Cyclicity versus center problem
The authors consider a one-parameter family of differential systems of the form \[ \begin{aligned} \dot x &=-y+ a^k x(x^2+ y^2)+ aP(x,y,a),\\ \dot y &=x+ a^ky(x^2+ y^2)+ aQ(x,y,a),\end{aligned} \] where \(P\) and \(Q\) are analytic functions, starting at least with terms of degree 4 in \(x\) and \(y\), and \(k\geq 1\) is an integer number. It is proved that the cyclicity of the origin is at most \(k- 1\) and there are analytic functions, \(P\) and \(Q\), for which this upper bound is sharp.
scientific article; zbMATH DE number 5826160
scientific article