On the Detection of Limit Cycles by the Variational Velocity Method

This paper summarizes, clarifies, and corrects some aspects of the variational velocity methodfor the detection of limit cycles. After definitions and statements of the most important theoremsassociated with this method, some aspects of the proof of the main theorem are corrected andreworked. An example from the original paper in Acta Appl. Math. 48 (1997),13–32, is then discussed and criticized. Finally, the limitations of this method are discussed,especially as it applies to systems involving multiple limit cycles (and therefore as it applies toHilberts XVIth Problem).