On the detection of limit cycles by the variational velocity method

This paper summarizes, clarifies, and corrects some aspects of the variatio nal velocity method for the detection of limit cycles. After definitions an d statements of the most important theorems associated with this method, so me aspects of the proof of the main theorem are corrected and reworked, An example from the original paper in Acta Appl. Math. 48 (1997), 13-32, is th en discussed and criticized. Finally, the limitations of this method are di scussed, especially as it applies to systems involving multiple limit cycle s (and therefore as it applies to Hilbert's XVIth Problem).