Linearization of Isochronous Centers

In this paper we study isochronous centers of polynomial systems. We first discuss isochronous centers of quadratic systems, cubic symmetric systems and reduced Kukles system. All these systems have rational first integrals. We give a unified proof of the isochronicity of these systems, by constructing algebraic linearizing changes of coordinates. We then study two other classes of systems with isochronous centers, namely the class of "complex"systems = iP(z), and the class of cubic systems symmetric with respect to a line and satisfying = 1. Both classes consist of Darboux integrable systems. We discuss their geometric properties and construct the linearizing changes of coordinates. We show that the class of polynomial isochronous systems carries a very rich geometry. Finally, we discuss the geometry of the linearizing changes of coordinates in the complex plane