Centers and isochronous centers for generalized quintic systems

In this paper we classify the centers and the isochronous centers of certain polynomial differential systems in R2R2 of degree d≥5d≥5 odd that in complex notation are ż=(λ+i)z+(zz̄)d−52(Az5+Bz4z̄+Cz3z̄2+Dz2z̄3+Ezz̄4+Fz̄5), where z=x+iyz=x+iy, λ∈Rλ∈R and A,B,C,D,E,F∈CA,B,C,D,E,F∈C. Note that if d=5d=5 we obtain the class of polynomial differential systems in the form of a linear system with homogeneous polynomial nonlinearities of degree 5.Due to the huge computations required for computing the necessary and sufficient conditions for the characterization of the centers and isochronous centers, our study uses algorithms of computational algebra based on the Gröbner basis theory and on modular arithmetics.