Global phase portraits of uniform isochronous centers with quartic homogeneous polynomial nonlinearities

Abstract

We classify the global phase portraits in the Poincare disc of the differential systems $\dot{x}=-y+xf(x,y),$ $\dot{y}=x+yf(x,y)$, where $f(x,y)$ is a homogeneous polynomial of degree 3. These systems have a uniform isochronous center at the origin. This paper together with the results presented in [9] completes the classification of the global phase portraits in the Poincare disc of all quartic polynomial differential systems with a uniform isochronous center at the origin.