Centers for generalized quintic polynomial differential systems

Abstract

We classify the centers of polynomial differential systems in $\mathbb {R}^2$ of odd degree $d \ge 5$, in complex notation, as $ \cdot z = i z + (z \overline z)^{({d-5})/{2}} (A z^5 + B z^4 \overline z + C z^3 \overline z^2+ D z^2 \overline z^3+ E z \overline z^4 + F \overline z^5)$, where $A, B, C, D, E, F \in \mathbb {C}$ and either $A=\Re (D)=0$, $A=\Im (D)=0$, $\Re (A)=D=0$ or $\Im (A)=D=0$.