Renormalization group and Lienard systems of differential equations

Abstract

Autonomous Lienard systems, which constitute a huge family of periodic motions, exhibit limit cycle behaviour in certain cases and centres in others. In the literature, the signature for the existence of these two different facets of periodic behaviour has been studied from different geometrical perspectives and not from a general viewpoint. Starting out from general considerations, we show in this work that a certain renormalization scheme is capable of unifying these two different aspects of periodic motion. We show that the renormalization group allows a unified analysis of the limit cycle and centre in a Lienard system of differential equations. While the approach is perturbative, it is possible to make a stronger statement in this regard. Two different classes of Lienard systems have been considered. The analysis provides clear insight into how the frequency gets corrected at different orders of perturbation as one flips the parity of the 'damping' term.