Recently we learned that the idea of "anti-secularity" in perturbation methods was known to Mathieu already by 1868, predating Lindstedt by several years. The Maple worksheet linked below recapitulates Mathieu's computations:
Nic Fillion and I wrote a more general introduction to perturbation methods using Maple and you can find that paper at
and the supporting Maple code in a workbook at
For instance, one of the problems solved is the lengthening pendulum and when we do so taking proper account of anti-secularity (we use renormalization for that one, I seem to remember) we get an error curve that is bounded over time.
Hope that some of you find this useful.