Asymptotic properties of Abelian integrals arising in quadratic systems
Is there a quadratic system with more than four limit cycles? This question is inspired by work on Hilbert's 16th problem, which asks for a bound on the number of limit cycles for planar vector fields of some fixed degree. In the paper under review, the author considers quadratic perturbations of the quadratic system \[ \dot x=-y+a x^2+by^2,\qquad \dot y=x(1+cy), \] which, for \(c\leq 0\), has two centers. The paper is motivated by its author's intuition that there might be a possibility to perturb this system so that two limit cycles appear from one center and three from the other. After a very interesting discussion that includes several theorems about the asymptotic properties of the Abelian integrals that arise naturally in this bifurcation problem, the author draws two conclusions: the unperturbed system is the only remaining unanalyzed case where ``relatively simple methods'' might be used to prove the existence of at least five limit cycles, but the asymptotic analysis of the Abelian integrals provides evidence that the actual upper bound is four. Although the exact upper bound is not obtained, the paper is still well worth reading for deep insights into the bifurcation theory of quadratic systems.
scientific article; zbMATH DE number 1512051