Classification of the centers, their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities

In this paper we classify the centers, the cyclicity of its Hopf bifurcation and their isochronicity for the polynomial differential systems in R2R2 of arbitrary degree d⩾3d⩾3 odd that in complex notation z=x+iyz=x+iy can be written asz˙=(λ+i)z+(zz¯)d−32(Az3+Bz2z¯+Czz¯2+Dz¯3), where λ∈Rλ∈R and A,B,C,D∈CA,B,C,D∈C. If d=3d=3 we obtain the well-known class of all polynomial differential systems of the form a linear system with cubic homogeneous nonlinearities.