The convexity of closed orbits of Liénard systems

Abstract

It is shown that each closed orbit (if exists) of the Liénard system x˙=F(x)−y, y˙=x is strictly convex under a mild condition on F(x). Specially the unique limit cycle of the Liénard system of the van der Pol equation x¨+μ(x2−1)x˙+x=0 is strictly convex for arbitrary μ>0. Some other examples are also provided.