Bifurcation of limit cycles in a particular class of quadratic systems with two centres
Suppose a differential equation in the plane has an invariant annulus of periodic solutions. A classic problem is to determine if any of these periodic solutions continue to exist after adding a small perturbation. The problem is solved by studying the fixed points of the Poincaré return map, parametrized by the perturbation, and defined on a curve transverse to the period annulus. In particular, the derivatives required to find the fixed points of the resulting parametrized family of maps are computable in terms of certain integrals around the unperturbed periodic orbits. However, in an application of the theory to a specific system, the analysis of the integrals so obtained is usually very difficult. In the paper under review, an analysis of this type is carried out for the perturbations, within the class of quadratic systems, for the period annuli surrounding the centers at the origin and at the point with coordinates \((0,1/n)\) of the quadratic system \[ \dot x= -y+ ny^ 2,\qquad \dot y= x- xy,\qquad 0< n< 1. \] Using clever analytic arguments and some numerical experiments, the authors are able to determine the number of periodic orbits that survive a quadratic perturbation in most cases. Their precise results are too numerous to formulate here. However, they find perturbations with at least two limit cycles. At present, within the class of quadratic systems, it seems that there is no known example of a system with a period annulus such that three of its periodic orbits continue to limit cycles after a perturbation.
scientific article; zbMATH DE number 703251