On saddle-loop bifurcations of limit cycles in perturbations of quadratic Hamiltonian systems
This paper studies the bifurcations (of saddle-loop connections) that arise from a perturbation of quadratic Hamiltonian vector fields on the plane. The problem is closely related to the second part of Hilbert's 16th problem concerning the number of limit cycles of polynomial vector fields. Guckenheimer et al. have conjectured that the number of limit cycles of vector fields of the form \(X_ \varepsilon (x,y) = (H_ y + \varepsilon f,-H_ x + \varepsilon g)\) (where \(\varepsilon\) is small and \(X_ 0\) is a quadratic Hamiltonian vector field) does not exceed two and gave numerical evidence for that. In this present paper this conjecture is proved.
quadratic Hamiltonian vector fields on the plane
scientific article; zbMATH DE number 665723