Area Preserving Maps

  • L. E. Reichl
Part of the Institute for Nonlinear Science book series (INLS)


Area preserving maps provide the simplest and most accurate means to visualize and quantify the behavior of conservative systems with two degrees of freedom. Such maps can be iterated on even the smallest computers with great accuracy, and provide beautiful pictures of the mechanisms at play during the transition to chaos. The class of area preserving maps we will study in this chapter are the so-called twist maps. When an integrable twist map is rendered non-integrable by a small perturbation, resonance can occur and degenerate lines of fixed points in the integrable map are changed to finite chains of alternating hyperbolic and elliptic fixed points surrounded by nonlinear resonance zones. As the strength of the perturbation is increased, the resonance zones grow, can overlap and form a chaotic sea.


Periodic Orbit Period Doubling Bifurcation Chaotic Region Bernoulli Shift Boundary Circle 
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  1. Aubry, S. (1978): in Solitons and Condensed Matter Physics, edited by A.R. Bishop and T. Schneider (Springer, Berlin) p.264.CrossRefGoogle Scholar
  2. Aubry, S. and LeDaeron, P.Y. (1983): Physica 8D381.MathSciNetzbMATHGoogle Scholar
  3. Benettin, G. Cergignani, C., Galgani, L., and Giorgilli, A. (1982a): Lettere Nuovo Cimento 281.CrossRefGoogle Scholar
  4. Benettin, G., Galgani, L., and Giorgilli, A. (1982b): Lettere Nuovo Cimento 29163.MathSciNetCrossRefGoogle Scholar
  5. Bensimon, D. and Kadanoff, L.P. (1984): Physics 13D82.MathSciNetzbMATHGoogle Scholar
  6. Berry, M.V. (1978): AIP Conference Proceedings 4616. Reprinted in [MacKay and Meiss 1987].ADSCrossRefGoogle Scholar
  7. Birkhoff, G.D. (1927): Acta Mathematica 50359. Reprinted in [MacKay and Meiss 1987].MathSciNetzbMATHCrossRefGoogle Scholar
  8. Bountis, T. (1981): Physica 3D577.MathSciNetzbMATHGoogle Scholar
  9. Channon, S.R. and Lebowitz, J.L. (1980): Ann. New York Acad. Sci. 108357.Google Scholar
  10. Chirikov, B. (1979): Phys. Rept. 52263.MathSciNetADSCrossRefGoogle Scholar
  11. Chirikov, B. and Shepelyansky, D.L. (1984): Physica 13D395.MathSciNetzbMATHGoogle Scholar
  12. Chirikov, B. and Shepelyansky, D.L. (1986): “Chaos Border and Statistical Anomalies”, Preprint 8–174, Institute of Nuclear Physics, Novosibirsk.Google Scholar
  13. Collet, P., Eckmann, J.-P., and Koch, H. (1981): Physica 3D457.MathSciNetzbMATHGoogle Scholar
  14. DeVogelaere, R. (1958): in Contributions to the Theory of Nonlinear Oscillations, Vol. IV, edited by S. Lefschetz, (Princeton University Press, Princeton) p.53Google Scholar
  15. Feigenbaum, M.J., (1978): J. Stat. Phys. 1925.MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. Feigenbaum, M.J., (1979): J. Stat. Phys. 216.MathSciNetCrossRefGoogle Scholar
  17. Fermi, E. (1949): Phys. Rev. 751169.ADSzbMATHCrossRefGoogle Scholar
  18. Greene, J. (1979): in Nonlinear Orbit Dynamics and the Beam-Beam Interaction, edited by M. Month and J.C. Herrera, American Institute of Physics Conference Proceedings, Vol.59(A.I.P., New York) p.257.Google Scholar
  19. Greene, J. (1968): J. Math. Phys. 9760.MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. Greene, J. (1979): J. Math. Phys. 201183.ADSCrossRefGoogle Scholar
  21. Greene, J., MacKay, R.S., Vivaldi, F., and Feigenbaum, M.J. (1981): Physica 3D 468.MathSciNetzbMATHGoogle Scholar
  22. Greene, J.M., MacKay, R.S., and Stark, J. (1986): Physica 21D267.MathSciNetGoogle Scholar
  23. Guckenheimer, J. and Holmes, P., (1983): Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields(Springer-Verlag, New York).zbMATHGoogle Scholar
  24. Hanson, J.D., Cary, J.R., and Meiss, J.D. (1985): J.Stat.Phys. 39327.MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. Hardy, G.H. and Wright, E.M. (1979): Introduction to the Theory of Numbers(Clarendon, Oxford).zbMATHGoogle Scholar
  26. Hasegawa, H.H. and Saphir, W.C. (1991): in Aspects of Nonlinear Dynamics: Solitons and Chaos, edited by J. Antoniou and F. Lambert (Springer-Verlag, Berlin).Google Scholar
  27. Hatori, T., Kamimura, T., and Ichikawa, Y.H. (1985): Physica 14D193.MathSciNetGoogle Scholar
  28. Helleman, R. (1983): “One Mechanism for the Onset of Large-Scale Chaos in Conservative and Dissipative Systems” in Long Time Prediction in Dynamics, edited by W. Horton, L.E. Reichl, and V. Szebehely, (J.Wiley and Sons, New York) p.95.Google Scholar
  29. Henon, M. (1969): Quartley of Applied Math. 27291.MathSciNetzbMATHGoogle Scholar
  30. Holmes, P.J. (1979): Philos. Trans. R. Soc. 292419.ADSzbMATHCrossRefGoogle Scholar
  31. Holmes, P.J. (1980): SIAM J. Appl Math. 3865.MathSciNetADSzbMATHCrossRefGoogle Scholar
  32. Holmes, P.J. and Marsden, J.E., (1982a): Comm. Math. Phys. 82523.MathSciNetADSzbMATHCrossRefGoogle Scholar
  33. Holmes, P.J. and Marsden, J.E., (1982b): J. Math. Phys. 23669.MathSciNetADSzbMATHCrossRefGoogle Scholar
  34. Kadanoff, L.P. (1981): Phys. Rev. Lett. 471641.MathSciNetADSCrossRefGoogle Scholar
  35. Karney, C.F.F. (1983): Physica 8D360.MathSciNetGoogle Scholar
  36. Katok, A. (1982): Ergodic Theory and Dyn. Sys.2185.MathSciNetzbMATHGoogle Scholar
  37. Koch, H. (1981): Physica 2D457.Google Scholar
  38. Lichtenberg, A.J. and Lieberman, M.A. (1983): Regular and Stochastic Motion(Springer-Verlag, New York)zbMATHCrossRefGoogle Scholar
  39. Lieberman, M.A. and Lichtenberg, A.J. (1972): Phys. Rev. A51852.ADSCrossRefGoogle Scholar
  40. Liu, J.-X., Chen, G.-Z., Wang, G.-R., and Chen, S.-G. (1989): Chinese Physics 9327.Google Scholar
  41. MacKay, R.S. (1982): “Renormalization in Area Preserving Maps”, Ph.D. Dissertation, Princeton, (University Microfilms, Int., Ann Arbor, Michigan).Google Scholar
  42. MacKay, R.S. (1983a): Physica 7D283.MathSciNetzbMATHGoogle Scholar
  43. MacKay, R.S. (1983b): “Period Doubling as a Universal Route to Stochasticity” in Long Time Prediction in Dynamics, edited by W. Horton, L.E. Reichl, and V. Szebehely, (J.Wiley and Sons, New York)Google Scholar
  44. MacKay, R.S. and Meiss, J.D. (1987): Hamiltonian Dynamical Systems(Adam Hilger, Bristol)zbMATHGoogle Scholar
  45. MacKay, R.S., Meiss, J.D., and Percival, I.C. (1984): Physica 13D55.MathSciNetzbMATHGoogle Scholar
  46. Mather, J.N. (1982): Topology 21457.MathSciNetzbMATHCrossRefGoogle Scholar
  47. Meiss, J.D. and Ott, E. (1986): Physica 20D387.MathSciNetzbMATHGoogle Scholar
  48. Meiss, J.D., Cary, J.R., Grebogi, C., Crawford, J.D., and Kaufman, A.N. (1983): Physica 6D375.MathSciNetzbMATHGoogle Scholar
  49. Melnikov, V.K., (1963): Trans. Moscow Math. Soc., 121.Google Scholar
  50. Meyer, K.R. (1970): Trans. Amer. Math. Soc. 14995.MathSciNetzbMATHCrossRefGoogle Scholar
  51. Morosov, A.D. (1976): Diff. Eqns.12164.Google Scholar
  52. Moser, J. (1973): Stable and Random Motions in Dynamical Systems(Princeton University Press, Princeton, N.J.)zbMATHGoogle Scholar
  53. Moser, J. (1968): Nachr. Akad. Wiss. Gottingen II, Math. Phys. Kd 1 1.Google Scholar
  54. Percival, I.C. (1979): in Nonlinear Dynamics and the Beam-Beam Interactionedited by M. Month and J.C. Herrera, American Institute of Physics, Conf.Proc. No.57 302.Google Scholar
  55. Petrosky, T.Y. (1986): Phys. Lett. A117328.CrossRefGoogle Scholar
  56. Petrosky, T.Y. and Schieve, W.C. (1985): Phys. Rev. A313907.ADSCrossRefGoogle Scholar
  57. Prasad, A.V. (1948): J. London Math. Soc. 23169.MathSciNetCrossRefGoogle Scholar
  58. Rechester, A.B. and White, R.B. (1980): Phys. Rev. Lett. 441586.MathSciNetADSCrossRefGoogle Scholar
  59. Rechester, A.B., Rosenbluth, M.N., and White, R.B. (1981): Phys. Rev. A232664.MathSciNetADSCrossRefGoogle Scholar
  60. Reichl, L.E. and Zheng, W.M. (1988): in Directions in Chaos, edited by Hao Bailin (World Scientific Pub.Co.)Google Scholar
  61. Shenker, S.J. and Kadanoff, L.P. (1982): J. Stat. Phys. 27631.MathSciNetADSCrossRefGoogle Scholar
  62. Tresser, C. and Coullet, P., (1978): C. R. Acad. Sci. Ser. A 287577.MathSciNetzbMATHGoogle Scholar
  63. Ulam, S.M. (1961): in the Proceedings of the Fourth, Berkeley Symposium on Mathematical Statistics and Probabilities, Vol. 3 (University of California Press, Berkeley).Google Scholar
  64. Vivaldi, F., Casati, G., and Guarneri, I. (1983): Phys. Rev. Lett. 51727.MathSciNetADSCrossRefGoogle Scholar
  65. Yamaguchi, Y. and Mishima, N. (1984): Phys. Lett. 104A179.MathSciNetCrossRefGoogle Scholar
  66. Yamaguchi, Y. (1985): Phys. Lett. 109A191.MathSciNetCrossRefGoogle Scholar
  67. Zaslavsky, G.M. and Chirikov, B.V. (1965): Sov. Phys. Doklady 9989.ADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • L. E. Reichl
    • 1
  1. 1.Center for Statistical Mechanics and Complex Systems, Department of PhysicsUniversity of Texas at AustinAustinUSA

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