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

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

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

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Issue date

2015-01-20

Author

García, I. A. (Isaac A.)

Llibre, Jaume

Maza Sabido, Susana

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

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