Normal form computation without central manifold reduction

An efficient method for calculating the normal form and the associated non-linear transformations for the semi-simple case without central manifold reduction is given in this paper. The one-step transformation concept is adopted for easy programming. This method can be used to calculate high order normal forms of high-dimensional ordinary differential equations of non-linear oscillators. A program in MATHEMATICA language is designed to perform the calculation. Three examples are given in order to verify the method and to show the efficiency of the program.