The three-dimensional center problem for the zero-Hopf singularity

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Issue date
2016-03-01
Author
García, I. A. (Isaac A.)
Valls, Claudia
Suggested citation
García, I. A. (Isaac A.); Valls, Claudia; . (2016) . The three-dimensional center problem for the zero-Hopf singularity. Discrete and Continuous Dynamical Systems Series A, 2016, vol. 36, núm. 4, p. 2027-2046. https://doi.org/10.3934/dcds.2016.36.2027.
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Abstract
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.
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http://hdl.handle.net/10459.1/58428
DOI
https://doi.org/10.3934/dcds.2016.36.2027
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Discrete and Continuous Dynamical Systems Series A, 2016, vol. 36, núm. 4, p. 2027-2046
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