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

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 R of degree d that in complex notation z = x+ iy can be written as ż = (λ+ i)z + (zz) d−5 2 ( Az4+jz1−j +Bzz + Cz2−jz3+j +Dz ) , where j is either 0 or 1, d is an arbitrary odd positive integer greater than or equal to five, λ ∈ R, and A,B,C,D ∈ C. Note that if d = 5 we obtain special families of quintic polynomial differential systems. Mathematics Subject Classification (2000). 34C05, 58F14.