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

View/ Open
Issue date
2015-01-20Suggested 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.
Metadata
Show full item recordAbstract
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.
Is part of
Journal of Differential Equations, 2015, vol. 258, p. 1990-2009European research projects
Related items
Showing items related by title, author, creator and subject.
-
Center cyclicity of a family of quartic polynomial differential system
García, I. A. (Isaac A.); Llibre, Jaume; Maza Sabido, Susanna (Springer, 2016-09-01)In this paper we study the cyclicity of the centers of the quartic polynomial family written in complex notation as z = i z + z z (A z^2 + B z z + C z^2 ), where A,B,C. We give an upper bound for the cyclicity of any ... -
Cyclicity of a simple focus via the vanishing multiplicity of inverse integrating factors
García, I. A. (Isaac A.); Llibre, Jaume; Maza Sabido, Susanna (Elsevier, 2013)First we provide new properties about the vanishing multiplicity of the inverse integrating factor of a planar analytic differential system at a focus. After we use this vanishing multiplicity for studying the cyclicity ... -
Center cyclicity of Lorenz, Chen and Lü systems
García, I. A. (Isaac A.); Maza Sabido, Susanna; Shafer, Douglas S. (Elsevier, 2018-11-09)This work provides upper bounds on the cyclicity of the centers on center manifolds in the well-known Lorenz family, and also in the Chen and Lü families. We prove that at most one limit cycle can be made to bifurcate from ...