## Centers for generalized quintic polynomial differential systems.(English)Zbl 1384.34038

The authors consider real planar polynomial differential systems $\dot z= iz+(z\overline z)^{(d-5)/2}(Az^5+ Bz^4\overline z+ Cz^3\overline z^2+ Dz^2\overline z^3+ Ez\overline z^4+ F\overline z^5),\tag{1}$ where $$z= x+iy$$, $$d\geq 5$$ is an arbitrary odd integer and $$A,B,C,D,E,F\in \mathbb{C}$$ satisfy one of the four conditions:
(c.1)
$$A= \text{Re}(D)= 0$$,
(c.2)
$$A=\text{Im}(D)= 0$$,
(c.3)
$$\text{Re}(A)= D= 0$$,
(c.4)
$$\text{Im}(A)= D= 0$$.
Sufficient center conditions of (1) are obtained.

### MSC:

 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34C25 Periodic solutions to ordinary differential equations
Full Text: