Abstract
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 nonintegrable 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, and can overlap and form a chaotic sea.
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References
Aubry, S. (1978): in Solitons and Condensed Matter Physics, edited by A.R. Bishop and T. Schneider (Springer, Berlin) p. 264.
Aubry, S. and LeDaeron, P.Y. (1983): Physica D 8 381.
Benettin, G., Cergignani, C., Galgani, L., and Giorgilli, A. (1982a): Lett. Nuovo Cimento 28 1.
Benettin, G., Galgani, L., and Giorgilli, A. (1982b): Lett. Nuovo Cimento 29 163.
Bensimon, D. and Kadanoff, L.P. (1984): Physics D 13 82.
Berry, M.V. (1978): AIP Conference Proceedings 46 (American Institute of Physics, New York), p. 16. Reprinted in [MacKay and Meiss 1987].
Birkhoff, G.D. (1927): Acta Math. 50 359. Reprinted in [MacKay and Meiss 1987].
Bountis, T. (1981): Physica D 3 577.
Casati, G., Guarneri, I., and Shepelyansky, D.L. (1988): IEEE J. Quantum Electron. 24 1420.
Channon, S.R. and Lebowitz, J.L. (1980): Ann. N. Y. Acad. Sci. 108 357.
Chirikov, B. (1979): Phys. Rep. 52 263.
Chirikov, B. and Shepelyansky, D.L. (1984): Physica D 13 395.
Chirikov, B. and Shepelyansky, D.L. (1986): “Chaos Border and Statistical Anomalies,” Preprint 86–174 (Institute of Nuclear Physics, Novosibirsk).
Cocke, S. and Reichl, L.E. (1990): Phys. Rev. A 41 3733.
Collet, P., Eckmann, J.-P., and Koch, H. (1981): Physica D 3 457.
del-Castillo-Negrete, D., Greene, J.M., and Morrison, P.J. (1996): Physica D 91 1.
del-Castillo-Negrete, D., Greene, J.M., and Morrison, P.J. (1996): Physica D 100 311.
del-Castillo-Negrete, D. and Morrison, P.J. (1993): Phys. Fluids A 5 948.
de Vogelaere, R. (1958): in Contributions to the Theory of Nonlinear Oscillations, Vol. IV, edited by S. Lefschetz (Princeton University Press, Princeton, N.J.) p. 53.
Driebe, D.J. (1999): Fully Chaotic Maps and Broken Time Symmetry (Kluwer Academic Publishers, Dordrecht).
Feigenbaum, M.J. (1978): J. Stat. Phys. 19 25.
Feigenbaum, M.J. (1979): J. Stat. Phys. 21 6.
Fermi, E. (1949): Phys. Rev. 75 1169.
Greene, J. (1968): J. Math. Phys. 9 760.
Greene, J. (1979a): J. Math. Phys. 20 1183.
Greene, J. (1979b): in Nonlinear Orbit Dynamics and the Beam-Beam Interaction, edited by M. Month and J.C. Herrera, AIP Conference Proceedings 59 (American Institute of Physics, New York) p. 257.
Greene, J., MacKay, R.S., Vivaldi, F., and Feigenbaum, M.J. (1981): Physica D 3 468.
Greene, J.M., MacKay, R.S., and Stark, J. (1986): Physica D 21 267.
Guckenheimer, J. and Holmes, P., (1983): Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York).
Hanson, J.D., Cary, J.R., and Meiss, J.D. (1985): J. Stat. Phys. 39 327.
Hardy, G.H. and Wright, E.M. (1979): Introduction to the Theory of Numbers (Clarendon, Oxford).
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).
Hatori, T., Kamimura, T., and Ichikawa, Y.H. (1985): Physica D 14 193.
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 (John Wiley and Sons, New York) p. 95.
Henon, M. (1969): Q. J. Appl. Math. 27 291.
Holmes, P.J. (1979): Philos. Trans. R. Soc. 292 419.
Holmes, P.J. (1980): SIAM J. Appl. Math. 38 65.
Holmes, P.J. and Marsden, J.E, (1982a): Commun. Math. Phys. 82 523.
Holmes, P.J. and Marsden, J.E. (1982b): J. Math. Phys. 23 669.
Ichikawa, Y.H., Kamimura, T., and Hatori, T. (1987): Physica D 29 247.
Ishizaki, R., Horita, T., Kobayashi, T., and Mori, H. (1991): Prog. Theor. Phys. 85 1013.
Kadanoff, L.P. (1981): Phys. Rev. Lett. 47 1641.
Karney, C.F.F. (1983): Physica D 8 360.
Katok, A. (1982): Ergodic Theor. Dyn. Syst. 2 185.
Lichtenberg, A.J. and Lieberman, M.A. (1983): Regular and Stochastic Motion (Springer-Verlag, New York).
Lieberman, M.A. and Lichtenberg, A.J. (1972): Phys. Rev. A 5 1852.
Liu, J.-X., Chen, G.-Z., Wang, G.-R., and Chen, S.-G. (1989): Chin. Phys. 9 327.
MacKay, R.S. (1982): “Renormalization in area-preserving Maps,” Ph.D. Dissertation, Princeton University (University Microfilms, Ann Arbor, Mich.).
MacKay, R.S. (1983a): Physica D 7 283.
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 (John Wiley and Sons, New York).
MacKay, R.S. and Meiss, J.D. (1987): Hamiltonian Dynamical Systems (Adam Hilger, Bristol).
MacKay, R.S., Meiss, J.D., and Percival, LC. (1984): Physica D 13 55.
Mather, J.N. (1982): Topology 21 457.
Meiss, J.D., Cary, J.R., Grebogi, C, Crawford, J.D., and Kaufman, A.N. (1983): Physica D 6 375.
Meiss, J.D. and Ott, E. (1986): Physica D 20 387.
Melnikov, V.K. (1963): Trans. Moscow Math. Soc. 12 1.
Meyer, K.R. (1970): Trans. Am. Math. Soc. 149 95.
Morosov, A.D. (1976): Diff. Eqns. 12 164.
Moser, J. (1973): Stable and Random Motions in Dynamical Systems (Princeton University Press, Princeton, N.J.).
Percival, I.C. (1979): in Nonlinear Dynamics and the Beam-Beam Interaction edited by M. Month and J.C. Herrera, AIP Conference Proceedings 57 (American Institute of Physics,, New York), p. 302.
Petrosky, T.Y. (1986): Phys. Lett. A 117 328.
Petrosky, T.Y. and Schieve, W.C. (1985): Phys. Rev. A 31 3907.
Prasad, A.V. (1948): J. London Math. Soc. 23 169.
Rechester, A.B., Rosenbluth, M.N., and White, R.B. (1981): Phys. Rev. A 23 2664.
Rechester, A.B. and White, R.B. (1980): Phys. Rev. Lett. 44 1586.
Reichl, L.E. (1998): A Modern Course in Statistical Physics, Second Edition (John Wiley and Sons, New York).
Reichl, L.E. and Zheng, W.M. (1988): in Directions in Chaos, edited by Hao Bailin (World Scientific, Singapore)
Rom-Kedar, V., and Zaslavsky, G.M. (1999): Chaos 9 697.
Shenker, S.J. and Kadanoff, L.R (1982): J. Stat. Phys. 27 631.
Stefancich, M., Allegrini, R, Bonci, L., Grigolini, P., and West, B.J. (1998): Phys. Rev. E 57 6625.
Tresser, C. and Coullet, P. (1978): C. R. Acad. Sci. Ser. A 287 577.
Ulam, S.M. (1961): in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probabilities, Vol. 3 (University of California Press, Berkeley).
Vivaldi, F., Casati, G., and Guarneri, I. (1983): Phys. Rev. Lett. 51 727.
Yamaguchi, Y. and Mishima, N. (1984): Phys. Lett. A 104 179.
Yamaguchi, Y. (1985): Phys. Lett. A 109 191.
Zaslavsky, G.M. (1998): Physics of Chaos in Hamiltonian Systems (Imperial College Press, London).
Zaslavsky, G.M. and Chirikov, B.V. (1965): Sov. Phys. Dokl. 9 989.
Zaslavsky, G.M., Edelman, M., and Niyazov, B.A. (1997): Chaos 7 159.
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Reichl, L.E. (2004). Area-Preserving Maps. In: The Transition to Chaos. Institute for Nonlinear Science. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4350-0_3
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