-
Previous Article
KdV waves in atomic chains with nonlocal interactions
- DCDS Home
- This Issue
-
Next Article
A combinatorial proof of the Kontsevich-Zorich-Boissy classification of Rauzy classes
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 |
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. |
[1] |
Victoriano Carmona, Emilio Freire, Soledad Fernández-García. Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 59-72. doi: 10.3934/dcds.2015.35.59 |
[2] |
Jaume Llibre, Ernesto Pérez-Chavela. Zero-Hopf bifurcation for a class of Lorenz-type systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1731-1736. doi: 10.3934/dcdsb.2014.19.1731 |
[3] |
Ming Zhao, Cuiping Li, Jinliang Wang, Zhaosheng Feng. Bifurcation analysis of the three-dimensional Hénon map. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 625-645. doi: 10.3934/dcdss.2017031 |
[4] |
Freddy Dumortier, Santiago Ibáñez, Hiroshi Kokubu, Carles Simó. About the unfolding of a Hopf-zero singularity. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4435-4471. doi: 10.3934/dcds.2013.33.4435 |
[5] |
Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623 |
[6] |
Paolo Perfetti. An infinite-dimensional extension of a Poincaré's result concerning the continuation of periodic orbits. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 401-418. doi: 10.3934/dcds.1997.3.401 |
[7] |
Naoki Hamamoto, Futoshi Takahashi. Sharp Hardy-Leray inequality for three-dimensional solenoidal fields with axisymmetric swirl. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3209-3222. doi: 10.3934/cpaa.2020139 |
[8] |
I. Baldomá, Tere M. Seara. The inner equation for generic analytic unfoldings of the Hopf-zero singularity. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 323-347. doi: 10.3934/dcdsb.2008.10.323 |
[9] |
Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247 |
[10] |
Biao Ou. Examinations on a three-dimensional differentiable vector field that equals its own curl. Communications on Pure and Applied Analysis, 2003, 2 (2) : 251-257. doi: 10.3934/cpaa.2003.2.251 |
[11] |
D. P. Demuner, M. Federson, C. Gutierrez. The Poincaré-Bendixson Theorem on the Klein bottle for continuous vector fields. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 495-509. doi: 10.3934/dcds.2009.25.495 |
[12] |
Anatoli F. Ivanov, Bernhard Lani-Wayda. Periodic solutions for three-dimensional non-monotone cyclic systems with time delays. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 667-692. doi: 10.3934/dcds.2004.11.667 |
[13] |
Jaume Llibre, Marco Antonio Teixeira. Regularization of discontinuous vector fields in dimension three. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 235-241. doi: 10.3934/dcds.1997.3.235 |
[14] |
Armengol Gasull, Víctor Mañosa. Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 651-670. doi: 10.3934/dcdsb.2019259 |
[15] |
Mário Bessa, Jorge Rocha. Three-dimensional conservative star flows are Anosov. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 839-846. doi: 10.3934/dcds.2010.26.839 |
[16] |
Alessandra Celletti, Sara Di Ruzza. Periodic and quasi--periodic orbits of the dissipative standard map. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 151-171. doi: 10.3934/dcdsb.2011.16.151 |
[17] |
C. Alonso-González, M. I. Camacho, F. Cano. Topological invariants for singularities of real vector fields in dimension three. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 823-847. doi: 10.3934/dcds.2008.20.823 |
[18] |
Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807 |
[19] |
Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239 |
[20] |
Chuanxin Zhao, Lin Jiang, Kok Lay Teo. A hybrid chaos firefly algorithm for three-dimensional irregular packing problem. Journal of Industrial and Management Optimization, 2020, 16 (1) : 409-429. doi: 10.3934/jimo.2018160 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]