Classification of the centers, their cyclicity and isochronicity for the generalized quadratic polynomial differential systems

In this paper we classify the centers, the cyclicity of their Hopf bifurcation and the isochronicity of the polynomial differential systems in R2R2 of degree d   that in complex notation z=x+iyz=x+iy can be written asz˙=(λ+i)z+(zz¯)d−22(Az2+Bzz¯+Cz¯2), where d⩾2d⩾2 is an arbitrary even positive integer, λ∈Rλ∈R and A,B,C∈CA,B,C∈C. Note that if d=2d=2 we obtain the well-known class of quadratic polynomial differential systems which can have a center at the origin.