Comptes Rendus
no. 4
Dynamical Systems
Unique normal forms for Hopf-zero vector fields
[Formes normales uniques des champs de vecteurs de type Hopf-zéro]

Présenté par : Bernard Malgrange

Guoting Chen1 ; Duo Wang2 ; Jiazhong Yang2
1 UFR de mathématiques, Université de Lille 1, 59655 Villeneuve d'Ascq, France
2 LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 345-348.

Nous étudions l'unicité des formes normales de champs de vecteurs de type Hopf-zéro dans R3. Des formes normales uniques dans le cas générique sont données par rapport aux changements de coordonnées et pour l'équivalence orbitale.

We consider normal forms of Hopf-zero vector fields in R3. Unique normal forms under conjugacy and orbital equivalence for the generic case are given.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00043-8

Guoting Chen 1 ; Duo Wang 2 ; Jiazhong Yang 2

1 UFR de mathématiques, Université de Lille 1, 59655 Villeneuve d'Ascq, France
2 LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China 
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Guoting Chen; Duo Wang; Jiazhong Yang. Unique normal forms for Hopf-zero vector fields. Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 345-348. doi : 10.1016/S1631-073X(03)00043-8. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00043-8/

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  • Majid Gazor; Nasrin Sadri Orbital and parametric normal forms for families of Hopf-zero singularity, Nonlinearity, Volume 37 (2024) no. 10, p. 105018 | DOI:10.1088/1361-6544/ad7662
  • B. W. Qin; K. W. Chung; A. Algaba; A. J. Rodríguez-Luis High-Order Approximation of Heteroclinic Bifurcations in Truncated 2D-Normal Forms for the Generic Cases of Hopf-Zero and Nonresonant Double Hopf Singularities, SIAM Journal on Applied Dynamical Systems, Volume 20 (2021) no. 1, p. 403 | DOI:10.1137/20m1325101
  • Antonio Algaba; Estanislao Gamero; Cristóbal García Orbital Hypernormal Forms, Symmetry, Volume 13 (2021) no. 8, p. 1500 | DOI:10.3390/sym13081500
  • Antonio Algaba; Natalia Fuentes; Estanislao Gamero; Cristóbal García Normal Form for a Class of Three-Dimensional Systems with Free-Divergence Principal Part, Nonlinear Systems, Vol. 1 (2018), p. 37 | DOI:10.1007/978-3-319-66766-9_2
  • Jing Li; Liying Kou; Duo Wang Unique Normal Form for a Class of Three-Dimensional Nilpotent Vector Fields, International Journal of Bifurcation and Chaos, Volume 27 (2017) no. 08, p. 1750131 | DOI:10.1142/s0218127417501310
  • Majid Gazor; Fahimeh Mokhtari Normal forms of Hopf-zero singularity, Nonlinearity, Volume 28 (2015) no. 2, p. 311 | DOI:10.1088/0951-7715/28/2/311
  • Majid Gazor; Fahimeh Mokhtari; Jan A. Sanders Normal forms for Hopf-Zero singularities with nonconservative nonlinear part, Journal of Differential Equations, Volume 254 (2013) no. 3, p. 1571 | DOI:10.1016/j.jde.2012.11.004
  • Majid Gazor; Fahimeh Mokhtari Volume-preserving normal forms of Hopf-zero singularity, Nonlinearity, Volume 26 (2013) no. 10, p. 2809 | DOI:10.1088/0951-7715/26/10/2809
  • JIANPING PENG; DUO WANG A SUFFICIENT CONDITION FOR THE UNIQUENESS OF NORMAL FORMS AND UNIQUE NORMAL FORMS OF GENERALIZED HOPF SINGULARITIES, International Journal of Bifurcation and Chaos, Volume 14 (2004) no. 09, p. 3337 | DOI:10.1142/s0218127404011247

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