The cyclicity of period annuli of some classes of reversible quadratic systems (Q862087)

The authors consider small one-parameter analytic quadratic perturbations of the reversible quadratic system with a center \[ \dot{x}=ax^2+by^2-2(b-1)y+(b-2),\;\;\dot{y}=-2xy, \;\;a,b\in\mathbb{R} \] and study the limit cycles bifurcating in the perturbed system. When \(a=-4\), the problem reduces to the elliptic case investigated by \textit{F. Dumortier} and \textit{C. Li} [J. Differ. Equations 176, 114--157 (2001; Zbl 1004.34018); ibid. 188, 512--554 (2003; Zbl 1057.34015)]. For this case, the total number of limit cycles bifurcating from periodic orbits is at most two. When \(a=2\) and \(0<b<2\), the same result is proved by estimating the number of zeros of the related generating function (Melnikov integral) \(M(h)\) taken from [the reviewer, Bull. Sci. Math. 122, 107--161 (1998; Zbl 0920.34037)]. To set up the bound, the authors use the Picard-Fuchs system for the components of \(M(h)\) and the argument principle in order to study the corresponding centroid curve. In addition, for any rational \(a\), the dimension of the Picard-Fuchs system is calculated as a function of \(a\).