Remarks on the ambiguity in multiple scales method

The method of multiple scales (MMS), developed forsystems with small non-linearities, is one of the most widely usedperturbation methods. Only particular solutions are sought for the higherorder approximate equations by using the ordinary MMS. An observation ismade inthis paper that the MMS works well only for the approximate solutions ofthe first two orders, while gives rise to a paradox in obtaining the thirdorder approximate solution of van der Pol equation. Taking the famous vander Pol equation as an illustrative example, it is proven that neglectingthe first order harmonic of the first order approximate solution may make the derivative sequence of the second order mixed partial derivativenot commutable. This leads to the ambiguity of the MMS and anothermathematical paradox. Unlike the ordinary MMS, the general solutioncontaining the first harmonic is adopted for the first order approximateequation, and then the ambiguity and the paradox are both eliminated. Theapproximate solutions are obtained by the proposed method and compared withthe numerical solutions. It is shown that the present technique is valid.