Classification of centers, their cyclicity and isochronicity for a class of polynomial differential systems of degree $d\geq 7$ odd

Abstract

In this paper we classify the centers, the cyclicity of its Hopf bifurcation and the isochronicity of the polynomial differential systems in $\mathbb{R}^2$ of degree $d\geq 7$ odd that in complex notation $z=x+ i y$ can be written as \[ \dot z = (\lambda+i) z + (z \overline z)^{\frac{d-7}2} (A z^6 \overline z + B z^4 \overline z^3 + C z^2 \overline z^5 +D \overline z^7), \] where $\lambda \in \mathbb{R}$, and $A,B,C,D \in \mathbb{C}$.