A COMPARISON OF THE METHODS OF NORMAL FORMS AND AVERAGING
范式理论与平均法的等价性分析

In this paper, a comparative study of the methods of averaging and normal forms is presented. The similarities and differences of these methods are discussed. It is noted that certain basic differences do exist between these methods. The method of normal forms, for example, aims at simplifying the governing differential equations, while the averaging method leads to ordered approximations for solutions as well as simplifying the equations. It is shown in this paper that the methods of normal forms and averaging produce identical results in lower order. However, the results obtained by the methods of normal forms and averaging are apparently different for some higher order. It is noted here that the normal form of a system is not unique and one can always find a near identity transformation that results in identical forms. Thus it is demonstrated in this paper that both methods lead to identical results, including the formal normal form as well as the associated coefficients. The illustrative examples are analyzed to support the conclusions. The analysis is carried out with regard to a two dimensional system with an imaginary pair; however, the results can be extended to higher dimensional systems.