The Hopf cyclicity of the centers of a class of quintic polynomial vector fields

Abstract

We consider families of planar polynomial vector fields having a singularity with purely imaginary eigenvalues for which a basis of its Bautin ideal B is known. We provide an algorithm for computing an upper bound of the Hopf cyclicity less than or equal to the Bautin depth of B. We also present a method for studying the cyclicity problem for the Hamiltonian and the time-reversible centers without the necessity of solving previously the Dulac complex center problem associated to the larger complexified family. As application we analyze the Hopf cyclicity of the quintic polynomial family written in complex notation as z = i z + zz (A z^3 + B z^2 z + C z z2 + D z3.

The first and third authors are partially supported by a MINECO grant number MTM2011-22877 and by a CIRIT grant number 2014 SGR 1204. The second author is partially supported by a MINECO/ FEDER grants numbers MTM2008-03437 and MTM2013-40998-P, an AGAUR grant number 2014SGR 568, ICREA Academia, two FP7- PEOPLE-2012-IRSES numbers 316338 and 318999, and FEDER-UNAB10- 4E-378.