Strongly isochronous centers of cubic systems with degenerate infinity (Q483748)

Consider the planar system \[ \dot{x}=-y-P(x,y),\, \dot{y}=x+Q(x,y), \] where \(P\) and \(Q\) are holomorphic in a neighborhood of the origin and with no constant or linear terms. The system is strongly isochronous of order \(n\) if all representation points at \(t_{0}\) on the rays \(I_{\nu}\), \(\nu=1,2,\dots,n\), and starting to move at times \(t_{0}\) along the trajectories of a center or a focus successively pass from one of these rays to another in the same time \(T=\frac{2 \pi}{n}\). The author considers the case in which \(P\) and \(Q\) contain only quadratic and cubic terms. It is shown that this system can have a strongly isochronous center of order two only. All second order strongly isochronous systems are given.