Phase Portraits of Uniform Isochronous Centers with Homogeneous Nonlinearities

We classify the phase portraits in the Poincaré disc of the differential equations of the form x′=−y+xf(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x^{\prime } = -y + x f(x,y)$\end{document}, ẏ=x+yf(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dot y =x + y f(x,y)$\end{document} where f(x,y) is a homogeneous polynomial of degree n − 1 when n = 2,3,4,5, and f has only simple zeroes. We also provide some general results on these uniform isochronous centers for all n ≥ 2. All our results have been revised by the program P4; see Chaps. 9 and 10 of Dumortier et al. (UniversiText, Springer-Verlag, New York, 2006).