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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
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