Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomial

In this work we study isochronous centers of two-dimensional autonomous sys tem in the plane with linear part of center type and non-linear pan, given by homogeneous polynomials of fourth degree. We first found necessary condi tions for such isochronous center in polar coordinates. Finally we give a p roof of the isochronicity of these systems using different methods. (C) Els evier, Paris.