On the derivation of amplitude equations for nonlinear oscillators

Amplitude equations for different zero--dimensional systems have been
obtained
by using a generalization of the multiple scales method. These systems
include
additive or/and multiplicative forcing terms, being the noise terms special
cases of the performed analysis. Examples of the use of this method are
applied
to the case of the van der Pol--Duffing oscillator. Expressions for the
amplitude equations in terms of a Lyapunov potential allow us to obtain a
theoretical expression for the probability distribution function which
reproduces the simulated results.