Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: 37G15, 37G10, 34C07.

 Citation:

•  [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.