Phase Portraits of Uniform Isochronous Centers with Homogeneous Nonlinearities

Abstract

We classify the phase portraits in the Poincaré disc of the differential equations of the form x = −y + xf (x, y), ẏ = x + yf (x, y) where f (x, y) is a homogeneous polynomial of degree n − 1, and f has only simple zeroes when n = 2, 3, 4, 5. We also provide some general results on these uniform isochronous centers for all n ≥ 2.