Analysis And Computation Of Some Problems About The Simplest Normal Form

The simplest normal form (SNF) is base on conventional normal form (CNF) and nonlinear transformation theory. It plays an important role in the study of the dynamical systems because it can be used to simplify the original differential equations or the classical normal form without changing the dynamical behavior of the original system. SNF's study and application are developing to high-dimension, but the computer of the SNF was very complicated, especially the nonsemi-simple systems related to great matrix operation, at present the study are limited to a kind of singularity system. So the following researchs were developed in order to facilitate the application.The form of SNF: one step transformation method was adopted and the SNF for the singularity of a pure imaginary pair and double zero eigenvalues by comparing the coefficients of similar terms successively based on matrix representation method; according to SNF theory, a new way named complex inner product to compute SNF was introduced. The solution of differential equation was assumed to the form of conjugate complex, then inner product was defined and the SNF of three pure imaginary pair eigenvalues system was obtained by intergraling the ptoduct.The computation of the coefficient of SNF: based on direct method of computing conventional normal form, the relationship between the SNF of semi-simple systems and the nonlinear transformation were obtained. A computer program in MATHEMATICA language was designed to calculate SNF of sime-simple systems, the method can calculate SNF for many kinds of high dimension singularity system without central manifold reduction.