lic language. However, it should be stressed that the program relies essent ially on recursive numerical computations, while symbolic calculations are used only for minor tasks. Some strategies are proposed to save computation time. Examples are presented to illustrate the application of the program to obtain high order normalization or to handle systems with large dimensio n. (C) 2001 Academic Press.