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

García, I. A. (Isaac A.); Llibre, Jaume; Maza Sabido, Susanna

doi:https://doi.org/10.1016/j.jde.2014.11.018

View/Open

Postprint (181.6Kb)

Issue date

2015-01-20

Author

García, I. A. (Isaac A.)

Llibre, Jaume

Maza Sabido, Susanna

Suggested citation

García, I. A. (Isaac A.); Llibre, Jaume; Maza Sabido, Susanna; . (2015) . The Hopf cyclicity of the centers of a class of quintic polynomial vector fields. Journal of Differential Equations, 2015, vol. 258, p. 1990-2009. https://doi.org/10.1016/j.jde.2014.11.018.

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.

URI

http://hdl.handle.net/10459.1/58370

DOI

https://doi.org/10.1016/j.jde.2014.11.018

Is part of

Journal of Differential Equations, 2015, vol. 258, p. 1990-2009

Collections

Except where otherwise noted, this item's license is described as cc-by-nc-nd (c) Elsevier, 2015