A topological study of planar vector field singularities

Robert Roussarie

doi:10.3934/dcds.2020226

Article Contents

2020, Volume 40, Issue 9: 5217-5245. Doi: 10.3934/dcds.2020226

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A topological study of planar vector field singularities

Robert Roussarie

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

Received: July 2019

Revised: March 2020

Published: June 2020

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Abstract

In this paper one extends results of Bendixson [1] and Dumortier [2] about the germs of vector fields at the origin of $ {{I\kern-0.3emR}}^2, $ which is assumed to be an singularity isolated from other singularities and periodic orbits as well. As a new tool, one uses minimal centred curves, which are curves surrounding the origin, with a minimal number of contact points with the vector field. A similar notion was introduced by Le Roux in [4]. It is noticeable that the arguments are essentially topological, with no use of a desingularization theory, as in [2] for instance.

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Mathematics Subject Classification: Primary: 34C05; Secondary: 34A26.

Citation:

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Figure 1. Sectors

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Figure 2. A sectoral decomposition

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Figure 9. Closed nodal region

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Figure 3. Contact points

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Figure 4.

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Figure 5.

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Figure 6. Elimination of contact points

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Figure 7.

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Figure 8.

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Figure 10. Pair of orbits crossing the origin

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Figure 11.

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Figure 12. A minimal centred annulus

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Figure 13.

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Figure 14.

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Figure 15.

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Figure 16.

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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).