The isochronous centers for Kukles homogeneous system of degree nine

Abstract

In this paper, we consider Kukles homogeneous systems ẋ=−y,ẏ=x+Qn(x,y), where Qn(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 Giné et al. (2015) and Giné 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 Gröbner basis of polynomial systems, we obtain that there is no isochronous center with the exception of the linear center.