LINEARIZATION OF ISOCHRONOUS CENTERS

In this paper we study isochronous centers of polynomial systems. We f irst discuss isochronous centers of quadratic systems, cubic symmetric systems and reduced Kukles system. All these systems have rational fi rst integrals. We give a unified proof of the isochronicity of these s ystems, by constructing algebraic linearizing changes of coordinates. We then study two other classes of systems with isochronous centers, n amely the class of ''complex'' systems z over dot = iP(z), and the cla ss of cubic systems symmetric with respect to a line and satisfying th eta over dot = 1. Both classes consist of Darboux integrable systems. We discuss their geometric properties and construct the linearizing ch anges of coordinates. We show that the class of polynomial isochronous systems carries a very rich geometry. Finally, we discuss the geometr y of the linearizing changes of coordinates in the complex plane. (C) 1995 Academic Press, Inc.