A Comparison of Different Versions of the Method of Multiple Scales for an Arbitrary Model of Odd Nonlinearities

A general model of cubic and fifth order nonlinearities is considered. The linear part as well as the nonlinearities are expressed in terms of arbitrary operators. Two different versions of the method of multiple scales are used in constructing the general transient and steady-state solutions of the model: Modified Rahman-Burton method and the Reconstitution method. It is found that the usual ordering of reconstitution can be used, if at higher orders of approximation, the time scale corresponding to that order is considered and all other time derivatives are ignored. Results are applied to an example and steady-state solutions are compared numerically for both methods.