Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials.

*(English)*Zbl 0978.34028The authors study plane time-reversible differential systems of the type $\dot{x}=-y+{X}_{5}(x,y)$, $\dot{y}=x+{Y}_{5}(x,y)$, where ${X}_{5}$, ${Y}_{5}$ are homogeneous polynomials of degree 5. They look for necessary and sufficient conditions for the origin to be an isochronous center. They find six classes of systems for which the origin is an isochronous center. For a seventh class it is proved that some necessary conditions for isochronicity hold.

The final section is devoted to prove the existence of nontrivial, nonreversible isochronous centers of systems of the type $\dot{x}=-y+{X}_{5}(x,y)$, $\dot{y}=x+{Y}_{5}(x,y)$.

Reviewer: Marco Sabatini (Trento)