Dynamics of Oscillator Chains

  • Allan J. Lichtenberg
  • Roberto Livi
  • Marco Pettini
  • Stefano Ruffo
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 728)

Abstract

The Fermi–Pasta–Ulam (FPU) nonlinear oscillator chain has proved to be a seminal system for investigating problems in nonlinear dynamics. First proposed as a nonlinear system to elucidate the foundations of statistical mechanics, the initial lack of confirmation of the researchers expectations eventually led to a number of profound insights into the behavior of high-dimensional nonlinear systems. The initial numerical studies, proposed to demonstrate that energy placed in a single mode of the linearized chain would approach equipartition through nonlinear interactions, surprisingly showed recurrences. Although subsequent work showed that the origin of the recurrences is nonlinear resonance, the question of lack of equipartition remained. The attempt to understand the regularity bore fruit in a profound development in nonlinear dynamics: the birth of soliton theory. A parallel development, related to numerical observations that, at higher energies, equipartition among modes could be approached, was the understanding that the transition with increasing energy is due to resonance overlap. Further numerical investigations showed that time-scales were also important, with a transition between faster and slower evolution. This was explained in terms of mode overlap at higher energy and resonance overlap at lower energy. Numerical limitations to observing a very slow approach to equipartition and the problem of connecting high-dimensional Hamiltonian systems to lower dimensional studies of Arnold diffusion, which indicate transitions from exponentially slow diffusion along resonances to power-law diffusion across resonances, have been considered. Most of the work, both numerical and theoretical, started from low frequency (long wavelength) initial conditions.

Coincident with developments to understand equipartition was another program to connect a statistical phenomenon to nonlinear dynamics, that of understanding classical heat conduction. The numerical studies were quite different, involving the excitation of a boundary oscillator with chaotic motion, rather than the excitation of the entire chain with regular motion. Although energy transitions are still important, the inability to reproduce exactly the law of classical heat conduction led to concern for the generiticity of the FPU chain and exploration of other force laws. Important concepts of unequal masses, and “anti-integrability,” i.e. isolation of some oscillators, were considered, as well as separated optical and acoustic modes that could only communicate through very weak interactions. The importance of chains that do not allow nonlinear wave propagation in producing the Fourier heat conduction law is now recognized.

A more recent development has been the exploration of energy placed on the FPU or related oscillator chains in high-frequency (short wavelength) modes and the existence of isolated structures (breathers). Breathers are found as solutions to partial differential equations, analogous to solitons at lower frequency. On oscillator chains, such as the FPU, energy initially in a single high-frequency mode is found, at higher energies, to self-organize in oscillator space to form compact structures. These structures are “chaotic breathers,” i.e. not completely stable, and disintegrate on longer time-scales. With the significant progress in understanding this evolution, we now have a rather complete picture of the nonlinear dynamics of the FPU and related oscillator chains, and their relation to a wide range of concepts in nonlinear dynamics.

This chapter’s purpose is to explicate these many concepts. After a historical perspective the basic chaos theory background is reviewed. Types of oscillators, numerical methods, and some analytical results are considered. Numerical results of studies of equipartition, both from low-frequency and high-frequency modes, are presented, together with numerical studies of heat conduction. These numerical studies are related to analytical calculations and estimates of energy transitions and time-scales to equipartition.

Keywords

Lyapunov Exponent Thermodynamic Limit Large Lyapunov Exponent Toda Lattice mKdV Equation 
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.
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References

  1. 1.
    E. Fermi, J.R. Pasta and S. Ulam, in Collected papers of Enrico Fermi, E. Segre (ed.). University of Chicago Press, Chicago, 2, 978, 1965.Google Scholar
  2. 2.
    H. Poincarée, Les méethodes Nouvelles de la Méecanique Celeste. Gauthier-Villars, Paris, 1892.Google Scholar
  3. 3.
    E. Fermi, Zeit. Phys. 24, 261 (1923).Google Scholar
  4. 4.
    E. Fermi, Nuovo Cimento 25, 267; 26, 105 (1923).Google Scholar
  5. 5.
    J. Ford and J. Waters, J. Math. Phys. 4, 1293 (1963).ADSCrossRefGoogle Scholar
  6. 6.
    E.A. Jackson, J. Math. Phys. 4, 551, 686 (1963).MATHADSCrossRefGoogle Scholar
  7. 7.
    R.L. Bivins, N. Metropolis and J.R. Pasta, J. Comp. Phys. 12, 62 (1972).Google Scholar
  8. 8.
    A.N. Kolmogorov, Dokl. Akad. Nauk. SSSR 98, 527 (1954).MATHMathSciNetGoogle Scholar
  9. 9.
    V.I. Arnold, Soviet Math. Dokl. 2, 501 (1961).Google Scholar
  10. 10.
    J. Moser, Nachr. Akad Wiss. Gottingen, K1, 1 (1962).Google Scholar
  11. 11.
    M. Henon and C. Heiles, Astr. J. 69, 73 (1964).ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    B.V. Chirikov, Plasma Phys. (J.N.E. Pt C) 1, 253 (1960).ADSGoogle Scholar
  13. 13.
    B.V. Chirikov, Phys. Rep. 52, 265 (1979).ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    V.I. Arnold, Russian Math. Surveys 18, 85 (1964).CrossRefADSGoogle Scholar
  15. 15.
    A.J. Lichtenberg and M.A. Lieberman, Regular and chaotic dynamics, 2nd edition Springer, New York, 1992.MATHGoogle Scholar
  16. 16.
    M. Falcioni, U.M.B. Marconi and A. Vulpiani, Phys. Rev. A 44, 2263 (1991).ADSCrossRefGoogle Scholar
  17. 17.
    N.N. Nekhorochev, Usp. Mat. Nauk. (USSR) 32, 6 (1977).Google Scholar
  18. 18.
    A. Giorgilli, Ann. Inst. H. Poincare Phys. Theor. 48, 423 (1988).MATHMathSciNetGoogle Scholar
  19. 19.
    P. Lochak, Phys. Lett. A 143, 39 (1990).ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    M. Toda, Prog. Theor. Phys. Suppl. 45, 174 (1970).ADSCrossRefGoogle Scholar
  21. 21.
    J. Ford, S.D. Stoddard and J.S. Turner, Prog. Theor. Phys. 50, 1547 (1973).ADSCrossRefGoogle Scholar
  22. 22.
    M. Henon, Phys. Rev. B 9, 1925 (1974).ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    A.N. Kolmogorov, Dokl. Akad. Nauk. SSSR 124, 754 (1959).MATHMathSciNetGoogle Scholar
  24. 24.
    Ya. G. Sinai, Dokl. Akad. Nauk. SSSR 124, 768 (1959).Google Scholar
  25. 25.
    G. Benettin, L. Galgani and J.M. Strelcyn, Phys. Rev. A 14, 2338 (1976).ADSCrossRefGoogle Scholar
  26. 26.
    V.I. Arnold and A. Avez, Ergodic problems of statistical mechanics. Benjamin, New York, 1968.Google Scholar
  27. 27.
    V.I. Arnold, Mathematical methods of classical mechanics. Springer-Verlag, Berlin, 1978.MATHGoogle Scholar
  28. 28.
    N.S. Krylov, Works on the foundations of statistical physics, Princeton University Press, Princeton, 1979.Google Scholar
  29. 29.
    J. Hadamard, J. Math. Pur. Appl. 4, 27 (1898).Google Scholar
  30. 30.
    G.A. Hedlund, Bull. Am. Math. Soc. 45, 241 (1939).CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    E. Hopf, Proc. Nat. Acad. Sci. 18, 263 (1932).CrossRefGoogle Scholar
  32. 32.
    V.I. Anosov, Proc. Steklov Math. Inst. 90, 1 (1967).MathSciNetGoogle Scholar
  33. 33.
    Ya.G. Sinai, Dokl. Akad. Nauk SSSR 153, 1261 (1963).Google Scholar
  34. 34.
    C.P. Ong, Adv. Math. 15, 269 (1975).MATHCrossRefGoogle Scholar
  35. 35.
    M.C. Gutzwiller, J. Math. Phys. 18, 806 (1977).ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    A. Knauf, Comm. Math. Phys. 110, 89 (1987).MATHADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    M. Szydlowski, M. Heller and W. Sasin, J. Math. Phys. 37, 346 (1996).MATHADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    H.E. Kandrup, Astrophys. J. 364, 420 (1990).ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    H.E. Kandrup, Physica A 169, 73 (1990).MATHADSCrossRefGoogle Scholar
  40. 40.
    H.E. Kandrup, Phys. Rev. E 56, 2722 (1997).ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    T.J. Hunt and MacKay, R.S., Nonlinearity 16, 1499 (2003).MATHADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    L. Casetti, M. Pettini and E.G.D. Cohen, Phys. Rep. 337, 237 (2000).MathSciNetGoogle Scholar
  43. 43.
    M.D. Kruskal and N.J. Zabusky, J. Math. Phys. 5, 231 (1964).ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    N.J. Zabusky and M.D. Kruskal Phys. Rev. Lett. 15, 240 (1965).ADSCrossRefMATHGoogle Scholar
  45. 45.
    C.F. Driscoll and T.M. O’Neil, Phys. Rev. Lett. 37, 69 (1976).ADSCrossRefGoogle Scholar
  46. 46.
    C.F. Driscoll and T.M. O’Neil, Rocky Mt. J. Math 5, 211 (1978).MathSciNetCrossRefGoogle Scholar
  47. 47.
    F.M. Izrailev and B.V. Chirikov, Sov. Phys. Dokl. 11, 30 (1966).Google Scholar
  48. 48.
    G. Benettin, L. Galgani and A. Giorgilli, Phys. Lett. A 120, 23 (1987).ADSCrossRefGoogle Scholar
  49. 49.
    G. Benettin, L. Galgani and A. Giorgilli, Commun. Math. Phys. 113, 87 (1987).Google Scholar
  50. 50.
    R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione and A. Vulpiani, Phys. Rev. A 28, 3544 (1983).ADSCrossRefGoogle Scholar
  51. 51.
    R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione and A. Vulpiani, Phys. Rev. A 31, 1039 (1985).ADSCrossRefGoogle Scholar
  52. 52.
    R. Livi, M. Pettini, S. Ruffo and A. Vulpiani, Phys. Rev. A 31, 2740 (1985).ADSCrossRefGoogle Scholar
  53. 53.
    M. Pettini and M. Landolfi, Phys. Rev. A 41, 768 (1990).ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    C.G. Goedde, A.J. Lichtenberg and M.A. Lieberman, Physica D 59, 200 (1992).MATHADSMathSciNetCrossRefGoogle Scholar
  55. 55.
    J. DeLuca, A.J. Lichtenberg and M.A. Lieberman, Chaos 5, 283 (1995).ADSCrossRefGoogle Scholar
  56. 56.
    H. Kantz, R. Livi and S. Ruffo, J. Stat. Phys. 76, 627 (1994).MathSciNetCrossRefADSGoogle Scholar
  57. 57.
    J. DeLuca, A.J. Lichtenberg and S. Ruffo, Phys. Rev. E 51, 2877 (1995).ADSCrossRefGoogle Scholar
  58. 58.
    J. DeLuca, A.J. Lichtenberg and S. Ruffo, Phys. Rev. E 60, 3781 (1999).ADSCrossRefGoogle Scholar
  59. 59.
    L. Berchialla, A. Giorgilli and S. Paleari, Phys. Lett. A 321, 167 (2004).MATHADSCrossRefGoogle Scholar
  60. 60.
    L. Casetti, C. Clementi and M. Pettini, Phys. Rev. E 54, 5969 (1996).ADSMathSciNetCrossRefGoogle Scholar
  61. 61.
    L. Casetti, M. Cerruti-Sola, M. Pettini and E.G.D. Cohen, Phys. Rev. E 55, 6566 (1997).ADSMathSciNetCrossRefGoogle Scholar
  62. 62.
    D.L. Shepelyansky, Nonlinearity 10, 1331 (1997).MATHADSMathSciNetCrossRefGoogle Scholar
  63. 63.
    M. Cerruti-Sola, M. Pettini and E.G.D. Cohen, Phys. Rev. E 62, 6078 (2000).ADSCrossRefGoogle Scholar
  64. 64.
    M. Pettini and M. Cerruti-Sola, Phys. Rev. A 41, 44, 975 (1991).ADSCrossRefGoogle Scholar
  65. 65.
    J. DeLuca and A.J. Lichtenberg, Phys. Rev. E 66, 026206 (2002).ADSMathSciNetCrossRefGoogle Scholar
  66. 66.
    A. Ponno, L. Galgani and F. Guerra, Phys. Rev. E 61, 7081 (2000).ADSCrossRefGoogle Scholar
  67. 67.
    N.J. Zabusky and G.S. Deem, J. Comp. Phys. 2, 126 (1967).ADSCrossRefGoogle Scholar
  68. 68.
    N. Budinsky and T. Bountis, Physica D 8, 445 (1983).MATHADSMathSciNetCrossRefGoogle Scholar
  69. 69.
    S. Flach, Physica D 91, 223 (1996).MATHMathSciNetCrossRefGoogle Scholar
  70. 70.
    P. Poggi and S. Ruffo, Physica D 103, 251 (1997).ADSMathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    G.M. Chechin, N.V. Novikova and A.A. Abramenko, Physica D 166, 208 (2002).MATHADSMathSciNetCrossRefGoogle Scholar
  72. 72.
    G.M. Chechin, D.S. Ryabov and K.G. Zhukov, Physica D 203, 121 (2005).MATHADSMathSciNetCrossRefGoogle Scholar
  73. 73.
    B. Rink, Physica D 175, 31 (2003).MATHADSMathSciNetCrossRefGoogle Scholar
  74. 74.
    T. Dauxois, S. Ruffo and A.Torcini, Phys. Rev. E 56, R 6229 (1997).Google Scholar
  75. 75.
    V.M. Burlakov, S.A. Kiselev and V.I. Rupasov Phys. Lett. A 147, 130 (1990).ADSCrossRefGoogle Scholar
  76. 76.
    K.W. Sandusky and J.B. Page, Phys. Rev. B 50, 866 (1994).ADSCrossRefGoogle Scholar
  77. 77.
    T. Dauxois, S. Ruffo and A.Torcini, J. Phys. IV 8, 147 (1998).Google Scholar
  78. 78.
    Yu.S. Kivshar and M. Peyrard, Phys. Rev. A 46, 3198 (1992).ADSCrossRefGoogle Scholar
  79. 79.
    T.B. Benjamin and J.E. Feir, J. Fluid Mech. 27, 417 (1967).MATHADSCrossRefGoogle Scholar
  80. 80.
    V.E. Zakharov and A.B. Shabat Zhurnal Eksperimentalnoi I Teoreticheskoi Fiziki 64, 1627 (1973).Google Scholar
  81. 81.
    G.P. Berman and A.R. Kolovskii, Zh. Eksp. Teor. Fiz. 87, 1938 and Sov. Phys. JETP 60, 1116 (1984).MathSciNetGoogle Scholar
  82. 82.
    A.J. Sievers and S. Takeno, Phys. Rev. Lett. 61, 970 (1988).ADSCrossRefGoogle Scholar
  83. 83.
    V.M. Burlakov and S.A. Kiselev, Sov. Phys. JETP 72, 854 (1991).Google Scholar
  84. 84.
    S. Flach and C.R. Willis, Phys. Rep. 295, 181 (1998).MathSciNetCrossRefADSGoogle Scholar
  85. 85.
    A.I. D’yachenko, V.E. Zackharov, A.N. Pushkarev, V.E. Shvets and V.V. Yan’kov, Sov. Phys. JETP 69, 1144 (1989).Google Scholar
  86. 86.
    O. Bang and M. Peyrard, Phys. Rev. E 53, 4143 (1996).ADSCrossRefGoogle Scholar
  87. 87.
    T. Cretegny, T. Dauxois, S. Ruffo and A. Torcini, Physica D 121, 109 (1998).ADSCrossRefGoogle Scholar
  88. 88.
    K. Ullman, A.J. Litchenberg and E. Corso, Phys. Rev E 61, 2471 (2000).ADSCrossRefGoogle Scholar
  89. 89.
    S. Flach, C.R. Willis and E. Olbrich, Phys. Rev. E 49, 836, (1994).ADSMathSciNetCrossRefGoogle Scholar
  90. 90.
    Yu.A. Kosevich, Phys. Rev. B 47, 3138 (1993).Google Scholar
  91. 91.
    Yu.A. Kosevich and S. Lepri, Phys. Rev. B 61, 299 (2000).ADSCrossRefGoogle Scholar
  92. 92.
    V.V. Mirnov, A.J. Lichtenberg and H. Guclu, Physica D 157, 251 (2001).MATHADSCrossRefGoogle Scholar
  93. 93.
    A.J. Lichtenberg and V.V. Mirnov, Physica D, 202, 116 (2005).MATHADSMathSciNetCrossRefGoogle Scholar
  94. 94.
    A.J. Lichtenberg, V.V. Mirnov and C. Day, Chaos, 15, 015109 (2005).ADSMathSciNetCrossRefGoogle Scholar
  95. 95.
    M.C. Forrest, C.G. Goedde and S. Sinha, Phys. Rev. Lett. 68, 2722 (1992).ADSCrossRefGoogle Scholar
  96. 96.
    J.L. Marin and S. Aubry, Physica D, 119, 163 (1998).MATHADSMathSciNetCrossRefGoogle Scholar
  97. 97.
    S.R. de Groot and P. Mazur, Non-equilibrium thermodynamics, Dover, NY, 1984.Google Scholar
  98. 98.
    P. Debye, Communication at the congress on the kinetic theory of matter. Gottingen, 1913.Google Scholar
  99. 99.
    R.E. Peierls, Quantum theory of solids. Oxford University Press (1955).Google Scholar
  100. 100.
    Z. Rieder, J.L. Lebowitz and E. Lieb, J. Math. Phys. 8, 1073 (1967).ADSCrossRefGoogle Scholar
  101. 101.
    S. Lepri, R. Livi and A. Politi, Phys. Rep. 377, 1 (2003).ADSMathSciNetCrossRefGoogle Scholar
  102. 102.
    D.N. Payton, M. Rich and W.M. Visscher, Phys. Rev. 160, 706 (1967).ADSCrossRefGoogle Scholar
  103. 103.
    E.A. Jackson, J.R. Pasta and J.F. Waters, J. Comput. Phys. 2, 207 (1968).ADSCrossRefGoogle Scholar
  104. 104.
    N. Nakazawa, Progr. Theor. Phys. (Suppl. 45), 231 (1970).ADSCrossRefGoogle Scholar
  105. 105.
    W.M. Visscher, Methods in computational physics Vol. 15, p. 371. Academic Press, New York, 1976.Google Scholar
  106. 106.
    H. Kaburaki and M. Machida, Phys. Lett. A 181, 85 (1993).ADSCrossRefGoogle Scholar
  107. 107.
    Y. Ohtsuboi, N. Nishiguchi and T. Sakuma, J. Phys. Condens. Matter 6, 3013 (1994).ADSCrossRefGoogle Scholar
  108. 108.
    F. Mokross and H. Buttner, J. Phys. C 16, 4539 (1983).ADSCrossRefGoogle Scholar
  109. 109.
    E.A. Jackson and A.D. Mistriotis, J. Phys. Cond. Mat. 1, 1223 (1984).ADSCrossRefGoogle Scholar
  110. 110.
    N. Nishiguchi and T. Sakuma, J. Phys. Condens. Matter 2, 7575 (1990).ADSCrossRefGoogle Scholar
  111. 111.
    O.V. Gendelman and L.I. Manevich, Sov. Phys. JETP 75, 271 (1992).Google Scholar
  112. 112.
    S. Lepri, R. Livi and A. Politi, Phys. Rev. Lett. 78, 1896 (1997).ADSCrossRefGoogle Scholar
  113. 113.
    S. Lepri, R. Livi and A. Politi, Physica D 119, 140 (1998).ADSCrossRefGoogle Scholar
  114. 114.
    S. Lepri, R. Livi and A. Politi, Europhys. Lett. 43, 271 (1998).ADSCrossRefGoogle Scholar
  115. 115.
    S. Lepri, Eur. Phys. J. B 18, 441 (2000).ADSCrossRefGoogle Scholar
  116. 116.
    T. Hatano, Phys. Rev. E 59, R1 (1999).Google Scholar
  117. 117.
    M. Vassalli, Diploma Thesis, University of Florence, 1999.Google Scholar
  118. 118.
    L. Galgani, A. Giorgilli, A. Martinoli and S. Vanzini, Physics D 59, 334 (1992).MATHADSMathSciNetCrossRefGoogle Scholar
  119. 119.
    G. Casati, J. Ford , F. Vivaldi and W.M. Visscher, Phys. Rev. Lett. 52, 1861 (1984).ADSCrossRefGoogle Scholar
  120. 120.
    T. Prozen and D.K. Campbell, Chaos 15, D15117 (2005).ADSCrossRefGoogle Scholar
  121. 121.
    J. De Luca and A.J. Lichtenberg, Phys. Rev. E. 66, 026206 (2002).ADSMathSciNetCrossRefGoogle Scholar
  122. 122.
    D. Sholl, Phys. Lett. A 149, 253 (1990).ADSCrossRefGoogle Scholar
  123. 123.
    R. Livi and A. Vulpiani, (eds), L’ Heritage de Kolmogorov en Physique, Editions Belin, Paris; The Kolmogorv Legacy in Physics Lectures Notes in Physics, Springer, Berlin and New York, 2003.Google Scholar
  124. 124.
    J.P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).ADSMathSciNetCrossRefGoogle Scholar
  125. 125.
    M. Tabor, Chaos and integrability in nonlinear dynamics, John Wiley & Sons. Inc., New York, 1989.MATHGoogle Scholar
  126. 126.
    E. Ott, Chaos in dynamical systems. Cambridge University Press, Cambridge 1993.MATHGoogle Scholar
  127. 127.
    Ya. B. Pesin, Russ. Math. Surveys 32, 55 (1977).Google Scholar
  128. 128.
    R. Livi, A. Politi and S. Ruffo, J. Phys. A 19, 2033 (1986).ADSMathSciNetCrossRefMATHGoogle Scholar
  129. 129.
    J.P. Eckmann and E. Wayne, J. Stat. Phys. 50, 853 (1988).MATHMathSciNetCrossRefADSGoogle Scholar
  130. 130.
    Ya.G. Sinai, Int. J. Bifur. Chaos 6, 1137 (1996).Google Scholar
  131. 131.
    L. Casetti, R. Livi and M. Pettini, Phys. Rev. Lett. 74, 375 (1995).ADSCrossRefGoogle Scholar
  132. 132.
    S. Isola, R. Livi, S. Ruffo and A. Vulpiani, Phys. Rev. A 33, 1163 (1986).ADSMathSciNetCrossRefGoogle Scholar
  133. 133.
    S. Flach, M.V. Ivanchenko and O.I. Kanakov, Phys. Rev. Lett. 95, 064102 (2005).ADSCrossRefGoogle Scholar
  134. 134.
    J. DeLuca, A.J. Lichtenberg and S. Ruffo, Phys. Rev. E 54, 2329 (1996).ADSCrossRefGoogle Scholar
  135. 135.
    A. Ponno and Bambusi D., Energy cascade in Fermi–Pasta–Ulam models, Proceedings of the International Conference on Symmetry and Perturbation Theory 2004, G. Gaeta et al. (eds), World Scientific Publishing, 263–270 and private communication, 2005.Google Scholar
  136. 136.
    J.A. Biello, P.R. Kramer, Y.V. L’vov, Discr. Cont. Dynam. Sys. (Suppl. 113) (2003).Google Scholar
  137. 137.
    L. Berchialla, Galgani L. and Giorgilli A., Discr. Cont. Dynam. Sys. (Suppl., 855), (2005).Google Scholar
  138. 138.
    T. Dauxois, R. Khomeriki, F. Piazza and S. Ruffo, Chaos 15, 015110 (2005).Google Scholar
  139. 139.
    A. Cafarella, M. Leo and R.A. Leo, Phys. Rev. E 69, 046604 (2004).ADSCrossRefGoogle Scholar
  140. 140.
    G.P. Berman, F.M. Izrailev, Chaos, 15, 015104 (2005).Google Scholar
  141. 141.
    L. Caiani, L. Casetti and M. Pettini, J. Phys. A 31, 3357 (1998).MATHADSMathSciNetCrossRefGoogle Scholar
  142. 142.
    M. Pettini, Phys. Rev. E 47, 828 (1993).ADSMathSciNetCrossRefGoogle Scholar
  143. 143.
    M. Cerruti-Sola and M. Pettini, Phys. Rev. E 51, 53 (1995).ADSMathSciNetCrossRefGoogle Scholar
  144. 144.
    M. Cerruti-Sola and M. Pettini, Phys. Rev. E 53, 179 (1996).ADSMathSciNetCrossRefGoogle Scholar
  145. 145.
    A. Abraham and J.E. Marsden, Foundations of mechanics. Addison-Wesley, Redwood City, 1987.Google Scholar
  146. 146.
    M.P. do Carmo, Riemannian geometry. Birkháuser, Boston-Basel, 1993.Google Scholar
  147. 147.
    L.P. Eisenhart, Ann. of Math. 30, 591 (1929).MathSciNetCrossRefGoogle Scholar
  148. 148.
    J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, ed. Springer, New York, 1983.MATHGoogle Scholar
  149. 149.
    M. Pettini and R. Valdettaro, Chaos 5, 646 (1995).MATHADSMathSciNetCrossRefGoogle Scholar
  150. 150.
    N.G. Van Kampen, Phys. Rep. 24, 71 (1976).CrossRefGoogle Scholar
  151. 151.
    R. Livi, M. Pettini, S. Ruffo and A. Vulpiani, J. Stat. Phys. 48, 539 (1987).MATHMathSciNetCrossRefADSGoogle Scholar
  152. 152.
    G. Benettin, L. Galgani, A. Giorgilli and J.M. Strelcyn, Meccanica 15, 9 and 21 (1980).MATHADSCrossRefGoogle Scholar
  153. 153.
    G. Benettin, L. Galgani and A. Giorgilli, Il Nuovo Cimento B 89, 89 (1985).Google Scholar
  154. 154.
    R. Kubo, M. Toda and N. Hashitsume, Statistical physics II. Springer Series in Solid State Sciences, Vol. 31, Springer, Berlin, 1991.Google Scholar
  155. 155.
    K. Aoki and D. Kusnezov, Phys. Rev. Lett. 86, 4029 (2001).ADSCrossRefGoogle Scholar
  156. 156.
    D. Escande, H. Kantz, R. Livi and S. Ruffo, J. Stat. Phys. 76, 605 (1994).MATHMathSciNetCrossRefADSGoogle Scholar
  157. 157.
    C. Giardinàa, R. Livi, A. Politi and M. Vassalli, Phys. Rev. Lett. 84, 2144 (2000).ADSCrossRefGoogle Scholar
  158. 158.
    O.V. Gendelman and A.V. Savin, Phys. Rev. Lett. 84, 2381 (2000).ADSCrossRefGoogle Scholar
  159. 159.
    J. Dawson, Phys. Fluids 5, 445 1962.MATHCrossRefADSGoogle Scholar
  160. 160.
    G. Casati, Found. Phys. 16, 51 (1986).ADSCrossRefGoogle Scholar
  161. 161.
    T. Prozen and M. Robnik, J. Phys. A 25, 3449 (1992).ADSCrossRefGoogle Scholar
  162. 162.
    D.J.R. Mimnagh and L.E. Ballentine, Phys. Rev. E 56, 5332 (1997).ADSCrossRefGoogle Scholar
  163. 163.
    H.A. Posch and W.G. Hoover, Phys. Rev. E 58, 4344 (1998).ADSCrossRefGoogle Scholar
  164. 164.
    M.J. Gillan and R.W. Holloway, J. Phys. C 18, 5705 (1985).ADSCrossRefGoogle Scholar
  165. 165.
    B. Hu, B. Li and H. Zhao, Phys. Rev. E 57, 2992 (1998).ADSCrossRefGoogle Scholar
  166. 166.
    A. Filippov, B. Hu, B. Li and A. Zelser, J. Phys. A 31, 7719 (1998).ADSCrossRefGoogle Scholar
  167. 167.
    B. Hu, B. Li and H. Zhao, Phys. Rev. E 61, 3828 (1999).ADSCrossRefGoogle Scholar
  168. 168.
    K. Aoki and D. Kusnezov, Phys. Lett. A 265, 250 (2000).MATHADSMathSciNetCrossRefGoogle Scholar
  169. 169.
    G.P. Tsironis, A.R. Bishop, A.V. Savin and A.V. Zolotaryuk, Phys. Rev. E 60, 6610 (1999).ADSCrossRefGoogle Scholar
  170. 170.
    G. Tsaur and J. Wang, Phys. Rev. E 54, 4657 (1996).ADSCrossRefGoogle Scholar
  171. 171.
    V.I. Oseledec, Trans. Moscow Math. Soc. 19, 197 (1968).MathSciNetGoogle Scholar
  172. 172.
    J.L. Lebowitz, Percus J.K. and Verlet L., Phys. Rev. 153, 250 (1967).ADSCrossRefGoogle Scholar
  173. 173.
    L. Caiani, L. Casetti, C. Clementi and M. Pettini, Phys. Rev. Lett. 79, 4361 (1997).ADSCrossRefGoogle Scholar
  174. 174.
    L. Caiani, L. Casetti, C. Clementi, G. Pettini, M. Pettini and R. Gatto, Phys. Rev. E 57, 3886 (1998).ADSCrossRefGoogle Scholar
  175. 175.
    M.-C. Firpo, Phys. Rev. E 57, 6599 (1998).ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Allan J. Lichtenberg
    • 1
  • Roberto Livi
    • 2
  • Marco Pettini
    • 3
  • Stefano Ruffo
    • 4
  1. 1.Electrical Engineering and Computer Science DepartmentUniversity of CaliforniaBerkeleyUSA
  2. 2.Dipartimento di Fisica and CSDCUniversità di Firenze via G. Sansone 150019 FirenzeItaly
  3. 3.INAF – Osservatorio Astro.sico di Arcetri Largo Enrico Fermi 550125 FirenzeItaly
  4. 4.Dipartimento di Energetica “S. Stecco” and CSDCUniversità di Firenze and INFN, via s. Marta 350019 FirenzeItaly

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