Quadratic systems with a 3rd-order (or 2nd-order) weak focus (Q2765012)

It is well known that a limit cycle of quadratic systems (QS) surrounds just one critical point which must be of focus type. Furthermore, only two different nests of limit cycles can coexist for QS. It is also known that if one of these singularities is a weak focus of order three (respectively, two) then it has no (respectively, at most one) limit cycles surrounding it.NEWLINENEWLINENEWLINEThe main result of this paper is that in both cases the other singularity that can also be surrounded by limit cycles, which is a strong focus, has at most one limit cycle around it. This result is coherent with the conjecture that asserts that QS have at most four limit cycles and that when this number is attained three of these limit cycles surround a focus and the fourth one surrounds the other one. The proof reduces the problem to prove the uniqueness of limit cycles for a Liénard differential equation of the form \(x''+f(x)x'+g(x)=0.\) This equation is studied by using Filippov's transformation. Nothing is said about the hyperbolicity of the limit cycle surrounding the strong focus. Similar results are presented in a paper of the same author and \textit{S. Zhao} [Appl. Math., Ser. B (Engl. Ed.) 16, No.~2, 127-132 (2001; Zbl 0994.34019)].