Advertisement

Torus Maps

  • R. S. MacKay
Part of the NATO ASI Series book series (NSSB, volume 280)

Abstract

Mathematically, the study of torus maps here developed is concerned with the dynamics of diffeomorphisms of the two-dimensional torus, isotopic to the identity.

Keywords

Periodic Orbit Bifurcation Diagram Periodic Point Rotation Number Resonance Region 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Arn]
    Arnol’d VI, Geometric methods in the theory of Ordinary Differential Equations, Springer (1983).Google Scholar
  2. [ACHM]
    Aronson DG, Chory MA, Hall GR, McGehee RP, Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer assisted study, Commun Math Phys 83 (1982) 303–354.MathSciNetADSMATHCrossRefGoogle Scholar
  3. [BGKM]
    Baesens C, Guckenheimer J, Kim S, MacKay RS, Three coupled oscillators: modelocking, global bifurcation and toroidal chaos, Physica D, to appear.Google Scholar
  4. [BGKM2]
    Baesens C, Guckenheimer J, Kim S, MacKay RS, Simple resonance regions of torus diffeomorphisms, IMA Conf Proc series (Springer), to appear.Google Scholar
  5. [BGOY]
    Battelino PM, Grebogi C, Ott E, Yorke JA, Chaotic attractors on a 3-torus and torus breakup, Physica D 39 (1989) 299–314.MathSciNetADSMATHCrossRefGoogle Scholar
  6. [BC]
    Benedicks M, Carleson L, The dynamics of the Hénon map, preprintGoogle Scholar
  7. [BH]
    Biswas DJ, Harrison RG, Experimental evidence of three-mode quasiperiodicity and chaos in a single longitudinal, multi-transverse mode cw CO2 laser, Phys Rev A 32 (1985) 3835–7.ADSCrossRefGoogle Scholar
  8. [BMS]
    Bogoljubov NN, Mitropolskii JuA, Samoilenko AM, Methods of accelerated convergence in nonlinear mechanics (Springer, 1976).Google Scholar
  9. [Boyd]
    Boyd C, On the structure of the family of Cherry fields on the torus, Ergod Th D y n Sys 5 (1985) 27–46.MATHGoogle Scholar
  10. [Cas]
    Casdagli MC, Periodic orbits for dissipative twist maps, Erg Th Dyn Sys 7 (1987) 165–173.MathSciNetMATHCrossRefGoogle Scholar
  11. [Chen]
    Chenciner A, Bifurcations de points fixes elliptiques III.-Orbites périodiques de petites périodes et élimination résonnante des couples de courbes invariantes, Publ Math IHES 66 (1988) 5–91.MathSciNetMATHGoogle Scholar
  12. [Cher]
    Cherry TM, Analytic quasi-periodic curves of discontinuous type on a torus, Proc Lond Math Soc 44 (1938) 175–215.CrossRefGoogle Scholar
  13. [Chir]
    Chirikov BV, A universal instability of many oscillator systems, Phys Repts 52 (1979) 265–379.MathSciNetADSCrossRefGoogle Scholar
  14. [DG]
    Dangelmayr G, Guckenheimer J, On a four parameter family of planar vector fields, Arch Rat Mech Anal 97 (1987) 321–352.MathSciNetMATHCrossRefGoogle Scholar
  15. [FLP]
    Fathi A, Laudenbach F, Poénaru V, Travaux de Thurston sur les surfaces, Astérisque 66–67 (1979).Google Scholar
  16. [Fra]
    Franks J, Realising rotation vectors for torus homeomorphisms, Trans Am Math Soc 311 (1989) 107–115.MathSciNetADSMATHCrossRefGoogle Scholar
  17. [FM]
    Franks J, Misiurewicz M, Rotation sets of toral flows, preprint.Google Scholar
  18. [Fri]
    Fried D, The geometry of cross sections to flows, Topology 21 (1982) 353–371.MathSciNetMATHCrossRefGoogle Scholar
  19. [GB]
    Gollub J, Benson SV, Many routes to turbulent convection, J Fluid Mech 100 (1980) 449–470.ADSCrossRefGoogle Scholar
  20. [GH]
    Guckenheimer J, Holmes PJ, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer (1983).Google Scholar
  21. [H]
    Herman MR, Sur la conjugation differentiable des difféomorphismes du cercle à des rotations, Publ Math IHES 49 (1979) 5–234.MathSciNetMATHGoogle Scholar
  22. [HJ]
    Hollinger F, Jung C, Single longitudinal-mode laser as a discrete dynamical system, J Opt Soc Am B 2 (1985) 218–225.ADSCrossRefGoogle Scholar
  23. [K]
    Katok A, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ Math IHES 51 (1980) 137–173.MathSciNetMATHGoogle Scholar
  24. [KMG]
    Kim S, MacKay RS, Guckenheimer J, Resonance regions for families of torus maps, Nonlinearity 2 (1989) 391–404.MathSciNetADSMATHCrossRefGoogle Scholar
  25. [EL]
    Iooss G, Los J, Quasi-genericity of bifurcations to high dimensional invariant tori for maps, Commun Math Phys 119 (1988) 453–500.MathSciNetADSMATHCrossRefGoogle Scholar
  26. [Lang]
    Langford WF, Periodic and steady-state mode interactions lead to tori, SLAM J Appl Math 37 (1979) 22–48.MathSciNetMATHCrossRefGoogle Scholar
  27. [LeC]
    Le Calvez P, Propriétés des attracteurs de Birkhoff, Ergod Th Dyn Sys 8 (1987) 241–310.Google Scholar
  28. [LC]
    Linsay PS, Cumming AW, Three-frequency quasiperiodic, phase-locking and the onset of chaos, Physica D 40 (1989) 196–217.MathSciNetADSMATHCrossRefGoogle Scholar
  29. [LM]
    Llibre J, MacKay RS, Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity, Erg Th Dyn Sys, to appear.Google Scholar
  30. [Mac]
    MacKay RS, An appraisal of the Ruelle-Takens route to turbulence, in The Global geometry of Turbulence, ed. Jimenez J (Plenum), to appear.Google Scholar
  31. [MT]
    MacKay RS, Tresser C, Transition to topological chaos for circle maps, Physica D 19 (1986) 206–237; and Erratum, Physica D 29 (1988) 427.MathSciNetADSMATHCrossRefGoogle Scholar
  32. [Mar]
    Markley NG, The Poincaré-Bendixson theorem for the Klein bottle, Trans Am Math Soc 135 (1969) 159–165.MathSciNetMATHGoogle Scholar
  33. [ML]
    Maurer J, Libchaber A, Effect of the Prandtl number on the onset pof turbulence in liquid 4He, J Physique Lett 41 (1980) L515–8.CrossRefGoogle Scholar
  34. [MZ]
    Misiurewicz M, Ziemian K, Rotation sets for maps of tori, preprint.Google Scholar
  35. [NRT]
    Newhouse SE, Ruelle D, Takens F, Occurrence of strange Axiom-A attractors near quasiperiodic flows on Tm, m ≥ 3, Commun Math Phys 64 (1978) 35–40.MathSciNetADSMATHCrossRefGoogle Scholar
  36. [Ox]
    Oxtoby JC, Ergodic sets, Bull Am Math Soc 58 (1952) 116–136.MathSciNetMATHCrossRefGoogle Scholar
  37. [PdM]
    Palis J, de Melo W, Geometric theory of dynamical systems, Springer (1982).Google Scholar
  38. [RhT]
    Rhodes F, Thompson CL, Topologies and rotation numbers for families of monotone functions on the circle, preprint.Google Scholar
  39. [Ru]
    Ruelle D, Chaotic evolution and strange attractors (CUP, 1989).Google Scholar
  40. [RT]
    Ruelle D, Takens F, On the nature of turbulence, Commun Math Phys 20 (1971) 167–192; and 23 (1971) 343-4.MathSciNetADSMATHCrossRefGoogle Scholar
  41. [Sell]
    Sell GR, Resonance and bifurcation in Hopf-Landau Dynamical Systems, in Nonlinear Dynamics and Turbulence, ed Barenblatt GI, Iooss G, Joseph DD, Pitman (1985).Google Scholar
  42. [Shub]
    Shub M, Global Stability of dynamical systems, Springer, 1987.Google Scholar
  43. [Soto]
    Sotomayor J, Generic bifurcations of dynamical systems, in Dynamical Systems, ed Peixoto MM (1973) 549–560.Google Scholar
  44. [Wal]
    Walters P, An introduction to ergodic theory, Springer, 1982.Google Scholar
  45. [W]
    Weil A, Les familles de courbes sur le tore, Mat Sb 43 (1936) 779–781.Google Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • R. S. MacKay
    • 1
  1. 1.Nonlinear Systems Laboratory Mathematics InstituteUniversity of WarwickCoventryEngland

Personalised recommendations