Limit cycles and Lie symmetries
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Given a planar vector ﬁeld U which generates the Lie symmetry of some other vector ﬁeld X, we prove a new criterion to control the stability of the periodic orbits of U. The problem is linked to a classical problem proposed by A.T. Winfree in the seventies about the existence of isochrons of limit cycles (the question suggested by the study of biological clocks), already answered by Guckenheimer using a diﬀerent terminology. We apply our criterion to give upper bounds of the number of limit cycles for some families of vector ﬁelds as well as to provide a class of vector ﬁelds with a prescribed number of hyperbolic limit cycles. Finally we show how this procedure solves the problem of the hyperbolicity of periodic orbits in problems where other criteria, like the classical one of the divergence, fail.