The fermi-pasta-ulam problem: Fifty years of progress

Chaos

Volumn 15, Issue 1, 2005, Pages

  Berman, G P ^a   Izrailev, F M ^a,b  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

^b BENEMÉRITA UNIVERSIDAD AUTÓNOMA DE PUEBLA   (Mexico)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 17744366868     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1855036     Document Type: Review

References (143)

1. E. Fermi, J. Pasta, and S. Ulam, Studies of the Nonlinear Problems, I, Los Alamos Report LA-1940, (1955); later published in Collected Papers of Enrico Fermi, edited by E. Segre, (University of Chicago Press, 1965), Vol. II, p. 978; also reprinted in Nonlinear Wave Motion, edited by A. C. Newell, Lecture Notes in Applied Mathematics (AMS, Providence, RI, 1974), Vol. 15; also in Many-Body Problems, edited by D. C. Mattis (World Scientific, Singapore, 1993).
2. E. Fermi, J. Pasta, and S. Ulam, Studies of the Nonlinear Problems, I, Los Alamos Report LA-1940, (1955); later published in Collected Papers of Enrico Fermi, edited by E. Segre, (University of Chicago Press, 1965), Vol. II, p. 978; also reprinted in Nonlinear Wave Motion, edited by A. C. Newell, Lecture Notes in Applied Mathematics (AMS, Providence, RI, 1974), Vol. 15; also in Many-Body Problems, edited by D. C. Mattis (World Scientific, Singapore, 1993).
3. E. Fermi, J. Pasta, and S. Ulam, Studies of the Nonlinear Problems, I, Los Alamos Report LA-1940, (1955); later published in Collected Papers of Enrico Fermi, edited by E. Segre, (University of Chicago Press, 1965), Vol. II, p. 978; also reprinted in Nonlinear Wave Motion, edited by A. C. Newell, Lecture Notes in Applied Mathematics (AMS, Providence, RI, 1974), Vol. 15; also in Many-Body Problems, edited by D. C. Mattis (World Scientific, Singapore, 1993).
4. E. Fermi, J. Pasta, and S. Ulam, Studies of the Nonlinear Problems, I, Los Alamos Report LA-1940, (1955); later published in Collected Papers of Enrico Fermi, edited by E. Segre, (University of Chicago Press, 1965), Vol. II, p. 978; also reprinted in Nonlinear Wave Motion, edited by A. C. Newell, Lecture Notes in Applied Mathematics (AMS, Providence, RI, 1974), Vol. 15; also in Many-Body Problems, edited by D. C. Mattis (World Scientific, Singapore, 1993).
5. E. Fermi, "Beweis dass ein mechanisches Normalsystem im allgemeinen quasi-ergodisch ist," Phys. Z. 24, 261 (1923).
6. J. L. Tuck, in Los Alamos Report LA-3990, 1968.
7. N. J. Zabusky and M. D. Kruskal, "Interaction of "solitons" in a collisionless plasma and the reccurence of initial states," Phys. Rev. Lett. 15, 240-243 (1965).
8. B. V. Chirikov, Atomnaya Energia 6, 630 (1959) [J. Nucl. Energy, Part C 1, 253 (1960)].
9. B. V. Chirikov, Atomnaya Energia 6, 630 (1959) [J. Nucl. Energy, Part C 1, 253 (1960)].
10. F. M. Izrailev and B. V. Chirikov, Statistical Properties of a Nonlinear String (Institute of Nuclear Physics, Novosibirsk, USSR, 1965) (in Russian); Dokl. Akad. Nauk SSSR 166, 57 (1966) [Sov. Phys. Dokl. 11, 30 (1966)].
11. F. M. Izrailev and B. V. Chirikov, Statistical Properties of a Nonlinear String (Institute of Nuclear Physics, Novosibirsk, USSR, 1965) (in Russian); Dokl. Akad. Nauk SSSR 166, 57 (1966) [Sov. Phys. Dokl. 11, 30 (1966)].
12. F. M. Izrailev and B. V. Chirikov, Statistical Properties of a Nonlinear String (Institute of Nuclear Physics, Novosibirsk, USSR, 1965) (in Russian); Dokl. Akad. Nauk SSSR 166, 57 (1966) [Sov. Phys. Dokl. 11, 30 (1966)].
13. F. M. Izrailev, A. I. Khisamutdinov, and B. V. Chirikov, "Numerical experiments with a chain of coupled anharmonic oscillators," Report 252, Institute of Nuclear Physics, Novosibirsk, USSR, 1968 (English translation: LA-4440-TR, Los Alamos, 1970).
14. S. M. Ulam, A Collection of Mathematical Problems (Interscience, New York, 1960).
15. N. J. Zabusky, "Computational synergetics and mathematical innovation," J. Comput. Phys. 43, 195-249 (1981).
16. T. P. Weissert, The Genesis of Simulation in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem (Springer-Verlag, New York, 1997).
17. I. Prigogine, Non-Equilibrium Statistical Mechanics (Interscience, New York, 1962).
18. J. Ford, "The Fermi-Pasta-Ulam problem: Paradox turns discovery," Phys. Rep. 213, 271 (1992).
19. J. L. Tuck and M. T. Menzel, "The superperiod of the nonlinear weighted string (FPU) problem," Adv. Math. 9, 399 (1972).
20. P. C. Hemmer, L. C. Maximon, and H. Wergeland, "Recurrence time of a dynamical system," Phys. Rev. 111, 689 (1958).
21. E. A. Jackson, "Nonlinearity and irreversibility in lattice dynamics," Rocky Mt. J. Math. 8, 12-197 (1978).
22. B. V. Chirikov, "Research concerning the theory of nonlinear resonances and stochasticity," Report 267, Institute of Nuclear Physics, Novosibirsk, USSR, 1969 (English translation: Report 71-40, CERN, Geneva, Switzerland, 1971).
23. B. V. Chirikov, "A universal instability of many-dimensional oscillator systems," Phys. Rep. 52, 263-379 (1979).
24. J. Ford, "Equipartition of energy for nonlinear systems," J. Math. Phys. 2, 387-393 (1961).
25. N. Kryloff and N. Bogolluboff, Introduction to Nonlinear Mechanics (Princeton University Press, Princeton, NJ, 1947).
26. E. A. Jackson, "Nonlinear coupled oscillators. U. Comparison of theory with computer solutions," J. Math. Phys. 4, 686-700 (1963).
27. E. A. Jackson, "Nonlinear coupled oscillators. I. Perturbation theory: Ergodic problem," J. Math. Phys. 4, 551-558 (1963).
28. N. J. Zabusky, "Exact solution for the vibrations of a nonlinear continuous model string," J. Math. Phys. 3, 1028-1039 (1962).
29. J. Ford and J. Waters, "Computer studies of energy sharing and ergodicity for nonlinear oscillator systems," J. Math. Phys. 4, 1293-1306 (1963).
30. A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 98, 527 (1954); "General theory of dynamical systems and classical mechanics," in Proceeding of the International Congress of Mathematics, Amsterdam, Vol. I, p. 315.
31. A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 98, 527 (1954); "General theory of dynamical systems and classical mechanics," in Proceeding of the International Congress of Mathematics, Amsterdam, Vol. I, p. 315.
32. V. I. Arnold, "A proof of the A. N. Kolmogorov's theorem on the conservation of conditional-periodic motions in a small change of the Hamiltonian function," Usp. Mat. Nauk 18, 13-40 (1963).
33. J. Moser, Nachr. Akad. Wiss. Goett. II, Math.-Phys. Kl. 1, 1 (1962); Annali della Scuola Normale Superiore di Pisa, Serie III Vol. 20, Fasc. III (1966), p. 499; Stable and Random Motion in Dynamical Systems, Annals of Math. Studies, No. 77 (University Press, Princeton, 1973).
34. J. Moser, Nachr. Akad. Wiss. Goett. II, Math.-Phys. Kl. 1, 1 (1962); Annali della Scuola Normale Superiore di Pisa, Serie III Vol. 20, Fasc. III (1966), p. 499; Stable and Random Motion in Dynamical Systems, Annals of Math. Studies, No. 77 (University Press, Princeton, 1973).
35. J. Moser, Nachr. Akad. Wiss. Goett. II, Math.-Phys. Kl. 1, 1 (1962); Annali della Scuola Normale Superiore di Pisa, Serie III Vol. 20, Fasc. III (1966), p. 499; Stable and Random Motion in Dynamical Systems, Annals of Math. Studies, No. 77 (University Press, Princeton, 1973).
36. V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, (Benjamin, New York, 1968).
37. T. Nishida, "A note on an existence of conditionally periodic ocsillation in a one-dimensional anharmonic lattice," Mem. Fac. Eng., Kyoto Univ. 33, 27-34 (1971).
38. B. Rink and F. Verhulst, "Near-integrability of periodic FPU-chains," Physica A 285, 467-482 (2000).
39. B. Rink, "Symmetry and resonance in periodic FPU chains," Commun. Math. Phys. 218, 665-685 (2001).
40. R. S. Northcote and R. B. Potts, "Energy sharing and equilibrium for nonlinear systems," J. Math. Phys. 5, 383-398 (1964).
41. Ya. G. Sinai, in Statistical Mechanics: Foundations and Application, Proceedings of the IUPAP meeting, Copenhagen, 1966, edited by T. A. Bak (Benjamin, New York, 1967), p. 559.
42. M. D. Kruskal and N. J. Zabusky, "Stroboscopic-perturbation procedure for treating a class of nonlinear wave equations," J. Math. Phys. 5, 231244 (1964).
43. M. Toda, "Studies of a non-linear lattices," Phys. Rep. 18C, 1-125 (1975).
44. D. J. Korteweg and G. de Vries, Philos. Mag. 39, 422 (1895).
45. N. J. Zabusky, "Solitons and bound states of the time-independent Schrödinger equation," Phys. Rev. 168, 124-128 (1968).
46. N. J. Zabusky, in Proceedings of the Symposium on Nonlinear Partial Differential Equations, edited by W. Ames (Academic, New York, 1967).
47. P. D. Lax, Commun. Pure Appl. Math. 21, 467 (1968).
48. Yu. A. Berezin and V. I. Karpman, Zh. Eksp. Teor. Fiz. 51, 1557 (1966) [Sov. Phys. JETP 24, 1049 (1967)].
49. Yu. A. Berezin and V. I. Karpman, Zh. Eksp. Teor. Fiz. 51, 1557 (1966) [Sov. Phys. JETP 24, 1049 (1967)].
50. R. M. Miura, "Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation," J. Math. Phys. 9, 1202-1204 (1968).
51. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, "Method for solving the Korteweg-de Vries equation," Phys. Rev. Lett. 19, 10951097 (1967).
52. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, "Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion," J. Math. Phys. 9, 1204-1209 (1968).
53. M. Toda, "Vibration of a chain with nonlinear interaction," J. Phys. Soc. Jpn. 22, 431 (1967).
54. M. Toda, J. Phys. Soc. Jpn. 26, 235 (1969); Prog. Theor. Phys. Suppl. 45, 174 (1970).
55. M. Toda, J. Phys. Soc. Jpn. 26, 235 (1969); Prog. Theor. Phys. Suppl. 45, 174 (1970).
56. J. Ford, S. D. Stoddard, and J. S. Turner, "On the integrability of the Toda lattice," Prog. Theor. Phys. 50, 1547-1560 (1973).
57. M. Hénon, 'Integrals of the Toda lattice," Phys. Rev. B 9, 1921-1923 (1974).
58. N. S. Krylov, Studies on the Foundation of Statistical Physics (Akad. Nauk SSSR, M.-L., 1950) [Engl. Transl., Princeton University Press, Princeton, NJ (1979)].
59. A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983).
60. K. R. Symon and A. M. Sessler, in Proceedings of the CERN Symposium on High Energy Accelerators and Pion Physics, Geneva 1956 (CERN, Geneva, 1956), p. 44, Vol. II.
61. G. N. Kulipanov, S. I. Mishnev, S. G. Popov, and G. M. Tumaikin, "Influence of nonlinearities on betatron oscillations in storage rings," Report 251, Institute of Nuclear Physics, Novosibirsk, USSR, 1968 (in Russian).
62. G. N. Kulipanov, S. I. Mishnev, and A. N. Skrinsky, "Study of the stochastic instability of betatron oscillations of electron beam in a storage ring," Report 323, Institute of Nuclear Physics, Novosibirsk, USSR, 1969 (in Russian).
63. F. M. Izrailev, "Investigation of chaotic oscillations for the string with quadratic nonlinearity," Report 77, Institute of Nuclear Physics, Novosibirsk, USSR, 1966 (in Russian).
64. F. M. Izrailev and V. A. Tayursky, "Numerical study of stochastic waves in a chain of coupled nonlinear oscillators," Report 78-70, Institute of Nuclear Physics, Novosibirsk, SSSR, 1970).
65. B. V. Chirikov, F. M. Izrailev, and V. A. Tayursky, "Numerical experiments on the statistical behaviour of dynamical systems with a few degrees of freedom," Comput. Phys. Commun. 5, 11-16 (1973).
66. R. L. Bivins, N. Metropolis, and J. R. Pasta, "Nonlinear coupled oscillators: Modal equation approach," J. Comput. Phys. 12, 65 (1973).
67. B. V. Chirikov, "When does the dynamical system turn into the statistical one?," Communication at the International Congress of Mathematicians, Moscow, 1966; Report of the Institute of Nuclear Physics, Novosibirsk, Russia, 1966.
68. H. Hirooka and N. Saito, "Computer studies on the approach to thermal equilibrium in coupled anharmonic oscillators. I. Two dimensional case," J. Phys. Soc. Jpn. 26, 624-630 (1969).
69. N. Ooyama, H. Hirooka, and N. Saito, "Computer studies on the approach to thermal equilibrium in coupled anharmonic oscillators. II. One-dimensional case," J. Phys. Soc. Jpn. 27, 815-824 (1969).
70. N. Saito and H. Hirooka, "Computer studies of ergodicity in coupled oscillators with anharmonic interaction," J. Phys. Soc. Jpn. 23, 167-171 (1967).
71. R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione, and A. Vulpiani, "Relaxation to different stationary states in the Fermi-Pasta-Ulam model," Phys. Rev. A 28, 3544-3552 (1983).
72. G. Benettin and A. Tenenbaum, "Ordered and stochastic behaviour in a two-dimensional Lennard-Jones system," Phys. Rev. A 28, 3020-3029 (1983).
73. R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione, and A, Vulpiani, "Equipartiion threshold in nonlinear large Hamiltonian systems: The Fermi-Pasta-Ulam model," Phys. Rev. A 31, 1039-1045 (1985).
74. R. Livi, M. Pettini, S. Ruffo, and A. Vulpiani, "Further results on the equipartition threshold in large nonlinear Hamiltonian systems," Phys. Rev. A 31, 2740-2742 (1985).
75. S. Isola, R. Livi, S. Ruffo, and A. Vulpiani, "Stability and chaos in Hamiltonian dynamics," Phys. Rev. A 33, 1163-1170 (1986).
76. N. Saito, N. Ooyama, Y. Aizawa, and H. Hirooka, Prog. Theor. Phys. Suppl. 45, 209 (1970).
77. C. F. Driscoll and T. M. O'Neil, "Explanation of instabilities observed on a Fermi-Pasta-Ulam lattice," Phys. Rev. Lett 37, 69 (1976).
78. N. Budinsky and T. Bountis, "Stability of nonlinear modes and chaotic properties of ID Fermi-Pasta-Ulam lattices," Physica D, 8, 445-452 (1983).
79. S. Flach, "Tangent bifurcation of band edge plane waves, dynamical symmetry breaking and vibrational localization," Physica D 91, 223-243 (1996).
80. G. P. Berman and A. R. Kolovsky, "The limit of stochasticity for a one-dimensional chain of interacting oscillators," Zh. Eksp. Teor. Fiz. 87, 1938 (1984); [Sov. Phys. JETP 60, 1116 (1984)].
81. G. P. Berman and A. R. Kolovsky, "The limit of stochasticity for a one-dimensional chain of interacting oscillators," Zh. Eksp. Teor. Fiz. 87, 1938 (1984); [Sov. Phys. JETP 60, 1116 (1984)].
82. P. Poggi and S. Ruffo, "Exact solutions in the FPU oscillator chain," Physica D 103, 251 (1997).
83. K. W. Sandusky and J. B. Page, "Interrelation between the stability of extended normal modes and the existence of intrinsic localized modes in nonlinear lattices with realistic potentials," Phys. Rev. B 50, 866-887 (1994).
84. Yu. A. Kosevich and S. Lepri, "Modulational instability and energy localization in anharmonic lattices at finite energy density," Phys. Rev. B 61, 299-307 (2000).
85. V. M. Burlakov, S. A. Kiselev, and V. I. Rupasov, "Localized vibrations of homogeneous anharmonic chains," Phys. Lett. A 147, 130-134 (1990).
86. B. Rink, "Direction-reversing traveling waves in the even Fermi-Pasta-Ulam lattice," J. Nonlinear Sci. 12, 479-504 (2002).
87. G. Benettin, L. Galgani, and J. M. Strelcyn, "Kolmogorov entropy and numerical experiments," Phys. Rev. A 14, 2338-2345 (1976).
88. M. Casartelli, E. Diana, L. Galgani, and A. Scotti, "Numerical computations on a stochastic parameter related to the Kolmogorov entropy," Phys. Rev. A 13, 1921-1925 (1976).
89. P. Bocchieri, A. Scotti, B. Bearzi, and A. Loinger, "Anharmonic chain with Lennard-Jones interaction," Phys. Rev. A 2, 2013-2019 (1970).
90. M. C. Carotta, C. Ferrario, G. Lo Vecchio, and L. Galgani, "New phenomenon in the stochastic transition of coupled oscillators," Phys. Rev. A 17, 786-794 (1978).
91. J. P. Eckmann and C. E. Wayne, J. Stat. Phys. 50, 853 (1988).
92. L. Casetti, R. Livi, and M. Pettini, "Gaussian model for chaotic instability of Hamiltonian Flows," Phys. Rev. Lett. 74, 375 (1995).
93. M. Pettini and M. Landolfi, "Relaxation properties and ergodicity breaking in nonlinear Hamiltonian dynamics," Phys. Rev. A 41, 768 (1990).
94. T. Dauxois, S. Ruffo, and A. Torcini, "Modulational estimate for the maximal Lyapunov exponent in Fermi-Pasta-Ulam chains," Phys. Rev. E 56, R6229 (1997).
95. G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, C.R. Seances Acad. Sci., Ser. A, 286, 431 (1978).
96. G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, Meccanica 9, 21 (1980).
97. R. Livi, A. Politi, and S. Ruffo, "Distribution of characteristic exponents in the thermodynamic limit," J. Phys. A 19, 2033 (1986).
98. L. Casetti, M. Cerruti-Sola, M. Pettini, and E. G. D. Cohen, "The Fermi-Pasta-Ulam problem revisited: Stochasticity thresholds in nonlinear Hamiltonian systems," Phys. Rev. E 55, 6566-6574 (1997).
99. V. E. Zakharov, "On the Fermi-Ulam-problem," Report 31-73, Institute of Nuclear Physics, Novosibirsk, USSR, 1973.
100. H. Kantz, R. Livi, and S. Ruffo, "Equipartition thresholds in chains of anharmonic oscillators," J. Stat. Phys. 76, 627-643 (1994).
101. J. Ford and G. H. Lunsford, "Stochastic behavior of resonant nearly linear oscillator systems in the limit of zero nonlinear coupling," Phys. Rev. A 1, 59-70 (1970).
102. F. M. Izraelev, "Nearly linear mappings and their applications," Physica D 1, 243-266 (1980).
103. D. L. Shepelyansky, "Low-energy chaos in the Fermi-Pasta-Ulam problem," Nonlinearity 10, 1331-1338 (1997).
104. H. Kantz, "Vanishing stability thresholds in the thermodynamic limit of nonintegrable conservative systems," Physica D 39, 322-335 (1989).
105. J. De Luca, A. J. Lichtenberg, and S. Ruffo, "Finite times to equipartition in the thermodynamic limit," Phys. Rev. E 60, 3781 (1999).
106. L. Galgani, A. Giorgilli, A. Martinoli, and S. Vanzini, "On the problem of energy equipartition for large systems of the Fermi-Pasta-Ulam type: analytical and numerical estimates," Physica D 59, 334 (1992).
107. N. N. Nekhoroshev, "Exponential estimate of the stability time of near-integrable Hamiltonian systems," Russ. Math. Surveys 32, 1-65 (1977).
108. N. N. Nekhoroshev, "Exponential estimate of the stability time of near-integrable Hamiltonian systems, II," Trudy Sem. Petrovs. No. 5 1979, p. 5-50 (in Russian).
109. G. Benettin, L. Galgani, and A. Giorgilli, "A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems," Celest. Mech. 37, 1-25 (1985).
110. V. I. Arnold, "On the nonstability of dynamical systems with many degrees of freedom," Dokl. Akad. Nauk SSSR 156, 9-12 (1964).
111. M. Pettini and M. Cerruti-Sola, "Strong Stochasticity threshold in nonlinear large Hamiltonian systems: Effect on mixing times," Phys. Rev. A 44, 975 (1991).
112. J. De Luca, A. J. Lichtenberg, and M. A. Lieberman, "Time scale to ergodicity hi the Fermi-Pasta-Ulam system," Chaos 5, 283 (1995).
113. J. De Luca, A. J. Lichtenberg, and S. Ruffo, "Energy transitions and time scales to equipartition in the Fermi-Pasta-Ulam oscillator chain," Phys. Rev. E 51, 2877 (1995).
114. J. De Luca, A. J. Lichtenberg, and S. Ruffo, "Universal evolution to equilibrium in oscillator chains," Phys. Rev. E 54, 2329 (1996).
115. K. Ullmann, A. J. Lichtenberg, and G. Corso, "Energy equaipartition starting from high-frequency modes in the Fermi-Pasta-Ulam beta-oscillator chain," Phys. Rev. E 61, 2471-2477 (2000).
116. R. Z. Sagdeev, D. A. Usikov, and G. M. Zaslavsky, Nonlinear Physics: From the Pendulum to Turbulence and Chaos, Chur, Switzerland (Harvard Academic, New York, 1988).
117. E. P. Gross, Nuovo Cimento 20, 454 (1961).
118. L. P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961).
119. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, New York, 1982).
120. P. Villain and M. Lewenstein, "Fermi-Pasta-Ulam problem revisited with a Bose-Einstein condensate," Phys. Rev. A 62, 043601 (2000). 109 Proceedings of the Nobel Symposia, "Quantum Chaos Y2K," Phys. Scr. T90 (2001).
121. P. Villain and M. Lewenstein, "Fermi-Pasta-Ulam problem revisited with a Bose-Einstein condensate," Phys. Rev. A 62, 043601 (2000). 109 Proceedings of the Nobel Symposia, "Quantum Chaos Y2K," Phys. Scr. T90 (2001).
122. G. P. Berman, A. M. Iomin, A. R. Kolovsky, and N. N. Tarkhanov, "On the dynamics of the four-wave interactions in a nonlinear quantum chain," Report N-377 F, Institute of Physics, Krasnoyarsk, 1986.
123. G. P. Berman and N. N. Tarkhanov, "Quantum dyanmics in the Fermi-Pasta-Ulam problem," Report 2004/05, Universität Potsdam, 2004.
124. G. Casati, B. V. Chirikov, F. M. Izraelev, and J. Ford, "Stochastic behavior of a quantum pendulum under a periodic perturbation," Lect. Notes Phys. 93, 334-352 (1979).
125. B. V. Chirikov, F. M. Izrailev, and D. L. Shepelyansky, "Dynamical stochasticity in classical and quantum mechanics," Sov. Sci. Rev. 2C, 209-267 (1981).
126. F. M. Izrailev, "Simple models of quantum chaos: spectrum and eigen-functions," Phys. Rep. 196, 299-393 (1990).
127. G. P. Berman, A. Smerzi, and A. R. Bishop, "Quantum instability of a Bose-Einstein condensate with attractive interaction," Phys. Rev. Lett. 88, 120402 (2002).
128. G. P. Berman and G. M. Zaslavsky, "Stochastic instability in a nonlinear quantum oscillator," Dokl. Akad. Nauk SSSR 240, 1082-1085 (1978).
129. G. P. Berman and G. M. Zaslavsky, "Condition of stochasticity in quantum non-linear systems," Physica A 91, 450-460 (1978).
130. G. P. Berman and M. Vishik, "Long time evolution of quantum averages near stationary points," Phys. Lett. A 319, 352-359 (2003).
131. G. P. Berman, F. Borgonovi, F. M. Izrailev, and A. Smerzi, "Irregular dynamics in a one-dimensional Bose system," Phys. Rev. Lett. 92, 030404 (2004).
132. S. Flach and C. R. Willis, "Discrete breathers," Phys. Rep. 295, 181-264 (1998).
133. N. J. Zabusky and G. S. Deem, "Dynamics of nonlinear lattices I. Localized optical excitations, acoustic radiation and strong nonlinear behavior," J. Comput. Phys. 2, 126-153 (1967).
134. P. Maniadis, A. V. Zolotaryuk, and G. P. Tsironis, "Existence and stability of discrete gap breathers in a diatomic β Fermi-Pasta-Ulam chain," Phys. Rev. E 67, 046612 (2003).
135. A. J. Sievers and S. Takeno, "Intrinsic localized modes in anharmonic crystals," Phys. Rev. Lett. 61, 970-973 (1988).
136. K. W. Sandusky, J. B. Pade, and K. E. Schmidt, "Stability and motion of intrinsic localized modes in nonlinear periodic lattices," Phys. Rev. B 46, 6161-6168 (1992).
137. G. James, "Existence of breathers on FPU lattices," Acad. Sci., Paris, C. R. 332, 581 (2001).
138. G. P. Tsironis and S. Aubry, "Slow relaxation phenomena induced by breathers in nonlinear lattices," Phys. Rev. Lett. 77, 5225-5228 (1996).
139. A. Bikaki, N. K. Voulgarakis, S. Aubry, and G. P. Tsironis, "Energy relaxation in discrete nonlinear lattices," Phys. Rev. E 59, 1234-1237 (1999).
140. T. Cretegny, T. Dauxois, S. Ruffo, and A. Torcini, "Localization and equipartition of energy in the beta-FPU chain: Chaotic breathers," Physica D 121, 109 (1998).
141. E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell, and C. E. Wieman, "Dynamics of collapse and exploiding Bose-Einsten condensates," Nature (London) 412, 295-299 (2001).
142. B. V. Chirikov, "Linear and nonlinear dynamical chaos," Open Syst. Inf. Dyn. 4, 241-280 (1997).
143. V. V. Flambaum and F. M. Izrailev, "Statistical theory of finite fermi-systems based on the structure of chaotic eigenstates," Phys. Rev. E 56, 5144-5159 (1997).