Center and isochronous center problems for quasi analytic systems (Q944082)

Consider the planar quasi-analytic systems \[ \begin{aligned} {dx\over dt} &=\delta x- y+ \sum^\infty_{k=2} (x^2+ y^2)^{(k-1)(\lambda- 1)/2}X_k(x,y),\\ {dy\over dt} &= x+\delta y+ \sum^\infty_{k=2} (x^2+ y^2)^{(k-1)(\lambda- 1)/2} Y_k(x, y),\end{aligned} \] where \[ \begin{aligned} X_k(x, y) &= \sum_{\alpha+\beta= k} A_{\alpha\beta} x^\alpha y^\beta,\\ Y_k(x, y) &= \sum_{\alpha+\beta= k} B_{\alpha\beta} x^\alpha y^\beta,\end{aligned} \] \(\lambda\) is a real constant and \(\lambda\neq 0\). Two recursive formulas to determine the focal values and period constants are given. Using general results on quasi-quadratic systems, the problem of an isochronous center of the origin is completely solved.