The three-dimensional center problem for the zero-Hopf singularity
MetadataShow full item record
In this work we extend well-known techniques for solving the Poincar\'e-Lyapunov nondegenerate analytic center problem in the plane to the 3-dimensional center problem at the zero-Hopf singularity. Thus we characterize the existence of a neighborhood of the singularity completely foliated by periodic orbits (including continua of equilibria) via an analytic Poincar\'e return map. The vanishing of the first terms in a Taylor expansion of the associated displacement map provides us with the necessary 3-dimensional center conditions in the parameter space of the family whereas the sufficiency is obtained through symmetry-integrability methods. Finally we use the proposed method to classify the 3-dimensional centers of some quadratic polynomial differential families possessing a zero-Hopf singularity.
Is part ofDiscrete and Continuous Dynamical Systems Series A, 2016, vol. 36, núm. 4, p. 2027-2046
European research projects
Showing items related by title, author, creator and subject.
Center problem and ν-cyclicity of polynomial zero-Hopf singularities with non-singular rotation axis García, I. A. (Isaac A.) (Elsevier, 2021-06-02)We consider three-dimensional polynomial families of vector fields parameterized by the admissible coefficients having a fixed zero-Hopf equilibrium and a non-singular rotation axis through it. We are interested in the ...
García, I. A. (Isaac A.); Valls, Claudia (Springer Science+Business Media New York, 2016-11-15)We study the 3-dimensional center problem at the zero-Hopf singularity in some families of polynomial vector fields arising from third-order polynomial differential equations. After proving some general properties we check ...
Period annulus of the harmonic oscillator with zero cyclicity under perturbations with a homogeneous polynomial field García, I. A. (Isaac A.); Maza Sabido, Susanna (Bolyai Institute. University of SzegedHungarian Academy of Sciences, 2019-01-14)In this work we prove, using averaging theory at any order in the small perturbation parameter, that the period annulus of the harmonic oscillator has cyclicity zero (no limit cycles bifurcate) when it is perturbed by any ...