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In this paper we classify the centers localized at the origin of coordinates, the cyclicity of their Hopf bifurcation and their isochronicity for the polynomial differential systems in $${\mathbb{R}^2}$$ of degree d that in complex notation z = x + iy can be written as$$ \dot z = (\lambda+i) z + (z \overline{z})^{\frac{d-5}{2}} \left(A z^{4+j} \overline{z}^{1-j} + B z^3 \overline{z}^2 + C z^{2-j} \overline{z}^{3+j}+D \overline{z}^5\right), $$where j is either 0 or 1, d is an arbitrary odd positive integer greater than or equal to five, $${\lambda \in \mathbb{R}}$$, and $${A,B,C,D \in \mathbb{C}}$$. Note that if d = 5 we obtain special families of quintic polynomial differential systems.

Classification of the centers, of their cyclicity and isochronicity for two classes of generalized quintic polynomial differential systems