A contribution to the theory and practice of multiple time scales expansion of nonlinear oscillators

A systematic method for the generation of multiple time scales expansion of the oscillator (d2x/dt2)+ omega 2x= epsilon Sigma n=1N Sigma m=1mgnmxn(dx/dt)m to any order is presented. The excessive freedom which is inherent in the process is conveniently controlled, thus allowing one to generate easily different expansions to the same problem. This option was used to study the extent by which different uses of this freedom can affect the accuracy of the expansion, concluding that the effect may be significant. The new method was applied to the Duffing and the van der Pol oscillators. The complicated algebraic computations involved were accomplished by a computer.