Small-Amplitude Limit Cycles in Liénard Systems via Multiplicity

A recent paper of Christopher and Lloyd reduces the computation of the order of degeneracy of a weak focus for a polynomial Liénard system x=y−F(x), y=−g(x) to the computation of the multiplicity of a polynomial map. In this paper, we first take advantage of that approach to obtain new lower and upper bounds for the maximum order of degeneracy of the origin in terms of the degrees of F′ and g. Later on, we implement an algorithm to compute this maximum order for concrete values of these degrees. As far as we know we enlarge the set of degrees for which this maximum order was known. Finally we extend Christopher and Lloydʹs result to analytic degenerate (or not) Liénard equations