Summary: A new perturbation method based on integrating vectors and multiple scales is presented for regularly perturbed systems of ordinary differential equations. Asymptotic approximations to first integrals are constructed on long time-scales, that is, on time-scales of order \(\varepsilon^{-n}\), where \(\varepsilon\) is a small parameter and \(n\geq 1\). In some cases, approximations to first integrals can be obtained which are valid for all times \(t\geq t_0\).
To show how this perturbation method works, the method is applied to the Van der Pol equation, a Mathieu equation, and to an equation for a harmonic oscillator with a cubic damping term.