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On the Cauchy problem for higher dimensional Benjamin-Ono and Zakharov-Kuznetsov equations
September
2020, 40(9): 5217-5245.
doi: 10.3934/dcds.2020226
A topological study of planar vector field singularities
Institut de Mathématique de Bourgogne, U.M.R. 5584 de C.N.R.S., Université de Bourgogne-Franche Comté, B.P. 47870, 21078-Dijon Cedex, France |
In this paper one extends results of Bendixson [
Keywords:
Planar vector field singularities,
phase portrait,
sectoral decomposition,
minimal centred curve,
contact index.
Mathematics Subject Classification:
Primary: 34C05; Secondary: 34A26.
Citation:
Robert Roussarie. A topological study of planar vector field singularities. Discrete & Continuous Dynamical Systems - A,
2020, 40
(9)
: 5217-5245.
doi: 10.3934/dcds.2020226
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References:
| [1] |
I. Bendixson,
Sur les courbes définies par des équations différentielles, Acta Math., 24 (1901), 1-88.
doi: 10.1007/BF02403068. |
| [2] |
F. Dumortier,
Singularities of vector fields on the plane, J. Differential Equations, 23 (1977), 53-106.
doi: 10.1016/0022-0396(77)90136-X. |
| [3] |
S. Lefschetz, Differential Equations: Geometric Theory, 2$^nd$ edition, Dover Publications Inc., New York, 1977. |
| [4] |
F. Le Roux, L'ensemble de rotation autour d'un point fixe, Astérisque, 350 (2013), x+109 pp. |
| [5] |
H. Poincaré, Sur les courbes définies par des équations différentielles, Journ. de Math. Pures et Appl., s.3 t.Ⅶ (1881) = [Oeuvres I, 3–44]; (Second part) Id., s. 3, t. Ⅷ (1882) = [Oeuvres I, 44–84]; (Third part) Id., s. 4, t. I (1885) = [Oeuvres I, 90–158]; (Fourth part) Id., s. 4, t. Ⅱ (1886) = [Oeuvres I, 167–222], Oeuvres de Henri Poincaré', Ⅰ-Ⅺ, Gauthier-Villars (new impression 1950–1965). Google Scholar |
show all references
References:
| [1] |
I. Bendixson,
Sur les courbes définies par des équations différentielles, Acta Math., 24 (1901), 1-88.
doi: 10.1007/BF02403068. |
| [2] |
F. Dumortier,
Singularities of vector fields on the plane, J. Differential Equations, 23 (1977), 53-106.
doi: 10.1016/0022-0396(77)90136-X. |
| [3] |
S. Lefschetz, Differential Equations: Geometric Theory, 2$^nd$ edition, Dover Publications Inc., New York, 1977. |
| [4] |
F. Le Roux, L'ensemble de rotation autour d'un point fixe, Astérisque, 350 (2013), x+109 pp. |
| [5] |
H. Poincaré, Sur les courbes définies par des équations différentielles, Journ. de Math. Pures et Appl., s.3 t.Ⅶ (1881) = [Oeuvres I, 3–44]; (Second part) Id., s. 3, t. Ⅷ (1882) = [Oeuvres I, 44–84]; (Third part) Id., s. 4, t. I (1885) = [Oeuvres I, 90–158]; (Fourth part) Id., s. 4, t. Ⅱ (1886) = [Oeuvres I, 167–222], Oeuvres de Henri Poincaré', Ⅰ-Ⅺ, Gauthier-Villars (new impression 1950–1965). Google Scholar |
Figure 2.
A sectoral decomposition
Figure 9.
Closed nodal region
Figure 3.
Contact points
Figure 6.
Elimination of contact points
Figure 10.
Pair of orbits crossing the origin
Figure 12.
A minimal centred annulus
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