The method of multiple scales (Scholarpedia) is a technique used to obtain approximate solutions to differential equations, most commonly when some of the more standard approaches to perturbation theory fail due to the appearance of secular terms, i.e., terms that grow unboundedly (over time, in the case of ordinary differential equations).

While this method seems to be useful and quite popular, all the sources I've seen describe the technique/algorithm and demonstrate its use (and observed accuracy) on various problems, but never mention any theorems guaranteeing accuracy. Are there any such theorems? Is there a good source on this topic in the mathematics literature (perhaps under a different title)?