多尺度法二义性的一种解释

多尺度法是为解决含小参数系统发展起来的应用最广泛的摄动法之一. 在求解高阶近 似方程时，多尺度法一般只求特解. 用多尺度法求解van der Pol 方程的三阶解时 将出现矛盾. 以van der Pol方程为例，证明了忽略一阶修正量中的一阶谐波 项使得混合偏导数不能交换顺序，从而导致了多尺度法的二义性和另一个数学矛盾. 在求解一阶修正量时采用含有一阶谐波项的全解，消除了二义性和该矛盾. 该 方法所求得的近似解与数值解进行了比较，结果非常吻合，验证了其合理性.

REMARKS ON THE AMBIGUITY IN MULTIPLE SCALES METHOD

The method of multiple scales (MMS), developed for systems with small non-linearities, is one of the most widely used perturbation methods. Only particular solutions are sought for the higher order approximate equations by using the ordinary MMS. An observation is made in this paper that the MMS works well only for the approximate solutions of the first two orders, while gives rise to a paradox in obtaining the third order approximate solution of van der Pol equation. Taking the famous van der Pol equation as an illustrative example, it is proven that neglecting the first order harmonic of the first order approximate solution may make the derivative sequence of the second order mixed partial derivative not commutable. This leads to the ambiguity of the MMS and another mathematical paradox. Unlike the ordinary MMS, the general solution containing the first harmonic is adopted for the first order approximate equation, and then the ambiguity and the paradox are both eliminated. The approximate solutions are obtained by the proposed method and compared with the numerical solutions. It is shown that the present technique is valid.