范式理论与平均法的等价性分析

分析了范式理论与平均法的等价性。得到的结论是：对含有一对纯虚根的二维非线性系统，使用两种方法得到的结果是等价的，并提供了两个算例来证实其结论的正确性。虽然分析是针对一类二维非线性系统，但其结论同样适合于高维非线性系统。

A COMPARISON OF THE METHODS OF NORMAL FORMS AND AVERAGING

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.