| For simplifying the analysis of instability and bifurcation of dynamical system, normal form theory is one of the most powerful tools for the study of nonlinear dynamical systems, in particular, of higher order nonlinear dynamical systems. In this dissertation a new recursive formula for computing normal forms based on the adjoint operator method of dynamical systems is first introduced. By solving a series of algebra equations with the aid of Maple software, we can obtain not only the normal form of the dynamical systems and the associated nonlinear transformations but also new higher order terms of dynamical systems after nonlinear transformations. The main contributions of the dissertation are listed as follows.Firstly, the review of normal form theory and methods is demonstrated, then the improved formula for computing normal form is presented based on the adjoint operator method. In this dissertation, a formula is firstly proposed to compute any k order normal form of dynamical system. By solving a series of algebra equations based on the adjoint operator method, we can obtain a new formula for computing normal form of dynamical system. The recursive formula can get not only the coefficients of the jth order(j = 2,...,k) normal forms and the associated nonlinear transformations but also higher (> j) order terms of dynamical system.Secondly, we not only give the associated proof of the new formula for computing normal forms of dynamical system but also apply this method to study the normal forms of two types of dynamical systems: one of dynamical systems has a double zero and a pair of pure imaginary eigenvalues, and the other is 4-dimensional nilpotent dynamical system, which shows the efficiency of the method.Lastly, the Maple program has been developed to compute the normal form and the associated nonlinear transformations of dynamical system. The detailed results of normal forms of original dynamical systems and the associated nonlinear transformations are also given.
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