DARBOUX LINEARIZATION AND ISOCHRONOUS CENTERS WITH A RATIONAL FIRST INTEGRAL

In this paper we study isochronous centers of polynomial systems. It i s known that a center is isochronous if and only if it is linearizable . We introduce the notion of Darboux linearizability of a center and g ive an effective criterion for verifying Darboux linearizability. If a center is Darboux linearizable. the method produces a linearizing cha nge of coordinates. Most of the known polynomial isochronous centers a re Darboux linearizable. Moreover, using this criterion we find a new two-parameter family of cubic isochronous centers and give the lineari zing changes of coordinates for centers belonging to that family. We a lso determine all Hamiltonian cubic systems which are Darboux lineariz able. In the second part of this work we restrict to the study of isoc hronous centers having a rational first integral We prove that, under certain conditions, the cycle vanishing at the isochronous center is e ither zero homologous in the closure of a generic fiber, or the functi on obtained from the first integral by eliminating the indeterminacy p oints has several critical points in the singular fiber passing throug h the isochronous center. (C) 1997 Academic Press.