Abstract:

We show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a  polynomial Liénard equation. The related vector field $X$ is Morse-Smale. Moreover it has the minimum number of singularities required for realizing the configuration in a Liénard equation. We provide an explicit upper bound on the degree of $X$, which is lower than the results obtained before, obtained in the context of general polynomial vector fields.