A perturbation method based on integrating vectors and multiple scales

In this paper a new perturbation method based on integrating vectors and multiple scales will be presented for regularly perturbed systems of ordinary differential equations. Asymptotic approximations of first integrals will be constructed on long time-scales, that is, on time-scales of order ε-n, where ε is a small parameter and n≥1. In some cases approximations of first integrals can be obtained which are valid for all times t≥t0. To show how this perturbation method works, the method is applied to the Van der Pol equation, a Mathieu equation, and an equation for a harmonic oscillator with a cubic damping term.