Method of multiple scales

Summary

The method of multiple scales builds approximations that stay valid over long times for problems with a small parameter. It treats a fast time $t$ and a slow time $T = \varepsilon t$ as independent variables. The slow dependence is chosen to remove secular terms, which would otherwise make the error grow. It handles damping, slow modulation of amplitude and phase, and other effects that the Poincaré–Lindstedt method cannot capture.