Archive for Rational Mechanics and Analysis

, Volume 90, Issue 2, pp 115–194 | Cite as

Knotted periodic orbits in suspensions of Smale's horseshoe: Torus knots and bifurcation sequences

  • Philip Holmes
  • R. F. Williams


We construct a suspension of Smale's horseshoe diffeomorphism of the two-dimensional disc as a flow in an orientable three manifold. Such a suspension is natural in the sense that it occurs frequently in periodically forced nonlinear oscillators such as the Duffing equation. From this suspension we construct a knot-hòlder or template—a branched two-manifold with a semiflow—in such a way that the periodic orbits are isotopic to those in the full three-dimensional flow. We discuss some of the families of knotted periodic orbits carried by this template. In particular we obtain theorems of existence, uniqueness and non-existence for families of torus knots. We relate these families to resonant Hamiltonian bifurcations which occur as horseshoes are created in a one-parameter family of area preserving maps, and we also relate them to bifurcations of families of one-dimensional ‘quadratic like’ maps which can be studied by kneading theory. Thus, using knot theory, kneading theory and Hamiltonian bifurcation theory, we are able to connect a countable subsequence of “one-dimensional” bifurcations with a subsequence of “area-preserving” bifurcations in a two parameter family of suspensions in which horseshoes are created as the parameters vary. One implication is that infinitely many bifurcation sequences are reversed as one passes from the one dimensional to the area-preserving family: there are no universal routes to chaos!


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. V. I. Arnold [1977] Fund. Anal. Appl. 11 (2) 1–10. Loss of stability of self oscillations close to resonances and versal deformations of equivariant vector fields.Google Scholar
  2. V. I. Arnold [1982] ‘Geometrical methods in the theory of ordinary differential equations’, Springer Verlag, Berlin, Heidelberg, New York (Russian original, Moscow, 1977).MathSciNetMATHGoogle Scholar
  3. V. I. Arnold & A. Avez [1968] ‘Ergodic Problems of Classical Mechanics’ New York, W. A. Benjamin Inc.Google Scholar
  4. J.-P. Babary & C. Mira [1969] C. R. Acad. Sci. Paris 268 Série A, 129–132 Sur un cas critique pour une récurrence autonome du deuxième ordre.Google Scholar
  5. R. E. Bedient [1984] Classifying 3-trip Lorenz knots. Preprint, Hamilton College, N.Y.Google Scholar
  6. P. Beiersdorfer, J.-M. Wersinger & Y. Treve, Phs. Lett. 96A, 269–272. Topology of the invariant manifolds of period-doubling attractors for some forced nonlinear oscillators.Google Scholar
  7. G. D. Birkhoff [1913] Trans. Am. Math. Soc. 14, 14–22, Proof of Poincaré's geometric theorem.Google Scholar
  8. G. D. Birkhoff [1927] ‘Dynamical Systems’, A.M.S. Publications, Providence, R.I.Google Scholar
  9. J. S. Birman [1975] ‘Braids, Links and Mapping Class Groups’, Princeton University Press, Princeton, N.J.Google Scholar
  10. J. Birman & R. F. Williams [1983 a] Topology, 22, 47–82. Knotted periodic orbits in dynamical systems I: Lorenz's equations.Google Scholar
  11. J. Birman & R. F. Williams [1983 b] Contemporary Mathematics, 20, 1–60, Knotted periodic orbits in dynamical systems II: knot holders for fibred knots.Google Scholar
  12. R. Bowen [1976] ‘On Axiom A Diffeomorphisms’, CBMS Regional Conference Series in Mathematics 35, AMS Publications, Providence, RI.Google Scholar
  13. A. Chenciner [1983] Bifurcations de difféomorphismes de R2 au voinsinage d'un point fixe elliptique. Les Houches Summer School Proceedings, ed. R. Helleman, G. Iooss, North Holland.Google Scholar
  14. P. Collet & J. P. Eckmann [1980] ‘Iterated Maps on the Interval as Dynamical Systems’, Birkhauser, Boston.Google Scholar
  15. J. D. Crawford &S. Omohundro [1984] Physica 13D, 161–180, On the global structure of period doubling flows.Google Scholar
  16. R. Devaney [1974] Personal communication.Google Scholar
  17. R. Devaney & Z. Nitecki [1979] Comm. Math. Phys. 67, 137–148, Shift automorphisms in the Hénon mapping.Google Scholar
  18. A. Douady & J. H. Hubbard [1982] C. R. Acad. Sci. Paris 294 Série I, 123–126, Itération des polynomes quadratiques complexes.Google Scholar
  19. H. El-Hamouly & C. Mira [1981] C. R. Acad. Sci. Paris 293, Série I, 525–528 Lien entres les propriétés d'un endomorphisme de dimension un et celles d'un difféomorphisme de dimension deux.Google Scholar
  20. M. J. Feigenbaum [1978] J. Stat. Phys. 19, 25–52, Quantitative universality for a class of nonlinear transformations.MATHGoogle Scholar
  21. J. Franks & R. F. Williams [1985] Positive braids via the Jones polynomial. Preprint, Northwestern University, Evanston, Ill.Google Scholar
  22. B. D. Greenspan & P. J. Holmes [1983] Homoclinic orbits, subharmonics, and global bifurcations in forced oscillations. Chapter 10, pp. 172–214 in Nonlinear Dynamics and Turbulence ed. G. Barenblatt, G. Iooss and D. D. Joseph, Pitman, London.Google Scholar
  23. B. D. Greenspan & P. J. Holmes [1984] SIAM J. on Math. Analysis 15, 69–97, Repeated resonance and homoclinic bifurcation in a periodically forced family of oscillators.Google Scholar
  24. J. Guckenheimer [1979] Comm. Math. Phys. 70, 133–160, Sensitive dependence on initial conditions for one dimensional maps.Google Scholar
  25. J. Guckenheimer [1980] ‘Bifurcations of Dynamical Systems’, in Dynamical Systems, ed. J. K. Moser, Birkhauser, Boston.Google Scholar
  26. J. Guckenheimer & P. J. Holmes [1983] ‘Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields’, Springer Verlag, New York.Google Scholar
  27. J. Guckenheimer & R. F. Williams [1979] Publ. Math. IHES 50, 59–72, Structural Stability of Lorenz Attractors.Google Scholar
  28. I. Gumowski & C. Mira [1974] C. R. Acad. Sci. Paris 278, Série A, 1591–1593. Bifurcation pour une récurrence du deuxième ordre, par traversée d'un cas critique avec deux multiplicateurs complexes conjugués.Google Scholar
  29. I. Gumowski & C. Mira [1980 a] ‘Dynamique Chaotique’, Editions Cepadue, Toulouse, France.Google Scholar
  30. I. Gumowski & C. Mira [1980 b] ‘Recurrences and Discrete Dynamical Systems’, Springer Lecture Notes in Mathematics 809, Springer Verlag, Berlin, Heidelberg, New York.Google Scholar
  31. C. Hayashi, [1964] ‘Nonlinear Oscillations in Physical Systems’, McGraw Hill, New York.Google Scholar
  32. M. Hénon [1976] Comm. Math. Phys. 50, 69–77, A two dimensional mapping with a strange attractor.Google Scholar
  33. M. W. Hirsch, C. C. Pugh & M. Shub [1977] ‘Invariant Manifolds’, Lecture Notes in Mathematics No. 583, Springer Verlag, Berlin, Heidelberg, New York.Google Scholar
  34. P. J. Holmes [1979] Phil Trans Roy Soc, Lond A 292, 419–448, A nonlinear oscillator with a strange attractor.Google Scholar
  35. P. J. Holmes & D. C. Whitley [1983] On the Attracting Set for Duffing's Equation II: A Geometrical Model for Moderate Force and Damping (Proc. ‘Order in Chaos’, Los Alamos National Laboratory, May 1982)—also Physica 7D, 111–123.Google Scholar
  36. P. J. Holmes & D. C. Whitley [1984a] Phil Trans Roy Soc Lond A 311, 43–102, Bifurcations of one and two dimensional maps.Google Scholar
  37. P. J. Holmes & D. C. Whitley [1984b] On the attracting set for Duffing's equation I: analytical methods for small force and damping; Proc. ‘Year of Concentration in Partial Differential Equations and Dynamical Systems’ University of Houston, W. E. Fitzgibbon III (ed), Pitman, London.Google Scholar
  38. G. Iooss [1979] ‘Bifurcation of Maps and Applications’, North-Holland, Amsterdam.Google Scholar
  39. M. V. Jacobson [1981] Comm. Math. Phys. 81, 39–88, Absolutely continuous invariant measures for one parameter families of one dimensional maps.PubMedGoogle Scholar
  40. L. Jonker [1979] Proc. Lond. Math. Soc. 39, 428–450. Periodic orbits and kneading invariants.Google Scholar
  41. L. Jonker & D. A. Rand [1981] Invent. Math. 62, 347–365 and 63, 1–15, Bifurcations in one dimension I: the nonwandering set and II: A versal model for bifurcations.Google Scholar
  42. K. Meyer [1970] Trans Am. Math. Soc. 149, 95–107, Generic bifurcation of periodic points.Google Scholar
  43. K. Meyer [1971] Trans Am. Math. Soc. 154, 273–277, Generic stability properties of periodic points.Google Scholar
  44. J. Milnor & R. Thurston [1977] ‘On Iterated Maps of the Interval I and II’ Unpublished notes, Princeton University, Princeton, NJ.Google Scholar
  45. C. Mira [1969a] C. R. Acad. Sci. Paris 268 Série A, 621–624, Traversée d'un cas critique, pour une récurrence du deuxième ordre, sous l'effet d'une variation de paramètre.Google Scholar
  46. C. Mira [1969b] C. R. Acad. Sci. Paris 269, Serie A, 1006–1009. Étude d'un premier cas d'exception pour une récurrence ou une transformation ponctuelle, autonome du deuxième ordre a variables réeles.Google Scholar
  47. C. Mira [1970] C. R. Acad. Sci. Paris 270, Série A, 332–335 and 466–469. Etude d'un second cas d'exception pour une recurrence ... and Sur les cas d'exception d'une recurrence ...Google Scholar
  48. C. Mira [1981] Proc IX International Conference on Nonlinear Oscillations, Kiev, Sept. 1981, Chaotic dynamics in point mappings.Google Scholar
  49. C. Mira [1982] C. R. Acad. Sci. Paris 294, Série I, 689–692 Ensembles rythmiques de suites de rotation pour un endomorphisme uni-dimensionnel et nombres de rotation d'un difféomorphisme conservatif bi-dimensionnel.Google Scholar
  50. M. Misiurewicz [1981 a] Publ. Math. IHES 53, 5–16. The structure of mappings of an interval with zero entropy.Google Scholar
  51. M. Misiurewicz [1981b] Publ. Math. IHES 53, 17–52, Absolutely continuous measures for certain maps of the interval.Google Scholar
  52. J. Moser [1973] ‘Stable and Random Motions in Dynamical Systems’, Princeton University Press, Princeton, NJ.Google Scholar
  53. A. H. Nayfeh & D. T. Mook [1979] ‘Nonlinear Oscillations’, Wiley, New York.Google Scholar
  54. S. E. Newhouse [1980] ‘Lectures on Dynamical Systems’ in ‘Dynamical Systems’ ed. J. K. Moser, Birkhauser, Boston.Google Scholar
  55. H. Poincaré [1899] ‘Les Méthodes Nouvelles de la Mécanique Céleste’ (3 Vols) GauthierVillars, Paris.Google Scholar
  56. D. Rolfsen [1977] ‘Knots and Links’. Publish or Perish, Berkeley, CA.Google Scholar
  57. O. E. Rossler [1979] Ann. N.Y. Acad Sci 316, 376–392, Continuous chaos: four prototype equations.Google Scholar
  58. D. Singer [1978] SIAM J. Appl. Math 35, 260–267, Stable orbits and bifurcations of maps of the interval.Google Scholar
  59. S. Smale [1963] Diffeomorphisms with many periodic points, in Differential and Combinatorial Topology, ed. S. S. Cairns, pp. 63–80, Princeton University Press, Princeton, N.J.Google Scholar
  60. S. Smale [1967] Bull Amer. Math. Soc. 73, 747–817, Differentiable dynamical systems.Google Scholar
  61. C. T. Sparrow [1982] ‘The Lorenz Equations: Bifurcations, Chaos and Strange Attractors’, Springer-Verlag; Berlin, Heidelberg, New York.Google Scholar
  62. F. Takens [1973 a] Ann. Inst. Fourier 23, 163–195, Normal forms for certain singularities of vectorfields.Google Scholar
  63. Y. Ueda [1980] Steady motions exhibited by Duffing's equation: a picture book of regular and chaotic motions, in ‘New Approaches to Nonlinear Problems in Dynamics’ ed. P. J. Holmes, SIAM Publications, Philadelphia.Google Scholar
  64. Y. Ueda [1981] Personal communication to P. Holmes.Google Scholar
  65. T. Ueza & Y. Aizawa [1982] Prog. Theor. Phys. 68, 1907–1916. Topological character of a periodic solution in three dimensional ordinary differential equation system.Google Scholar
  66. S. M. Ulam & J. von Neumann [1947] Bull Am. Math. Soc. 53, 1120, On combinations of stochastic and deterministic processes.Google Scholar
  67. S. J. Van Strien [1981] On the bifurcations creating horseshoes in Dynamical Systems and Turbulence, D. A. Rand and L. S. Young (eds), Springer Lecture Notes in Mathematics 898, Springer Verlag, Berlin, Heidelberg, New York.Google Scholar
  68. Y. H. Wan [1978 a] SIAM J. Appl. Math 34, 167–175. Computation of the stability condition for the Hopf bifurcation of diffeomorphisms on ℝ2.Google Scholar
  69. Y. H. Wan [1978 b] Arch. Rational Mech. Anal. 68, 343–357, Bifurcation into invariant tori at points of resonance.Google Scholar
  70. D. C. Whitley [1982] The Bifurcations and Dynamics of Certain Quadratic Maps of the Plane, Ph. D. Thesis, University of Southampton, U.K.Google Scholar
  71. R. F. Williams [1974] Publ. Math. IHES 43, 169–203, Expanding attractors.Google Scholar
  72. R. F. Williams [1977] Springer Lectures Notes in Math. No. 615, Turbulence Seminar, 94–112, The Structure of Lorenz attractors.Google Scholar
  73. R. F. Williams [1979] Publ. Math. IHES 50, 73–99, The structure of Lorenz attractors.Google Scholar
  74. J. A. Yorke & K. T. Alligood [1983] Bull. Am. Math. Soc. 9, 319–322, Cascades of period-doubling bifurcations: a prerequisite for horseshoes.Google Scholar

Copyright information

© Springer-Verlag GmbH & Co 1985

Authors and Affiliations

  • Philip Holmes
    • 1
    • 2
  • R. F. Williams
    • 1
    • 2
  1. 1.Departments of Theoretical and Applied Mechanics and MathematicsCornell UniversityIthaca
  2. 2.Department of MathematicsNorthwestern UniversityEvanston

Personalised recommendations