Bifurcations and distribution of limit cycles which appear from two nests of periodic orbits (Q988132)

Some real planar autonomous differential equations \[ x'=P(x,y),\quad y'=Q(x,y) \] can be written as complex differential equations of the form \[ z'=f(z), \] where \(z=x+i\,y\). This paper deals with the equation \[ z'=iz+z^3. \] Thought as a planar system, it has three critical points of center type, given by the origin and of \(z^\pm=\pm \exp(3\pi i/4)\). Hence, it is natural to study the number of limit cycles that remain when we consider the perturbed differential equation \[ z'=iz+z^3+\varepsilon R_n(z,\bar z), \] where \(\varepsilon\) is real and small enough, and \(R_n\) is a polynomial of degree \(n\) coming from a real polynomial perturbation of the planar system. In fact, this problem has already been studied for the period annulus of the origin [see \textit{A. Gasull, W. Li, J. Llibre} and \textit{Z. Zhang}, Pac. J. Math. 202, No.~2, 341--361 (2002; Zbl 1086.34523)]. The paper under review studies the number of limit cycles that appear simultaneously from the period annuli of the points \(z^\pm\) when the polynomial \(R_m\) has some special shape.