Computation of normal forms for higher dimensional semi-simple systems

Abstract

An efficient method to calculate the normal form and the associated nonlinear transformations for the semi-simple case is given in this paper. The one step transformation concept is adopted to make the approach very easy to be programmed. An intelligent judgement is used to simplify the tedious calculation. This method can be used to calculate high order normal form (without limitation, up to the capacity of the computer) of high dimensional (until dimension 9) ordinary differential equations of the nonlinear oscillators. A program in Mathmatica language is designed to perform the calculation. Six examples are given in order to verify the method and to show the efficiency of the program.