Centers and isochronous centers for two classes of generalized seventh and ninth systems
The authors study centers and isochronous centers for planar polynomial systems of odd degree \(d\geq 7\) that, in complex notation, can be written as \[ \dot z= (\lambda+i)z+(z\bar z)^{(d-7-2j)/2}\left(Az^{5+j}\bar z^{2+j} +B z^{4+j}\bar z^{3+j}+Cz^{3+j}\bar z^{4+j}+D\bar z^{7+2j}\right), \] where \(j\in\{0,1\}\), \(\lambda\) is a real parameter and \(A,B,C\) and \(D\) are complex parameters. Note that it is a family of systems with homogeneous non-linearities. Therefore, by using the Cherkas transformation, it can be transformed into an Abel equation. Both, the expression of the system in polar coordinates and the associated Abel equation, are used along the paper to study the centers and the isochronous centers. All centers found in the family are either Hamiltonian or reversible with respect to a straight line.
scientific article; zbMATH DE number 5833834