Continuation of Approximate Transformation Groups via Multiple Time Scales Method

Recently, the theory of approximate symmetries was developedfor tackling differential equations with a small parameter. This theoryfurnishes us with a tool, e.g. for constructing approximate groupinvariant solutions. Usually, these solutions are determined by powerseries in the small parameter and hence they are well defined only in asmall region of independent variables. In this paper, we modify theapproximate symmetry analysis by combining it with the multiple timescales method. In this way, we can extend the domain of definition ofapproximate symmetries of differential equations with a small parameterand of their invariant solutions. The method is illustrated by the vander Pol equation. It is shown that, in this example, our approachprovides a group theoretical background of ad hoc methods widelyused in perturbation techniques.