Japanese Journal of Mathematics

, Volume 4, Issue 1, pp 47–61 | Cite as

Right-handed vector fields & the Lorenz attractor



The main purpose of this paper is to introduce a class of vector fields on the 3-sphere that we call “right-handed”. Roughly speaking, they are characterized by the fact that any two orbits link positively. We give various natural examples and provide some kind of homological characterization. We then describe some of the main dynamical properties of these flows.

Keywords and phrases:

dynamical systems Lorenz attractor positive braids 

Mathematics Subject Classification (2000):

37C70 57M25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V.I. Arnold, The asymptotic Hopf invariant and its applications. Selected translations, Selecta Math. Soviet., 5 (1986), 327–345.MATHMathSciNetGoogle Scholar
  2. 2.
    V.I. Arnold and B.A. Khesin, Topological Methods in Hydrodynamics, Appl. Math. Sci., 125, Springer-Verlag, New York, 1998.Google Scholar
  3. 3.
    J.S. Birman and R.F. Williams, Knotted periodic orbits in dynamical systems. I. Lorenz’s equations, Topology, 22 (1983), 47–82.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    D. DeTurck and H. Gluck, The Gauss linking integral on the 3-sphere and in hyperbolic 3-space, e-print, arXiv:math.GT/0406276.Google Scholar
  5. 5.
    D. Fried, The geometry of cross sections to flows, Topology, 21 (1982), 353–371.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    D. Fried, Transitive Anosov flows and pseudo-Anosov maps, Topology, 22 (1983), 299–303.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    É. Ghys, Knots and dynamics, In: International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 247–277.Google Scholar
  8. 8.
    D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., 36 (1976), 225–255.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    W. Tucker, A rigorous ODE solver and Smale’s 14th problem, Found. Comput. Math., 2 (2002), 53–117.MATHMathSciNetGoogle Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2009

Authors and Affiliations

  1. 1.Unité de Mathématiques, Pures et Appliquées, de l’ École Normale Supérieure de Lyon, U.M.R. 5669 du Centre national de la recherche scientifiqueLyon Cedex 07France

Personalised recommendations