Parallelization of the Lyapunov Constants and Cyclicity for Centers of Planar Polynomial Vector Fields

Christopher in 2006 proved that under some assumptions the linear parts of the Lyapunov constants with respect to the parameters give the cyclicity of an elementary center. This paper is devote to establish a new approach, namely parallelization, to compute the linear parts of the Lyapunov constants. More concretely, it is showed that parallelization computes these linear parts in a shorter quantity of time than other traditional mechanisms. To show the power of this approach, we study the cyclicity of the holomorphic center ż = iz + z + z + · · · + z under general polynomial perturbations of degree n, for n ≤ 13. We also exhibit that, from the point of view of computation, among the Hamiltonian, time-reversible, and Darboux centers, the holomorphic center is the best candidate to obtain high cyclicity examples of any degree. For n = 4, 5, . . . , 13, we prove that the cyclicity of the holomorphic center is at least n +n−2. This result give the highest lower bound for M(6),M(7), . . . ,M(13) among the existing results, where M(n) is the maximum number of limit cycles bifurcating from an elementary monodromic singularity of polynomial systems of degree n. As a direct corollary we also obtain the highest lower bound for the Hilbert numbers H(6) ≥ 40, H(8) ≥ 70, and H(10) ≥ 108, because until now the best result was H(6) ≥ 39, H(8) ≥ 67, and H(10) ≥ 100.