Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems

Abstract

We classify new classes of centers and of isochronous centers for polynomial differential systems in ${\mathbb R^2}$ of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as

$$\dot z = (\lambda+i) z + (z \overline z)^{\frac{d-7-2j}2} \left(A z^{5+j} \overline z^{2+j} + B z^{4+j} \overline z^{3+j} + C z^{3+j} \overline z^{4+j}+D \overline z^{7+2j} \right),$$

where j is either 0 or ${1, \lambda \in \mathbb R}$ and ${A,B,C \in \mathbb C }$ . Note that if j = 0 and d = 7 we obtain a special case of seventh polynomial differential systems which can have a center at the origin, and if j = 1 and d = 9 we obtain a special case of ninth polynomial differential systems which can have a center at the origin.