On the number of limit cycles of planar quadratic vector fields (Q1048553)

The paper is devoted to restricted Hilbert's 16th problem for quadratic vector fields which can be transformed to the Kapteyn normal form \[ \dot{x}=\lambda_1x-y-\lambda_3x^2+(2\lambda_2+\lambda_5)xy+\lambda_6y^2,\quad \dot{y}=x+\lambda_1y+\lambda_2x^2+(2\lambda_3+\lambda_4)xy-\lambda_2y^2, \] where \(\lambda_1\in\mathbb{R},\) \(\bar{\lambda}=(\lambda_2,\dots,\lambda_6)\in\mathbb{S}^5.\) The author considers limit cycles surrounding the origin which do not pass the \(\delta\)-neighborhoods of all singular points and that are contained in a disc \(\sqrt{x^2+y^2}\leq1/\delta,\) \(\delta>0\) is arbitrary. The main result estimates the Hilbert number \(H(\delta,\lambda)\) under certain conditions uniformly in \(\lambda\).