April  2016, 36(4): 2027-2046. doi: 10.3934/dcds.2016.36.2027

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

1. 

Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Spain

2. 

Departamento de Matemática, Instituto Superior Técnico , Universidade Técnica de Lisboa, Av. Rovisco Pais 1049-001, Lisboa

Received  January 2015 Revised  July 2015 Published  September 2015

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.
Citation: Isaac A. García, Claudia Valls. The three-dimensional center problem for the zero-Hopf singularity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2027-2046. doi: 10.3934/dcds.2016.36.2027
References:
[1]

N. N. Bautin, On the number of limit cycles which appear with the variation of the coefficients from an equilibrium position of focus or center type,, Math. USSR-Sb, 1954 (1954), 397. Google Scholar

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C. Christopher and C. Li, Limit Cycles of Differential Equations,, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, (2007). Google Scholar

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W. Kapteyn, On the midpoints of integral curves of differential equations of the first degree,, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland (1911), (1911), 1446. Google Scholar

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W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree,, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, 100 (1912), 1354. Google Scholar

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A. M. Liapunov, Problème Général de la Stabilité du Mouvement,, Ann. of Math. Studies 17, (1947). Google Scholar

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J. Llibre, C. Pantazi and S. Walcher, First integrals of local analytic differential systems,, Bull. Sci. Math., 136 (2012), 342. doi: 10.1016/j.bulsci.2011.10.003. Google Scholar

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J. Llibre and C. Valls, Classification of the centers, their cyclicity and isocronicity for the generalized quadratic polynomial differential systems,, J. Math. Anal. Appl., 357 (2009), 427. Google Scholar

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R. Moussu, Symétrie et forme normale des centres et foyers dégénérés,, Ergodic Theory Dynam. Systems, 2 (1982), 241. Google Scholar

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H. Poincaré, Mémoire sur les courbes définies par les équations différentielles,, Oeuvres de Henri Poincaré, (1051), 95. Google Scholar

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H. .Zołądek, Quadratic systems with center and their perturbations,, J. Differential Equations, 109 (1994), 223. doi: 10.1006/jdeq.1994.1049. Google Scholar

show all references

References:
[1]

N. N. Bautin, On the number of limit cycles which appear with the variation of the coefficients from an equilibrium position of focus or center type,, Math. USSR-Sb, 1954 (1954), 397. Google Scholar

[2]

C. Christopher and C. Li, Limit Cycles of Differential Equations,, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, (2007). Google Scholar

[3]

W. Kapteyn, On the midpoints of integral curves of differential equations of the first degree,, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland (1911), (1911), 1446. Google Scholar

[4]

W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree,, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, 100 (1912), 1354. Google Scholar

[5]

A. M. Liapunov, Problème Général de la Stabilité du Mouvement,, Ann. of Math. Studies 17, (1947). Google Scholar

[6]

J. Llibre, C. Pantazi and S. Walcher, First integrals of local analytic differential systems,, Bull. Sci. Math., 136 (2012), 342. doi: 10.1016/j.bulsci.2011.10.003. Google Scholar

[7]

J. Llibre and C. Valls, Classification of the centers, their cyclicity and isocronicity for the generalized quadratic polynomial differential systems,, J. Math. Anal. Appl., 357 (2009), 427. Google Scholar

[8]

R. Moussu, Symétrie et forme normale des centres et foyers dégénérés,, Ergodic Theory Dynam. Systems, 2 (1982), 241. Google Scholar

[9]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles,, Oeuvres de Henri Poincaré, (1051), 95. Google Scholar

[10]

H. .Zołądek, Quadratic systems with center and their perturbations,, J. Differential Equations, 109 (1994), 223. doi: 10.1006/jdeq.1994.1049. Google Scholar

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