On constructing accurate approximations of first integrals for difference equations

In this paper, a perturbation method based on invariance factors and multiple scales will be presented for weakly nonlinear, regularly perturbed systems of ordinary difference equations. Asymptotic approximations of first integrals will be constructed on long iteration-scales, that is, on iteration-scales of order ε - 1 , where ε is a small parameter. It will be shown that all invariance factors have to satisfy a functional equation. To show how this perturbation method works, the method is applied to a Van der Pol equation, and a Rayleigh equation. It will be explicitly shown for the first time in the literature how these multiple scales should be introduced for systems of difference equations to obtain very accurate approximations of first integrals on long iteration-scales.