The isochronous centers for Kukles homogeneous system of degree nine

Abstract In this paper, we consider Kukles homogeneous systems x = − y , y = x + Q n ( x , y ) , where Q n ( x , y ) is a homogeneous polynomial of degree n. There are two conjectures on the center-focus problem and isochronous center problem of the above systems. These two conjectures are claimed to be proven in Gine et al. (2015) and Gine et al. (2017). However, the proofs may have some gaps, hence they are still open. In this paper, we consider isochronous centers in the family of Kukles homogeneous systems of degree nine. By using Period Abel constants and Grobner basis of polynomial systems, we obtain that there is no isochronous center with the exception of the linear center.