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

In this work we study isochronous centers of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fourth degree. We first found necessary conditions for such isochronous center in polar coordinates. Finally we give a proof of the isochronicity of these systems using different methods.

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