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

Isaac A. García; Claudia  Valls

doi:10.3934/dcds.2016.36.2027

Article Contents

2016, Volume 36, Issue 4: 2027-2046. Doi: 10.3934/dcds.2016.36.2027

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

Isaac A. García^1, and
Claudia Valls^2,

1.
Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida
2.
Departamento de Matemática, Instituto Superior Técnico , Universidade Técnica de Lisboa, Av. Rovisco Pais 1049-001, Lisboa

Received: December 31, 2014

Revised: June 30, 2015

Published: May 2017

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Abstract

In this work we extend well-known techniques for solving the Poincaré-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 Poincaré 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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Mathematics Subject Classification: 37G15, 37G10, 34C07.

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References

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