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

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, 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.
Keywords: continua of periodic orbits, Zero-Hopf singularity, Poincaré map., three-dimensional vector fields.
Mathematics Subject Classification: 37G15, 37G10, 34C0.
Citation: Isaac A. García, Claudia Valls. The three-dimensional center problem for the zero-Hopf singularity. Discrete and Continuous Dynamical Systems, 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), 397-413.

[2]

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

[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), 1446-1457 (Dutch).

[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), 1354-1365, 21 27-33 (Dutch).

[5]

A. M. Liapunov, Problème Général de la Stabilité du Mouvement, Ann. of Math. Studies 17, Princeton Univ. Press, 1947.

[6]

J. Llibre, C. Pantazi and S. Walcher, First integrals of local analytic differential systems, Bull. Sci. Math., 136 (2012), 342-359. doi: 10.1016/j.bulsci.2011.10.003.

[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), 427-437.

[8]

R. Moussu, Symétrie et forme normale des centres et foyers dégénérés, Ergodic Theory Dynam. Systems, 2 (1982), 241-251.

[9]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, Oeuvres de Henri Poincaré, Vol. I, Gauthiers-Villars, Paris, 1051, 95-114.

[10]

H. .Zołądek, Quadratic systems with center and their perturbations, J. Differential Equations, 109 (1994), 223-273. doi: 10.1006/jdeq.1994.1049.

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), 397-413.

[2]

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

[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), 1446-1457 (Dutch).

[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), 1354-1365, 21 27-33 (Dutch).

[5]

A. M. Liapunov, Problème Général de la Stabilité du Mouvement, Ann. of Math. Studies 17, Princeton Univ. Press, 1947.

[6]

J. Llibre, C. Pantazi and S. Walcher, First integrals of local analytic differential systems, Bull. Sci. Math., 136 (2012), 342-359. doi: 10.1016/j.bulsci.2011.10.003.

[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), 427-437.

[8]

R. Moussu, Symétrie et forme normale des centres et foyers dégénérés, Ergodic Theory Dynam. Systems, 2 (1982), 241-251.

[9]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, Oeuvres de Henri Poincaré, Vol. I, Gauthiers-Villars, Paris, 1051, 95-114.

[10]

H. .Zołądek, Quadratic systems with center and their perturbations, J. Differential Equations, 109 (1994), 223-273. doi: 10.1006/jdeq.1994.1049.

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