Phase Portraits of Uniform Isochronous Centers with Homogeneous Nonlinearities

We classify the phase portraits in the Poincare disc of the differential equations of the form $x^{\prime } = -y + x f(x,y)$ , $\dot y =x + y f(x,y)$ 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).