Darboux Linearization and Isochronous Centers with a Rational First Integral

Abstract

In this paper we study isochronous centers of polynomial systems. It is known that a center is isochronous if and only if it is linearizable. We introduce the notion of Darboux linearizability of a center and give an effective criterion for verifying Darboux linearizability. If a center is Darboux linearizable, the method produces a linearizing change of coordinates. Most of the known polynomial isochronous centers are Darboux linearizable. Moreover, using this criterion we find a new two-parameter family of cubic isochronous centers and give the linearizing changes of coordinates for centers belonging to that family. We also determine all Hamiltonian cubic systems which are Darboux linearizable. In the second part of this work we restrict to the study of isochronous centers having a rational first integral. We prove that, under certain conditions, the cycle vanishing at the isochronous center is either zero homologous in the closure of a generic fiber, or the function obtained from the first integral by eliminating the indeterminacy points has several critical points in the singular fiber passing through the isochronous center.