• Previous Article
    Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices
  • DCDS Home
  • This Issue
  • Next Article
    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

Received  July 2019 Revised  March 2020 Published  June 2020

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.

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
References:
[1]

I. Bendixson, Sur les courbes définies par des équations différentielles, Acta Math., 24 (1901), 1-88.  doi: 10.1007/BF02403068.  Google Scholar

[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.  Google Scholar

[3]

S. Lefschetz, Differential Equations: Geometric Theory, 2$^nd$ edition, Dover Publications Inc., New York, 1977.  Google Scholar

[4]

F. Le Roux, L'ensemble de rotation autour d'un point fixe, Astérisque, 350 (2013), x+109 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[3]

S. Lefschetz, Differential Equations: Geometric Theory, 2$^nd$ edition, Dover Publications Inc., New York, 1977.  Google Scholar

[4]

F. Le Roux, L'ensemble de rotation autour d'un point fixe, Astérisque, 350 (2013), x+109 pp.  Google Scholar

[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 1.  Sectors
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
[1]

Tien-Tsan Shieh. From gradient theory of phase transition to a generalized minimal interface problem with a contact energy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2729-2755. doi: 10.3934/dcds.2016.36.2729

[2]

Hebai Chen, Xingwu Chen, Jianhua Xie. Global phase portrait of a degenerate Bogdanov-Takens system with symmetry. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1273-1293. doi: 10.3934/dcdsb.2017062

[3]

Antonio Garijo, Armengol Gasull, Xavier Jarque. Local and global phase portrait of equation $\dot z=f(z)$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 309-329. doi: 10.3934/dcds.2007.17.309

[4]

Yi Shi, Kai Bao, Xiao-Ping Wang. 3D adaptive finite element method for a phase field model for the moving contact line problems. Inverse Problems & Imaging, 2013, 7 (3) : 947-959. doi: 10.3934/ipi.2013.7.947

[5]

Roberto Avanzi, Nicolas Thériault. A filtering method for the hyperelliptic curve index calculus and its analysis. Advances in Mathematics of Communications, 2010, 4 (2) : 189-213. doi: 10.3934/amc.2010.4.189

[6]

Huaiyu Jian, Hongjie Ju, Wei Sun. Traveling fronts of curve flow with external force field. Communications on Pure & Applied Analysis, 2010, 9 (4) : 975-986. doi: 10.3934/cpaa.2010.9.975

[7]

Dawei Yang, Shaobo Gan, Lan Wen. Minimal non-hyperbolicity and index-completeness. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1349-1366. doi: 10.3934/dcds.2009.25.1349

[8]

Haiyan Yin. The stability of contact discontinuity for compressible planar magnetohydrodynamics. Kinetic & Related Models, 2017, 10 (4) : 1235-1253. doi: 10.3934/krm.2017047

[9]

Honghu Liu. Phase transitions of a phase field model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 883-894. doi: 10.3934/dcdsb.2011.16.883

[10]

Alexander Krasnosel'skii, Jean Mawhin. The index at infinity for some vector fields with oscillating nonlinearities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 165-174. doi: 10.3934/dcds.2000.6.165

[11]

Cruz Vargas-De-León, Alberto d'Onofrio. Global stability of infectious disease models with contact rate as a function of prevalence index. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1019-1033. doi: 10.3934/mbe.2017053

[12]

Pavel Krejčí, Elisabetta Rocca, Jürgen Sprekels. Phase separation in a gravity field. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 391-407. doi: 10.3934/dcdss.2011.4.391

[13]

George Dassios, Michalis N. Tsampas. Vector ellipsoidal harmonics and neuronal current decomposition in the brain. Inverse Problems & Imaging, 2009, 3 (2) : 243-257. doi: 10.3934/ipi.2009.3.243

[14]

Jaume Llibre, Ricardo Miranda Martins, Marco Antonio Teixeira. On the birth of minimal sets for perturbed reversible vector fields. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 763-777. doi: 10.3934/dcds.2011.31.763

[15]

Franz W. Kamber and Peter W. Michor. The flow completion of a manifold with vector field. Electronic Research Announcements, 2000, 6: 95-97.

[16]

Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247

[17]

Xinfu Chen, G. Caginalp, Christof Eck. A rapidly converging phase field model. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1017-1034. doi: 10.3934/dcds.2006.15.1017

[18]

A.M. Krasnosel'skii, Jean Mawhin. The index at infinity of some twice degenerate compact vector fields. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 207-216. doi: 10.3934/dcds.1995.1.207

[19]

Xiao-Song Yang. Index sums of isolated singular points of positive vector fields. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1033-1039. doi: 10.3934/dcds.2009.25.1033

[20]

Gianluca Crippa, Milton C. Lopes Filho, Evelyne Miot, Helena J. Nussenzveig Lopes. Flows of vector fields with point singularities and the vortex-wave system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2405-2417. doi: 10.3934/dcds.2016.36.2405

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (71)
  • HTML views (85)
  • Cited by (0)

Other articles
by authors

[Back to Top]