Normal Modes, Symmetries and Stability

  • Tassos Bountis
  • Haris Skokos
Chapter
Part of the Springer Series in Synergetics book series (SSSYN, volume 10)

Abstract

The present Chapter studies nonlinear normal modes (NNMs) of coupled oscillators from an altogether different perspective. Focusing entirely on periodic boundary conditions and using the Fermi Pasta Ulam β (FPU − β) and FPU − α models as examples, we demonstrate the importance of discrete symmetries in locating and analyzing exactly a class of NNMs called one-dimensional “bushes”, depending on a single periodic function \(\hat{q}(t)\). Using group theoretical arguments one can similarly identify n-dimensional bushes described by \(\hat{{q}}_{1}(t),\ldots,\hat{{q}}_{n}(t)\), which represent quasiperiodic orbits characterized by n incommensurate frequencies. Expressing these solutions as linear combinations of single bushes, it is possible to simplify the linearized equations about them and study their stability analytically. We emphasize that these results are not limited to monoatomic particle chains, but can apply to more complicated molecular structures in two and three spatial dimensions, of interest to solid state physics.

Keywords

Symmetry Group Normal Mode Vibrational State Discrete Symmetry Modal Space 
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.

References

  1. 1.
    F. Abdullaev, O. Bang, M.P. Sørensen (eds.), Nonlinearity and Disorder: Theory and Applications. NATO Science Series II: Mathematics, Physics and Chemistry, vol. 45 (Springer, Heidelberg, 2002)Google Scholar
  2. 2.
    M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series, vol. 149 (Cambridge University Press, Cambridge, 1991)Google Scholar
  3. 3.
    M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981)MATHGoogle Scholar
  4. 4.
    M.J. Ablowitz, B. Prinari, A.D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems (Cambridge University Press, Cambridge, 2004)MATHGoogle Scholar
  5. 5.
    E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Scaling theory of localization: absence of quantum diffusion in two dimensions. Phys. Rev. Lett. 42, 673–676 (1979)ADSGoogle Scholar
  6. 6.
    M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965)Google Scholar
  7. 7.
    O. Afsar, U. Tirnakli, Probability densities for the sums of iterates of the sine-circle map in the vicinity of the quasiperiodic edge of chaos. Phys. Rev. E 82, 046210 (2010)ADSGoogle Scholar
  8. 8.
    Y. Aizawa, Symbolic dynamics approach to the two-dimensional chaos in area-preserving maps. Prog. Theor. Phys. 71, 1419–1421 (1984)MathSciNetADSMATHGoogle Scholar
  9. 9.
    D. Alonso, R. Artuso, G. Casati, I. Guarneri, Heat conductivity and dynamical instability. Phys. Rev. Lett. 82, 1859–1862 (1999)ADSGoogle Scholar
  10. 10.
    P.W. Anderson, Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958)ADSGoogle Scholar
  11. 11.
    D.V. Anosov, Geodesic flows on a compact Riemann manifold of negative curvature. Trudy Mat. Inst. Steklov 90, 3–210 (1967). English translation, Proc. Steklov Math. Inst. 90, 3–210 (1967)Google Scholar
  12. 12.
    D.V. Anosov, Y.G. Sinai, Some smooth Ergodic systems. Russ. Math. Surv. 22(5), 103–167 (1967)MathSciNetGoogle Scholar
  13. 13.
    Ch. Antonopoulos, T. Bountis, Stability of simple periodic orbits and chaos in a Fermi-Pasta-Ulam lattice. Phys. Rev. E 73, 056206 (2006)MathSciNetADSGoogle Scholar
  14. 14.
    Ch. Antonopoulos, T. Bountis, Detecting order and chaos by the linear dependence index (LDI) method. ROMAI J. 2, 1–13 (2006)MathSciNetMATHGoogle Scholar
  15. 15.
    Ch. Antonopoulos, H. Christodoulidi, Weak chaos detection in the Fermi-Pasta-Ulam-α system using q-Gaussian statistics. Int. J. Bifurc. Chaos 21, 2285–2296 (2011)Google Scholar
  16. 16.
    Ch. Antonopoulos, T.C. Bountis, Ch. Skokos, Chaotic dynamics of N-degree of freedom Hamiltonian systems. Int. J. Bifurc. Chaos 16, 1777–1793 (2006)MathSciNetMATHGoogle Scholar
  17. 17.
    Ch. Antonopoulos, V. Basios, T. Bountis, Weak chaos and the “melting transition” in a confined microplasma system. Phys. Rev. E. 81, 016211 (2010)ADSGoogle Scholar
  18. 18.
    Ch. Antonopoulos, T. Bountis, V. Basios, Quasi-stationary chaotic states of multidimensional Hamiltonian systems. Phys. A 390, 3290–3307 (2011)MathSciNetGoogle Scholar
  19. 19.
    V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1989)Google Scholar
  20. 20.
    V.I. Arnold, A. Avez, Problèmes Ergodiques de la Mécanique Classique (Gauthier-Villars, Paris, 1967 / Benjamin, New York, 1968)Google Scholar
  21. 21.
    S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization. Phys. D 103, 201–250 (1997)MathSciNetMATHGoogle Scholar
  22. 22.
    F. Baldovin, E. Brigatti, C. Tsallis, Quasi-stationary states in low-dimensional Hamiltonian systems. Phys. Lett. A 320, 254–260 (2004)MathSciNetADSMATHGoogle Scholar
  23. 23.
    F. Baldovin, L.G. Moyano, A.P. Majtey, A. Robledo, C. Tsallis, Ubiquity of metastable-to-stable crossover in weakly chaotic dynamical systems. Phys. A 340, 205–218 (2004)MathSciNetGoogle Scholar
  24. 24.
    D. Bambusi, A. Ponno, On metastability in FPU. Commun. Math. Phys. 264, 539–561 (2006)MathSciNetADSMATHGoogle Scholar
  25. 25.
    D. Bambusi, A. Ponno, Resonance, Metastability and Blow Up in FPU. Lecture Notes in Physics, vol. 728 (Springer, New York/Berlin, 2008), pp. 191–205Google Scholar
  26. 26.
    R. Barrio, Sensitivity tools vs. Poincaré sections. Chaos Soliton Fract. 25, 711–726 (2005)MathSciNetADSMATHGoogle Scholar
  27. 27.
    R. Barrio, Painting chaos: a gallery of sensitivity plots of classical problems. Int. J. Bifurc. Chaos 16, 2777–2798 (2006)MathSciNetMATHGoogle Scholar
  28. 28.
    R. Barrio, W. Borczyk, S. Breiter, Spurious structures in chaos indicators maps. Chaos Soliton Fract. 40, 1697–1714 (2009)MathSciNetADSMATHGoogle Scholar
  29. 29.
    C. Beck, Brownian motion from deterministic dynamics. Phys. A 169, 324–336 (1990)MathSciNetGoogle Scholar
  30. 30.
    G. Benettin, A. Ponno, Time-scales to equipartition in the Fermi-Pasta-Ulam problem: finite size effects and thermodynamic limit. J. Stat. Phys. 144, 793–812 (2011)MathSciNetADSMATHGoogle Scholar
  31. 31.
    G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica 15, 9–20 (1980)Google Scholar
  32. 32.
    G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 2: Numerical application. Meccanica 15, 21–30 (1980)Google Scholar
  33. 33.
    G. Benettin, L. Galgani, A. Giorgilli, A proof of Nekhoroshev’s theorem for the stability times in nearly integrable Hamiltonian systems. Celest. Mech. 37, 1–25 (1985)MathSciNetADSMATHGoogle Scholar
  34. 34.
    G. Benettin, A. Carati, L. Galgani, A. Giorgilli, The Fermi-Pasta-Ulam problem and the metastability perspective. Lecture Notes in Physics, vol. 728 (Springer, New York/Berlin, 2008), pp. 151–189Google Scholar
  35. 35.
    G. Benettin, R. Livi, A. Ponno, The Fermi-Pasta-Ulam problem: scaling laws vs. initial conditions. J. Stat. Phys. 135, 873–893 (2009)Google Scholar
  36. 36.
    D. Benisti, D.F. Escande, Nonstandard diffusion properties of the standard map. Phys. Rev. Lett. 80, 4871–4874 (1998)MathSciNetADSMATHGoogle Scholar
  37. 37.
    L. Berchialla, A. Giorgilli, S. Paleari, Exponentially long times to equipartition in the thermodynamic limit. Phys. Lett. A, 321, 167–172 (2004)ADSMATHGoogle Scholar
  38. 38.
    L. Berchialla, L. Galgani, A. Giorgilli, Localization of energy in FPU chains. Discret. Contin. Dyn. Syst. 11, 855–866 (2004)MathSciNetMATHGoogle Scholar
  39. 39.
    J.M. Bergamin, Numerical approximation of breathers in lattices with nearest-neighbor interactions, Phys. Rev. E 67, 026703 (2003)MathSciNetADSGoogle Scholar
  40. 40.
    J.M. Bergamin, Localization in nonlinear lattices and homoclinic dynamics. Ph.D. Thesis, University of Patras, 2003Google Scholar
  41. 41.
    J.M. Bergamin, T. Bountis, C. Jung, A method for locating symmetric homoclinic orbits using symbolic dynamics. J. Phys. A-Math. Gen. 33, 8059–8070 (2000)MathSciNetADSMATHGoogle Scholar
  42. 42.
    J.M. Bergamin, T. Bountis, M.N. Vrahatis, Homoclinic orbits of invertible maps. Nonlinearity 15, 1603–1619 (2002)MathSciNetMATHGoogle Scholar
  43. 43.
    G.P. Berman, F.M. Izrailev, The Fermi-Pasta-Ulam problem: fifty years of progress. Chaos 15, 015104 (2005)MathSciNetADSGoogle Scholar
  44. 44.
    P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968)MATHGoogle Scholar
  45. 45.
    J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht,  P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, A. Aspect, Direct observation of Anderson localization of matter-waves in a controlled disorder. Nature 453, 891–894 (2008)ADSGoogle Scholar
  46. 46.
    G. Birkhoff, G.-C. Rota, Ordinary Differential Equations (Wiley, New York, 1978)MATHGoogle Scholar
  47. 47.
    J.D. Bodyfelt, T.V. Laptyeva, Ch. Skokos, D.O. Krimer, S. Flach, Nonlinear waves in disordered chains: probing the limits of chaos and spreading. Phys. Rev. E 84, 016205 (2011)ADSGoogle Scholar
  48. 48.
    J.D. Bodyfelt, T.V. Laptyeva, G. Gligoric, D.O. Krimer, Ch. Skokos, S. Flach, Wave interactions in localizing media – a coin with many faces. Int. J. Bifurc. Chaos 21, 2107–2124 (2011)MATHGoogle Scholar
  49. 49.
    J. Boreux, T. Carletti, Ch. Skokos, M. Vittot, Hamiltonian control used to improve the beam stability in particle accelerator models. Commun. Nonlinear Sci. Numer. Simul. (2011) 17, 1725–1738 (2012)Google Scholar
  50. 50.
    J. Boreux, T. Carletti, Ch. Skokos, Y. Papaphilippou, M. Vittot, Efficient control of accelerator maps. Int. J. Bifurc. Chaos (2012, In Press) E-print arXiv:1103.5631Google Scholar
  51. 51.
    T. Bountis, Investigating non-integrability and Chaos in complex time. Phys. D 86, 256–267 (1995)MathSciNetMATHGoogle Scholar
  52. 52.
    T. Bountis, Stability of motion: From Lyapunov to the dynamics N-degree of freedom Hamiltonian systems. Nonlinear Phenomena and Complex Systems 9, 209–239 (2006)MathSciNetGoogle Scholar
  53. 53.
    T. Bountis, J.M. Bergamin, Discrete Breathers in Nonlinear Lattices: A Review and Recent Results. Lecture Notes in Physics, vol. 626 (Springer, New York/Berlin, 2003)Google Scholar
  54. 54.
    T. Bountis, M. Kollmann, Diffusion rates in a 4-dimensional mapping model of accelerator dynamics. Phys. D 71, 122–131 (1994)MATHGoogle Scholar
  55. 55.
    T. Bountis, K.E. Papadakis, The stability of vertical motion in the N-body circular Sitnikov problem. Celest. Mech. Dyn. Astron. 104, 205–225 (2009)MathSciNetADSMATHGoogle Scholar
  56. 56.
    T. Bountis, H. Segur, in Logarithmic Singularities and Chaotic Behavior in Hamiltonian Systems, ed. by M. Tabor, Y. Treves. A.I.P. Conference Proceedings, vol. 88, 279–292 (A.I.P., New York, 1982)Google Scholar
  57. 57.
    T. Bountis, Ch. Skokos, Application of the SALI chaos detection method to accelerator mappings. Nucl. Instrum. Methods A 561, 173–179 (2006)ADSGoogle Scholar
  58. 58.
    T. Bountis, Ch. Skokos, Space charges can significantly affect the dynamics of accelerator maps. Phys. Lett. A 358, 126–133 (2006)ADSMATHGoogle Scholar
  59. 59.
    T. Bountis, S. Tompaidis, Strong and weak instabilities in a 4-D mapping model of accelerator dynamics, in Nonlinear Problems in Future Particle Accelerators, ed. by W. Scandale, G. Turchetti (World Scientific, Singapore, 1991), pp. 112–127Google Scholar
  60. 60.
    T. Bountis, H. Segur, F. Vivaldi, Integrable Hamiltonian systems and the Painlevé property. Phys. Rev. A 25, 1257–1264 (1982)MathSciNetADSGoogle Scholar
  61. 61.
    T. Bountis, H.W. Capel, M. Kollmann, J.C. Ross, J.M. Bergamin, J.P. van der Weele, Multibreathers and homoclinic orbits in one-dimensional nonlinear lattices. Phys. Lett. A 268, 50–60 (2000)MathSciNetADSMATHGoogle Scholar
  62. 62.
    T. Bountis, J.M. Bergamin, V. Basios, Stabilization of discrete breathers using continuous feedback control. Phys. Lett. A 295, 115–120 (2002)ADSGoogle Scholar
  63. 63.
    T. Bountis, T. Manos, H. Christodoulidi, Application of the GALI method to localization dynamics in nonlinear systems. J. Comput. Appl. Math. 227, 17–26 (2009)MathSciNetADSMATHGoogle Scholar
  64. 64.
    T. Bountis, G. Chechin, V. Sakhnenko, Discrete symmetries and stability in Hamiltonian dynamics. Int. J. Bifurc. Chaos 21, 1539–1582 (2011)MathSciNetMATHGoogle Scholar
  65. 65.
    V.A. Brazhnyi, V.V. Konotop, Theory of nonlinear matter waves in optical lattices. Mod. Phys. Lett. B 18, 627–651 (2004)ADSGoogle Scholar
  66. 66.
    N. Budinsky, T. Bountis, Stability of nonlinear modes and chaotic properties of 1D Fermi-Pasta-Ulam lattices. Phys. D 8, 445–452 (1983)MathSciNetMATHGoogle Scholar
  67. 67.
    A. Cafarella, M. Leo, R.A. Leo, Numerical analysis of the one-mode solutions in the Fermi-Pasta-Ulam system. Phys. Rev. E 69, 046604 (2004)ADSGoogle Scholar
  68. 68.
    P. Calabrese, A. Gambassi, Slow dynamics in critical ferromagnetic vector models relaxing from a magnetized initial state. J. Stat. Mech.-Theory Exp. 2007, P01001 (2007)Google Scholar
  69. 69.
    F. Calogero, D. Gomez-Ullate, P.M. Santini, M. Sommacal, On the transition from regular to irregular motions, explained as travel on Riemann surfaces. J. Phys. A 38, 8873–8896 (2005)MathSciNetADSMATHGoogle Scholar
  70. 70.
    F. Calogero, D. Gomez-Ullate, P.M. Santini, M. Sommacal, Towards a theory of chaos explained as travel on Riemann surfaces. J. Phys. A 42, 015205 (2009)MathSciNetADSGoogle Scholar
  71. 71.
    D.K. Campbell, P. Rosenau, G.M. Zaslavsky (eds.), The Fermi-Pasta-Ulam problem: the first 50 Years. Chaos, Focus Issue 15, 015101 (2005)Google Scholar
  72. 72.
    R. Capuzzo-Dolcetta, L. Leccese, D. Merritt, A. Vicari, Self-consistent models of cuspy triaxial galaxies with dark matter haloes. Astrophys. J. 666, 165–180 (2007)ADSGoogle Scholar
  73. 73.
    J.R. Cary, D.F. Escande, A.D. Verga, Nonquasilinear diffusion far from the chaotic threshold. Phys. Rev. Lett. 65, 3132–3135 (1990)ADSGoogle Scholar
  74. 74.
    G. Casati, B. Li, Heat conduction in one dimensional systems: Fourier law, chaos, and heat control, in Nonlinear Dynamics and Fundamental Interactions. NATO Science Series, Springer, New York/Berlin, vol. 213, Part 1, 1–16 (2006)Google Scholar
  75. 75.
    G. Casati, T. Prosen, Mixing property of triangular billiards. Phys. Rev. Lett. 83, 4729–4732 (1999)ADSGoogle Scholar
  76. 76.
    G. Casati, J. Ford, F. Vivaldi, W.M. Visscher, One-dimensional classical many-body system having a normal thermal conductivity. Phys. Rev. Lett. 52, 1861–1864 (1984)ADSGoogle Scholar
  77. 77.
    A. Celikoglu, U. Tirnakli, S.M. Duarte Queirós, Analysis of return distributions in the coherent noise model. Phys. Rev. E 82, 021124 (2010)ADSGoogle Scholar
  78. 78.
    J. Chabé, G. Lemarié, B. Grémaud, D. Delande, P. Szriftgiser, J.-C. Garreau, Experimental observation of the Anderson metal-insulator transition with atomic matter waves. Phys. Rev. Lett. 101, 255702 (2008)ADSGoogle Scholar
  79. 79.
    C.W. Chang, D. Okawa, A. Majumdar, A. Zettl, Solid-state thermal rectifier. Science 314, 1121 (2006)Google Scholar
  80. 80.
    G.M. Chechin, Computers and group-theoretical methods for studying structural phase transition. Comput. Math. Appl. 17, 255–278 (1989)MathSciNetMATHGoogle Scholar
  81. 81.
    G.M. Chechin, V.P. Sakhnenko, Interactions between normal modes in nonlinear dynamical systems with discrete symmetry. Exact results. Phys. D 117, 43–76 (1998)MathSciNetMATHGoogle Scholar
  82. 82.
    G.M. Chechin, K.G. Zhukov, Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries. Phys. Rev. E 73, 036216 (2006)MathSciNetADSGoogle Scholar
  83. 83.
    G.M. Chechin, T.I. Ivanova, V.P. Sakhnenko, Complete order parameter condensate of low-symmetry phases upon structural phase transitions. Phys. Status Solidi B 152, 431–446 (1989)ADSGoogle Scholar
  84. 84.
    G.M. Chechin, E.A. Ipatova, V.P. Sakhnenko, Peculiarities of the low-symmetry phase structure near the phase-transition point. Acta Crystallogr. A 49, 824–831 (1993)Google Scholar
  85. 85.
    G.M. Chechin, N.V. Novikova, A.A. Abramenko, Bushes of vibrational modes for Fermi-Pasta-Ulam chains. Phys. D 166, 208–238 (2002)MathSciNetMATHGoogle Scholar
  86. 86.
    G.M. Chechin, A.V. Gnezdilov, M.Yu. Zekhtser, Existence and stability of bushes of vibrational modes for octahedral mechanical systems with Lennard-Jones potential. Int. J. Nonlinear Mech. 38, 1451–1472 (2003) *********Google Scholar
  87. 87.
    G.M. Chechin, D.S. Ryabov, K.G. Zhukov, Stability of low-dimensional bushes of vibrational modes in the Fermi-Pasta-Ulam chains. Phys. D 203, 121–166 (2005)MathSciNetMATHGoogle Scholar
  88. 88.
    B.V. Chirikov, A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263–379 (1979)MathSciNetADSGoogle Scholar
  89. 89.
    B.V. Chirikov, D.L. Shepelyansky, Correlation properties of dynamical chaos in Hamiltonian systems. Phys. D 13, 395–400 (1984)MathSciNetMATHGoogle Scholar
  90. 90.
    S.-N. Chow, M. Yamashita, Geometry of the Melnikov vector, in Nonlinear Equations in Applied Sciences, ed. by W.F. Ames, C. Rogers (Academic Press, San Diego, 1991), pp. 79–148Google Scholar
  91. 91.
    D.N. Christodoulides, F. Lederer, Y. Silberberg, Discretizing light behavious in linear and nonlinear waveguide lattices. Nature 424, 817 (2003)ADSGoogle Scholar
  92. 92.
    H. Christodoulidi, Dynamics on low-dimensional tori and chaos in Hamiltonian systems. Ph.D. Thesis, University of Patras, 2010Google Scholar
  93. 93.
    H. Christodoulidi, T. Bountis, Low-dimensional quasiperiodic motion in Hamiltonian systems. ROMAI J. 2, 37–44 (2006)MathSciNetMATHGoogle Scholar
  94. 94.
    H. Christodoulidi, C. Efthymiopoulos, T. Bountis, Energy localization on q-tori, long-term stability, and the interpretation of Fermi-Pasta-Ulam recurrences. Phys. Rev. E 81, 016210 (2010)ADSGoogle Scholar
  95. 95.
    P.M. Cincotta, C. Simó, Simple tools to study global dynamics in non-axisymmetric galactic potentials-I. Astron. Astrophys. Suppl. 147, 205–228 (2000)ADSGoogle Scholar
  96. 96.
    P.M. Cincotta, C.M. Giordano, C. Simó, Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits. Phys. D 182, 151–178 (2003)MathSciNetMATHGoogle Scholar
  97. 97.
    E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955)MATHGoogle Scholar
  98. 98.
    R.M. Conte, M. Musette, The Painlevé Handbook (Springer, Heidelberg, 2008)MATHGoogle Scholar
  99. 99.
    G. Contopoulos, Order and Chaos in Dynamical Astronomy (Springer, Heidelberg, 2002)MATHGoogle Scholar
  100. 100.
    G. Contopoulos, B. Barbanis, Lyapunov characteristic numbers and the structure of phase-space. Astron. Astrophys. 222, 329–343 (1989)MathSciNetADSGoogle Scholar
  101. 101.
    G. Contopoulos, P. Magnenat, Simple three-dimensional periodic orbits in a galactic-type potential. Celest. Mech. 37, 387–414 (1985)MathSciNetADSMATHGoogle Scholar
  102. 102.
    G. Contopoulos, N. Voglis, Spectra of stretching numbers and helicity angles in dynamical systems. Celest. Mech. Dyn. Astr. 64, 1–20 (1996)MathSciNetADSMATHGoogle Scholar
  103. 103.
    G. Contopoulos, N. Voglis, A fast method for distinguishing between ordered and chaotic orbits. Astron. Astrophys. 317, 73–81 (1997)ADSGoogle Scholar
  104. 104.
    G. Contopoulos, L. Galgani, A. Giorgilli, On the number of isolating integrals in Hamiltonian systems. Phys. Rev. A 18, 1183–1189 (1978)ADSGoogle Scholar
  105. 105.
    T. Cretegny, T. Dauxois, S. Ruffo, A. Torcini, Localization and equipartition of energy in the beta-FPU chain: chaotic breathers. Phys. D 121, 109–126 (1998)Google Scholar
  106. 106.
    F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463–512 (1999)ADSGoogle Scholar
  107. 107.
    H.T. Davis, Introduction to Nonlinear Differential and Integral Equations (Dover, New York, 1962)MATHGoogle Scholar
  108. 108.
    T. Dauxois, Non-Gaussian distributions under scrutiny. J. Stat. Mech.-Theory Exp. 2007, N08001 (2007)Google Scholar
  109. 109.
    J. De Luca, A.J. Lichtenberg, Transitions and time scales to equipartition in oscillator chains: low-frequency initial conditions. Phys. Rev. E 66, 026206 (2002)MathSciNetADSGoogle Scholar
  110. 110.
    J. De Luca, A.J. Lichtenberg, M.A. Lieberman, Time scale to ergodicity in the Fermi-Pasta-Ulam system. Chaos 5, 283–297 (1995)ADSGoogle Scholar
  111. 111.
    J. De Luca, A.J. Lichtenberg, S. Ruffo, Energy transitions and time scales to equipartition in the Fermi-Pasta-Ulam oscillator chain. Phys. Rev. E 51, 2877–2885 (1995)ADSGoogle Scholar
  112. 112.
    J. De Luca, A.J. Lichtenberg, S. Ruffo, Finite times to equipartition in the thermodynamic limit. Phys. Rev. E 60, 3781–3786 (1999)ADSGoogle Scholar
  113. 113.
    L. Drossos, T. Bountis, Evidence of natural boundary and nonintegrability of the mixmaster universe model. J. Nonlinear Sci. 7, 1–11 (1997)MathSciNetGoogle Scholar
  114. 114.
    W.E. Drummond, D. Pines, Nonlinear stability of plasma oscillations. Nucl. Fusion Suppl. 3, 1049–1057 (1962)Google Scholar
  115. 115.
    S.M. Duarte Queirós, The role of ergodicity and mixing in the central limit theorem for Casati-Prosen triangle map variables. Phys. Lett. A 373, 1514–1518 (2009)MATHGoogle Scholar
  116. 116.
    G. Duffing, Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung (Vieweg & Sohn, Braunschweig, 1918)MATHGoogle Scholar
  117. 117.
    J.-P. Eckmann, D. Ruelle, Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985)MathSciNetADSGoogle Scholar
  118. 118.
    J.T. Edwards, D.J. Thouless, Numerical studies of localization in disordered systems. J. Phys. C Solid 5, 807–820 (1972)ADSGoogle Scholar
  119. 119.
    N.K. Efremidis, D.N. Christodoulides, Lattice solitons in Bose-Einstein condensates. Phys. Rev. A 67, 063608 (2003)ADSGoogle Scholar
  120. 120.
    L.H. Eliasson, Absolutely convergent series expansions for quasi periodic motions. Math. Phys. Electron. J. 2, 4 (1996)MathSciNetGoogle Scholar
  121. 121.
    E. Fermi, J. Pasta, S. Ulam, Studies of nonlinear problems. Los Alamos Sci. Lab. Rep. No. LA-1940 (1955), in Nonlinear Wave Motion, ed. by A.C. Newell. Lectures in Applied Mathematics, vol. 15 (Amer. Math. Soc., Providence, 1974), pp. 143–155Google Scholar
  122. 122.
    S. Flach, Conditions on the existence of localized excitations in nonlinear discrete systems. Phys. Rev. E 50, 3134–3142 (1994)MathSciNetADSGoogle Scholar
  123. 123.
    S. Flach, Obtaining breathers in nonlinear Hamiltonian lattices. Phys. Rev. E 51, 3579–3587 (1995)ADSGoogle Scholar
  124. 124.
    S. Flach, Spreading of waves in nonlinear disordered media. Chem. Phys. 375, 548–556 (2010)ADSGoogle Scholar
  125. 125.
    S. Flach, A.V. Gorbach, Discrete breathers – Advances in theory and applications. Phys. Rep. 467, 1–116 (2008)ADSGoogle Scholar
  126. 126.
    S. Flach, A. Ponno, The Fermi-Pasta-Ulam problem: periodic orbits, normal forms and resonance overlap criteria. Phys. D 237, 908–917 (2008)MathSciNetMATHGoogle Scholar
  127. 127.
    S. Flach, C. Willis, Discrete breathers. Phys. Rep. 295, 181–264 (1998)MathSciNetGoogle Scholar
  128. 128.
    S. Flach, M.V. Ivanchenko, O.I. Kanakov, q-Breathers and the Fermi-Pasta-Ulam problem. Phys. Rev. Lett. 95, 064102 (2005)ADSGoogle Scholar
  129. 129.
    S. Flach, M.V. Ivanchenko, O.I. Kanakov, q-breathers in Fermi-Pasta-Ulam chains: existence, localization, and stability. Phys. Rev. E 73, 036618 (2006)MathSciNetADSGoogle Scholar
  130. 130.
    S. Flach, O.I. Kanakov, M.V. Ivanchenko, K. Mishagin, q-breathers in FPU-lattices – scaling and properties for large systems. Int. J. Mod. Phys. B 21, 3925–3932 (2007)ADSGoogle Scholar
  131. 131.
    S. Flach, D.O. Krimer, Ch. Skokos, Universal spreading of wavepackets in disordered nonlinear systems. Phys. Rev. Lett. 102, 024101 (2009)ADSGoogle Scholar
  132. 132.
    A.S. Fokas, T. Bountis, Order and the ubiquitous occurrence of Chaos. Phys. A 228, 236–244 (1996)MathSciNetGoogle Scholar
  133. 133.
    J. Ford, The Fermi-Pasta-Ulam problem: paradox turns discovery. Phys. Rep. 213, 271–310 (1992)MathSciNetADSGoogle Scholar
  134. 134.
    F. Freistetter, Fractal dimensions as chaos indicators. Celest. Mech. Dyn. Astron. 78, 211–225 (2000)MathSciNetADSMATHGoogle Scholar
  135. 135.
    C. Froeschlé, E. Lega, On the structure of symplectic mappings. The fast Lyapunov indicator: A very sensitive tool. Celest. Mech. Dyn. Astron. 78, 167–195 (2000)ADSMATHGoogle Scholar
  136. 136.
    C. Froeschlé, Ch. Froeschlé, E. Lohinger, Generalized Lyapunov characteristic indicators and corresponding Kolmogorov like entropy of the standard mapping. Celest. Mech. Dyn. Astron. 56, 307–314 (1993)ADSMATHGoogle Scholar
  137. 137.
    C. Froeschlé, E. Lega, R. Gonczi, Fast Lyapunov indicators. Application to asteroidal motion. Celest. Mech. Dyn. Astron. 67, 41–62 (1997)ADSMATHGoogle Scholar
  138. 138.
    C. Froeschlé, R. Gonczi, E. Lega, The fast Lyapunov indicator: a simple tool to detect weak chaos. Application to the structure of the main asteroidal belt. Planet. Space Sci. 45, 881–886 (1997)Google Scholar
  139. 139.
    F. Fucito, F. Marchesoni, E. Marinari, G. Parisi, L. Peliti, S. Ruffo, A. Vulpiani, Approach to equilibrium in a chain of nonlinear oscillators. J. Phys.-Paris 43, 707–713 (1982)MathSciNetGoogle Scholar
  140. 140.
    L. Galgani, A. Scotti, Planck-like distributions in classical nonlinear mechanics. Phys. Rev. Lett. 28, 1173–1176 (1972)ADSGoogle Scholar
  141. 141.
    Z. Galias, Rigorous investigation of the Ikeda map by means of interval arithmetic. Nonlinearity 15, 1759–1779 (2002)MathSciNetADSMATHGoogle Scholar
  142. 142.
    G. Gallavotti, Twistless KAM tori. Commun. Math. Phys. 164, 145–156 (1994)MathSciNetADSMATHGoogle Scholar
  143. 143.
    G. Gallavotti, Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems: a review. Rev. Math. Phys. 6, 343–411 (1994)MathSciNetMATHGoogle Scholar
  144. 144.
    I. García-Mata, D.L. Shepelyansky, Delocalization induced by nonlinearity in systems with disorder. Phys. Rev. E 79, 026205 (2009)ADSGoogle Scholar
  145. 145.
    P. Gaspard, Lyapunov exponent of ion motion in microplasmas. Phys. Rev. E 68, 056209 (2003)ADSGoogle Scholar
  146. 146.
    E. Gerlach, Ch. Skokos, Comparing the efficiency of numerical techniques for the integration of variational equations. Discr. Cont. Dyn. Sys.-Supp. September, 475–484 (2011)Google Scholar
  147. 147.
    E. Gerlach, S. Eggl, Ch. Skokos, Efficient integration of the variational equations of multi-dimensional Hamiltonian systems: application to the Fermi-Pasta-Ulam lattice. Int. J. Bifurc. Chaos (2012, In Press) E-print arXiv:1104.3127Google Scholar
  148. 148.
    A. Giorgilli, U. Locatelli, Kolmogorov theorem and classical perturbation theory. Z. Angew. Math. Phys. 48, 220–261 (1997)MathSciNetMATHGoogle Scholar
  149. 149.
    A. Giorgilli, U. Locatelli, A classical self-contained proof of Kolmogorov’s theorem on invariant tori, in Hamiltonian Systems of Three or More Degrees of Freedom, ed. by C. Simó. NATO Advanced Study Institute, vol. 533 (Kluwer, Dordrecht, 1999), pp. 72–89Google Scholar
  150. 150.
    A. Giorgilli, D. Muraro, Exponentially stable manifolds in the neighbourhood of elliptic equilibria. Boll. Unione Mate. Ital. B 9, 1–20 (2006)MathSciNetMATHGoogle Scholar
  151. 151.
    M.L. Glasser, V.G. Papageorgiou, T.C. Bountis, Mel’nikov’s function for two-dimensional mappings. SIAM J. Appl. Math. 49, 692–703 (1989)MathSciNetMATHGoogle Scholar
  152. 152.
    A. Goriely, Integrability and Nonintegrability of Dynamical Systems (World Scientific, Singapore, 2001)MATHGoogle Scholar
  153. 153.
    G.A. Gottwald, I. Melbourne, A new test for chaos in deterministic systems. Proc. R. Soc. Lond. A Math. 460, 603–611 (2004)MathSciNetADSMATHGoogle Scholar
  154. 154.
    G.A. Gottwald, I. Melbourne, Testing for chaos in deterministic systems with noise. Phys. D 212, 100–110 (2005)MathSciNetMATHGoogle Scholar
  155. 155.
    E. Goursat, Cours d’ Analyse Mathématique vol. 2 (Gauthier-Villars, Paris, 1905)Google Scholar
  156. 156.
    B. Grammaticos, B. Dorizzi, R. Padjen, Painlevé property and integrals of motion for the Hénon-Heiles system. Phys. Lett. A 89, 111–113 (1982)MathSciNetADSGoogle Scholar
  157. 157.
    P.E. Greenwood, M.S. Nikulin, A Guide to Chi-Squared Testing, (Wiley, New York, 1996)Google Scholar
  158. 158.
    P. Grassberger, Proposed central limit behavior in deterministic dynamical systems. Phys. Rev. E 79, 057201 (2009)ADSGoogle Scholar
  159. 159.
    W. Greub, Multilinear Algebra, 2nd edn. (Springer, Heidelberg, 1978)MATHGoogle Scholar
  160. 160.
    J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983)MATHGoogle Scholar
  161. 161.
    M.G. Hahn, X. Jiang, S. Umarov, On q-Gaussians and exchangeability. J. Phys. A-Math. Theor. 43, 165208 (2010)MathSciNetADSGoogle Scholar
  162. 162.
    E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Comput. Math., vol. 31 (Springer, Berlin, 2002)Google Scholar
  163. 163.
    P. Hemmer, Dynamic and stochastic type of motion by the linear chain. Det Physiske Seminar i Trondheim 2, 66 (1959)Google Scholar
  164. 164.
    M. Hénon, C. Heiles, The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73–79 (1964)ADSGoogle Scholar
  165. 165.
    R.C. Hilborn, Chaos and Nonlinear Dynamics (Oxford University Press, New York, 1994)MATHGoogle Scholar
  166. 166.
    H.J. Hilhorst, Note on a q-modified central limit theorem. J. Stat. Mech.-Theory Exp. 2010, P10023 (2010)MathSciNetGoogle Scholar
  167. 167.
    H.J. Hilhorst, G. Schehr, A note on q-Gaussians and non-Gaussians in statistical mechanics. J. Stat. Mech.-Theory Exp. 2007, P06003 (2007)MathSciNetGoogle Scholar
  168. 168.
    T.L. Hill Thermodynamics of Small Systems (Dover, New York, 1994)Google Scholar
  169. 169.
    E. Hille, Lectures on Ordinary Differential Equations (Addison-Wesley, Reading, 1969)MATHGoogle Scholar
  170. 170.
    M.W. Hirsch, S. Smale, R.L. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos (Elsevier, New York, 2004)MATHGoogle Scholar
  171. 171.
    J.E. Howard, Discrete virial theorem. Celest. Mech. Dyn. Astron. 92, 219–241 (2005)ADSMATHGoogle Scholar
  172. 172.
    B. Hu, B. Li, H. Zhao, Heat conduction in one-dimensional chains. Phys. Rev. E 57, 2992 (1998)ADSGoogle Scholar
  173. 173.
    H. Hu, A. Strybulevych, J. Page, S. Skipetrov, B. van Tiggelen, Localization of ultrasound in a three-dimensional elastic network. Nat. Phys. 4, 945–948 (2008)Google Scholar
  174. 174.
    J.H. Hubbard, B.B. Hubbard, Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach (Prentice Hall, Upper Saddle River, 1999)MATHGoogle Scholar
  175. 175.
    M.C. Irwin, Smooth Dynamical Systems (Academic, New York, 1980)MATHGoogle Scholar
  176. 176.
    N. Jacobson, Lectures in Abstract Algebra, vol. II (van Nostrand, Princeton, 1951)Google Scholar
  177. 177.
    M. Johansson, G. Kopidakis, S. Lepri, S. Aubry, Transmission thresholds in time-periodically driven nonlinear disordered systems. Europhys. Lett. 86, 10009 (2009)ADSGoogle Scholar
  178. 178.
    M. Johansson, G. Kopidakis, S. Aubry, KAM tori in 1D random discrete nonlinear Schrödinger model? Europhys. Lett. 91, 50001 (2010)ADSGoogle Scholar
  179. 179.
    O.I. Kanakov, S. Flach, M.V. Ivanchenko, K.G. Mishagin, Scaling properties of q-breathers in nonlinear acoustic lattices. Phys. Lett. A 365, 416–420 (2007)ADSGoogle Scholar
  180. 180.
    H. Kantz, P. Grassberger, Internal Arnold diffusion and chaos thresholds in coupled symplectic maps. J. Phys. A-Math. Gen. 21, L127–L133 (1988)MathSciNetADSMATHGoogle Scholar
  181. 181.
    G.I. Karanis, Ch.L. Vozikis, Fast detection of chaotic behavior in galactic potentials. Astron. Nachr. 329, 403–412 (2008)ADSGoogle Scholar
  182. 182.
    Y.V. Kartashov, V.A. Vysloukh, L. Torner, Soliton shape and mobility control in optical lattices. Prog. Opt. 52, 63–148 (2009)Google Scholar
  183. 183.
    Y.V. Kartashov, B.A. Malomed, L. Torner, Solitons in nonlinear lattices. Rev. Mod. Phys. 83, 247 (2011)ADSGoogle Scholar
  184. 184.
    A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. IHÉS 51, 137–173 (1980)MathSciNetMATHGoogle Scholar
  185. 185.
    A. Katok, J.-M. Strelcyn, Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities. Lecture Notes in Mathematics, vol. 1222 (Springer, Berlin, 1986)Google Scholar
  186. 186.
    A.N. Kaufman, Quasilinear diffusion of an axisymmetric toroidal plasma. Phys. Fluids 15, 1063 (1972)ADSGoogle Scholar
  187. 187.
    W. Ketterle, D.S. Durfee, D.M. Stamper-Kurn, Making, probing and understanding Bose-Einstein condensates, in Bose-Einstein Condensation in Atomic Gases. Proceedings of the International School of Physics “Enrico Fermi”, ed. by M. Inguscio, S. Stringari, C.E. Wieman (IOS Press, Amsterdam, 1999), pp. 67–176Google Scholar
  188. 188.
    P.G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation. Tracts in Modern Physics, vol. 232 (Springer, Heidelberg, 2009)Google Scholar
  189. 189.
    Y.S. Kivshar, Intrinsic localized modes as solitons with a compact support. Phys. Rev. E 48, R43–R45 (1993)MathSciNetADSGoogle Scholar
  190. 190.
    Y.S. Kivshar, G.P. Agrawal, Optical Solitons. From Fibers to Photonic Crystals (Academic, Amsterdam, 2003)Google Scholar
  191. 191.
    Y.S. Kivshar, B.A. Malomed, Dynamics of solitons in near-integrable systems. Rev. Mod. Phys. 61, 763–915 (1989)ADSGoogle Scholar
  192. 192.
    W. Kobayashi, Y. Teraoka, I. Terasaki, An oxide thermal rectifier. Appl. Phys. Lett. 95, 171905 (2009)ADSGoogle Scholar
  193. 193.
    Y. Kominis, Analytical solitary wave solutions of the nonlinear Kronig-Penney model in photonic structures. Phys. Rev. E 73, 066619 (2006)ADSGoogle Scholar
  194. 194.
    Y. Kominis, T. Bountis, Analytical solutions of systems with piecewise linear dynamics. Int. J. Bifurc. Chaos 20, 509–518 (2010)MathSciNetGoogle Scholar
  195. 195.
    Y. Kominis, K. Hizanidis, Lattice solitons in self-defocusing optical media: analytical solutions of the nonlinear Kronig-Penney model. Opt. Lett. 31, 2888–2890 (2006)ADSGoogle Scholar
  196. 196.
    Y. Kominis, K. Hizanidis, Power dependent soliton location and stability in complex photonic structures. Opt. Expr. 16, 12124–12138 (2008)ADSGoogle Scholar
  197. 197.
    Y. Kominis, K. Hizanidis, Power-dependent reflection, transmission and trapping dynamics of lattice solitons at interfaces. Phys. Rev. Lett. 102, 133903 (2009)ADSGoogle Scholar
  198. 198.
    Y. Kominis, A. Papadopoulos, K. Hizanidis, Surface solitons in waveguide arrays: analytical solutions. Opt. Expr. 15, 10041–10051 (2007)ADSGoogle Scholar
  199. 199.
    Y. Kominis, A.K. Ram, K. Hizanidis, Quasilinear theory of electron transport by radio frequency waves and non-axisymmetric perturbations in toroidal plasmas. Phys. Plasmas 15, 122501 (2008)ADSGoogle Scholar
  200. 200.
    Y. Kominis, T. Bountis, K. Hizanidis, Breathers in a nonautonomous Toda lattice with pulsating coupling. Phys. Rev. E 81, 066601 (2010)MathSciNetADSGoogle Scholar
  201. 201.
    Y. Kominis, A.K. Ram, K. Hizanidis, Kinetic theory for distribution functions of wave-particle interactions in plasmas. Phys. Rev. Lett. 104, 235001 (2010)ADSGoogle Scholar
  202. 202.
    G. Kopidakis, S. Komineas, S. Flach, S. Aubry, Absence of wave packet diffusion in disordered nonlinear systems. Phys. Rev. Lett. 100, 084103 (2008)ADSGoogle Scholar
  203. 203.
    Y.A. Kosevich, Nonlinear sinusoidal waves and their superposition in anharmonic lattices. Phys. Rev. Lett. 71, 2058–2061 (1993)ADSGoogle Scholar
  204. 204.
    T. Kotoulas, G. Voyatzis, Comparative study of the 2:3 and 3:4 resonant motion with Neptune: an application of symplectic mappings and low frequency analysis. Celest. Mech. Dyn. Astron. 88, 343–363 (2004)MathSciNetADSMATHGoogle Scholar
  205. 205.
    I. Kovacic, M.J. Brennan (eds.), The Duffing Equation: Nonlinear Oscillators and Their Behaviour (Wiley, Hoboken, 2011)MATHGoogle Scholar
  206. 206.
    B. Kramer, A. MacKinnon, Localization: theory and experiment. Rep. Prog. Phys. 56, 1469–1564 (1993)ADSGoogle Scholar
  207. 207.
    D.O. Krimer, S. Flach, Statistics of wave interactions in nonlinear disordered systems. Phys. Rev. E 82, 046221 (2010)ADSGoogle Scholar
  208. 208.
    Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D.N. Christodoulides, Y. Silberberg, Anderson localization and nonlinearity in one-dimensional disordered photonic lattices. Phys. Rev. Lett. 100, 013906 (2008)ADSGoogle Scholar
  209. 209.
    L.D. Landau, E.M. Lifshitz, Mechanics, Third edn, Volume 1 of Course of Theoretical Physics (Butterworth-Heinemann, Amsterdam, 1976)Google Scholar
  210. 210.
    T.V. Laptyeva, J.D. Bodyfelt, D.O. Krimer, Ch. Skokos, S. Flach, The crossover from strong to weak chaos for nonlinear waves in disordered systems. Europhys. Lett. 91, 30001 (2010)ADSGoogle Scholar
  211. 211.
    J. Laskar, The chaotic motion of the Solar System: a numerical estimate of the size of the chaotic zones. Icarus 88, 266–291 (1990)ADSGoogle Scholar
  212. 212.
    J. Laskar, Frequency analysis of multi-dimensional systems. Global dynamics and diffusion. Phys. D 67, 257–281 (1993)MathSciNetMATHGoogle Scholar
  213. 213.
    J. Laskar, Introduction to frequency map analysis, in Hamiltonian Systems of Three or More Degrees of Freedom, ed. by C. Simó. NATO Advanced Study Institute, vol. 533 (Kluwer, Dordrecht, 1999), pp. 134–150Google Scholar
  214. 214.
    J. Laskar, C. Froeschlé, A. Celletti, The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard map. Phys. D 56, 253–269 (1992)MATHGoogle Scholar
  215. 215.
    M. Lax, W.H. Louisell, W.B. McKnight, From Maxwell to paraxial wave optics. Phys. Rev. A 11, 1365–1370 (1975)ADSGoogle Scholar
  216. 216.
    F. Lederer, G.I. Stegeman, D.N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, Discrete solitons in optics. Phys. Rep. 463, 1–126 (2008)Google Scholar
  217. 217.
    E. Lega, C. Froeschlé, Comparison of convergence towards invariant distributions for rotation angles, twist angles and local Lyapunov characteristic numbers. Planet. Space Sci. 46, 1525–1534 (1998)ADSGoogle Scholar
  218. 218.
    M. Leo, R.A. Leo, Stability properties of the N ∕ 4 (π ∕ 2-mode) one-mode nonlinear solution of the Fermi-Pasta-Ulam-β system. Phys. Rev. E 76, 016216 (2007)ADSGoogle Scholar
  219. 219.
    S. Lepri, R. Livi, A. Politi, Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377(1), 1–80 (2003)MathSciNetADSGoogle Scholar
  220. 220.
    S. Lepri, R. Livi, A. Politi, Studies of thermal conductivity in Fermi Pasta Ulam-like lattices. Chaos 15, 015118 (2005)ADSGoogle Scholar
  221. 221.
    B. Li, J. Wang, Anomalous heat conduction and anomalous diffusion in one-dimensional systems. Phys. Rev. Lett. 91, 044301 (2003)ADSGoogle Scholar
  222. 222.
    B. Li, L. Wang, B. Hu, Finite thermal conductivity in 1D models having zero Lyapunov exponents. Phys. Rev. Lett. 88, 223901 (2002)ADSGoogle Scholar
  223. 223.
    B. Li, G. Casati, J. Wang, Heat conductivity in linear mixing systems. Phys. Rev. E 67, 021204 (2003)ADSGoogle Scholar
  224. 224.
    B. Li, G. Casati, J. Wang, T. Prosen, Fourier Law in the alternate-mass hard-core potential chain. Phys. Rev. Lett. 92, 254301 (2004)ADSGoogle Scholar
  225. 225.
    B. Li, J. Wang, G. Casati, Thermal diode: rectification of heat flux. Phys. Rev. Lett. 93, 184301 (2004)ADSGoogle Scholar
  226. 226.
    A.J. Lichtenberg, M.A. Lieberman, Regular and Chaotic Dynamics, Second edn. (Springer, New York, 1992)MATHGoogle Scholar
  227. 227.
    A. Lichtenberg, R. Livi, M. Pettini, S. Ruffo, Dynamics of oscillator chains. Lect. Notes Phys. 728, 21–121 (2008)MathSciNetADSGoogle Scholar
  228. 228.
    R. Livi, M. Pettini, S. Ruffo, A. Vulpiani, Further results on the equipartition threshold in large nonlinear Hamiltonian systems. Phys. Rev. A 31, 2740–2742 (1985)ADSGoogle Scholar
  229. 229.
    R. Livi, A. Politi, S. Ruffo, Distribution of characteristic exponents in the thermodynamic limit. J. Phys. A-Math. Gen. 19, 2033–2040 (1986)MathSciNetADSMATHGoogle Scholar
  230. 230.
    W.C. Lo, L. Wang, B. Li, Thermal transistor: heat flux switching and modulating. J. Phys. Soc. Jpn, 77(5), 054402 (2008)Google Scholar
  231. 231.
    E. Lohinger, C. Froeschlé, R. Dvorak, Generalized Lyapunov exponents indicators in Hamiltonian dynamics: an application to a double star system. Celest. Mech. Dyn. Astron. 56, 315–322 (1993)ADSMATHGoogle Scholar
  232. 232.
    A.M. Lyapunov, The General Problem of the Stability of Motion (Taylor and Francis, London, 1992) (English translation from the French: A. Liapounoff, Problème général de la stabilité du mouvement. Annal. Fac. Sci. Toulouse 9, 203–474 (1907). The French text was reprinted in Annals Math. Studies Vol.17 Princeton Univ. Press (1947). The original was published in Russian by the Mathematical Society of Kharkov in 1892)Google Scholar
  233. 233.
    M. Macek, P. Stránský, P. Cejnar, S. Heinze, J. Jolie, J. Dobeš, Classical and quantum properties of the semiregular arc inside the Casten triangle. Phys. Rev. C 75, 064318 (2007)ADSGoogle Scholar
  234. 234.
    M. Macek, J. Dobeš, P. Stránský, P. Cejnar, Regularity-induced separation of intrinsic and collective dynamics. Phys. Rev. Lett. 105, 072503 (2010)ADSGoogle Scholar
  235. 235.
    R.S. MacKay, S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7, 1623–1843 (1994)MathSciNetADSMATHGoogle Scholar
  236. 236.
    R.S. Mackay, J.D. Meiss, Hamiltonian Dynamical Systems (Adam Hilger, Bristol, 1986)Google Scholar
  237. 237.
    M.C. Mackey, M. Tyran-Kaminska, Deterministic Brownian motion: the effects of perturbing a dynamical system by a chaotic semi-dynamical system. Phys. Rep. 422, 167–222 (2006)MathSciNetADSGoogle Scholar
  238. 238.
    R.S. Mackay, J.D. Meiss, I.C. Percival, Transport in Hamiltonian systems. Phys. D 13, 55–81 (1984)MathSciNetMATHGoogle Scholar
  239. 239.
    N.P. Maffione, L.A. Darriba, P.M. Cincotta, C.M. Giordano, A comparison of different indicators of chaos based on the deviation vectors: application to symplectic mappings. Celest. Mech. Dyn. Astron. 111, 285–307 (2011)MathSciNetADSGoogle Scholar
  240. 240.
    W. Magnus, S. Winkler, Hill’s Equation (Wiley, New York, 1969) and 2nd edn. (Dover, New York, 2004)Google Scholar
  241. 241.
    P. Maniadis, T. Bountis, Quasiperiodic and chaotic breathers in a parametrically driven system without linear dispersion. Phys. Rev. E 73, 046211 (2006)MathSciNetADSGoogle Scholar
  242. 242.
    T. Manos, E. Athanassoula, Regular and chaotic orbits in barred galaxies – I. Applying the SALI/GALI method to explore their distribution in several models. Mon. Not. R. Astron. Soc. 415, 629–642 (2011)Google Scholar
  243. 243.
    T. Manos, S. Ruffo, Scaling with system size of the Lyapunov exponents for the Hamiltonian Mean Field model. Transp. Theor. Stat. 40, 360–381 (2011)MathSciNetGoogle Scholar
  244. 244.
    T. Manos, Ch. Skokos, T. Bountis, Application of the Generalized Alignment Index (GALI) method to the dynamics of multi-dimensional symplectic maps, in Chaos, Complexity and Transport: Theory and Applications. Proceedings of the CCT07, ed. by C. Chandre, X. Leoncini, G. Zaslavsky (World Scientific, Singapore, 2008), pp. 356–364Google Scholar
  245. 245.
    T. Manos, Ch. Skokos, E. Athanassoula, T. Bountis, Studying the global dynamics of conservative dynamical systems using the SALI chaos detection method. Nonlinear Phenom. Complex Syst. 11, 171–176 (2008)MathSciNetGoogle Scholar
  246. 246.
    T. Manos, Ch. Skokos, T. Bountis, Global dynamics of coupled standard maps, in Chaos in Astronomy. Astrophysics and Space Science Proceedings, ed. by G. Contopoulos, P.A. Patsis (Springer, Berlin/Heidelberg, 2009), pp. 367–371Google Scholar
  247. 247.
    T. Manos, Ch. Skokos, Ch. Antonopoulos, Probing the local dynamics of periodic orbits by the generalized alignment index (GALI) method. Int. J. Bifurc. Chaos (2012, In Press) E-print arXiv:1103.0700Google Scholar
  248. 248.
    J.L. Marín, S. Aubry, Breathers in nonlinear lattices: numerical calculation from the anticontinuous limit. Nonlinearity 9, 1501–1528 (1996)MathSciNetADSMATHGoogle Scholar
  249. 249.
    J.D. Meiss, E. Ott, Markov tree model of transport in area-preserving maps. Phys. D 20, 387–402 (1986)MathSciNetMATHGoogle Scholar
  250. 250.
    D.R. Merkin, Introduction to the Theory of Stability. Series: Texts in Applied Mathematics, vol. 24 (Springer, New York, 1997)Google Scholar
  251. 251.
    G. Miritello, A. Pluchino, A. Rapisarda, Central limit behavior in the Kuramoto model at the “edge of chaos”. Phys. A 388, 4818–4826 (2009)Google Scholar
  252. 252.
    M. Molina, Transport of localized and extended excitations in a nonlinear Anderson model. Phys. Rev. B 58, 12547–12550 (1998)ADSGoogle Scholar
  253. 253.
    M. Mulansky, A. Pikovsky, Spreading in disordered lattices with different nonlinearities. Europhys. Lett. 90, 10015 (2010)ADSGoogle Scholar
  254. 254.
    M. Mulansky, K. Ahnert, A. Pikovsky, D.L. Shepelyansky, Dynamical thermalization of disordered nonlinear lattices. Phys. Rev. E 80, 056212 (2009)ADSGoogle Scholar
  255. 255.
    M. Mulansky, K. Ahnert, A. Pikovsky, Scaling of energy spreading in strongly nonlinear disordered lattices. Phys. Rev. E 83, 026205 (2011)ADSGoogle Scholar
  256. 256.
    N.N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrable Hamiltoninan systems. Russ. Math. Surv. 32(6), 1–65 (1977)MATHGoogle Scholar
  257. 257.
    Z. Nitecki, Differentiable Dynamics (M.I.T., Cambridge, MA, 1971)MATHGoogle Scholar
  258. 258.
    J.A. Núñez, P.M. Cincotta, F.C. Wachlin, Information entropy. An indicator of chaos. Celest. Mech. Dyn. Astron. 64, 43–53 (1996)ADSMATHGoogle Scholar
  259. 259.
    V.I. Oseledec, A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc. 19, 197–231 (1968)MathSciNetGoogle Scholar
  260. 260.
    E.A. Ostrovskaya, Y.S. Kivshar, Matter-wave gap vortices in optical lattices. Phys. Rev. Lett. 93, 160405 (2004)ADSGoogle Scholar
  261. 261.
    A.A. Ovchinnikov, Localized long-lived vibrational states in molecular crystals. Sov. Phys. JETP-USSR 30, 147 (1970)ADSGoogle Scholar
  262. 262.
    P. Panagopoulos, T.C. Bountis, Ch. Skokos, Existence and stability of localized oscillations in one-dimensional lattices with soft spring and hard spring potentials. J. Vib. Acoust. 126, 520–527 (2004)Google Scholar
  263. 263.
    P. Papagiannis, Y. Kominis, K. Hizanidis, Power- and momentum-dependent soliton dynamics in lattices with longitudinal modulation. Phys. Rev. A 84, 013820 (2011)ADSGoogle Scholar
  264. 264.
    R.E. Peierls, Quantum theory of solids, in Theoretical Physics in the Twentieth Century, ed. by M. Fierz, V.F. Weisskopf (Wiley, New York, 1961) 140–160Google Scholar
  265. 265.
    L. Perko, Differential Equations and Dynamical Systems (Springer, New York, 1995)Google Scholar
  266. 266.
    J.B. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents. Math. USSR Izv. 10, 1261–1305 (1976)Google Scholar
  267. 267.
    Ya.B. Pesin, Lyapunov characteristic indexes and ergodic properties of smooth dynamic systems with invariant measure. Dokl. Acad. Nauk. SSSR 226, 774–777 (1976)Google Scholar
  268. 268.
    Ya.B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv. 32(4), 55–114 (1977)Google Scholar
  269. 269.
    Y.G. Petalas, C.G. Antonopoulos, T.C. Bountis, M.N. Vrahatis, Evolutionary methods for the approximation of the stability domain and frequency optimization of conservative maps. Int. J. Bifurc. Chaos 18, 2249–2264 (2008)MathSciNetMATHGoogle Scholar
  270. 270.
    M. Peyrard, The design of a thermal rectifier. Europhys. Lett. 76, 49 (2006)ADSGoogle Scholar
  271. 271.
    A. Pikovsky, D. Shepelyansky, Destruction of Anderson localization by a weak nonlinearity. Phys. Rev. Lett. 100, 094101 (2008)ADSGoogle Scholar
  272. 272.
    P. Poggi, S. Ruffo, Exact solutions in the FPU oscillator chain. Phys. D 103, 251–272 (1997)MathSciNetMATHGoogle Scholar
  273. 273.
    H. Poincaré, Sur les Propriétés des Functions Définies par les Équations aux Différences Partielles (Gauthier-Villars, Paris, 1879)Google Scholar
  274. 274.
    H. Poincaré Les Méthodes Nouvelles de la Mécanique Céleste, vol. 1 (Gauthier Villars, Paris, 1892) (English translation by D.L. Goroff, New Methods in Celestial Mechanics (American Institute of Physics, 1993))Google Scholar
  275. 275.
    A. Ponno, D. Bambusi, Korteweg-de Vries equation and energy sharing in Fermi-Pasta-Ulam. Chaos 15, 015107 (2005)MathSciNetADSGoogle Scholar
  276. 276.
    A. Ponno, E. Christodoulidi, Ch. Skokos, S. Flach, The two-stage dynamics in the Fermi-Pasta-Ulam problem: from regular to diffusive behavior. Chaos, 21, 043127 (2011)ADSGoogle Scholar
  277. 277.
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flanney, Numerical Recipes in Fortran 77. The Art of Scientific Computing, Second edn. (Cambridge University Press, Cambridge/New York, 2001)Google Scholar
  278. 278.
    K. Pyragas, Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421–428 (1992)ADSGoogle Scholar
  279. 279.
    A. Ramani, B. Grammaticos, T. Bountis, The Painlevé property and singularity analysis of integrable and non-integrable systems. Phys. Rep. 180, 159–245 (1989)MathSciNetADSGoogle Scholar
  280. 280.
    A.B. Rechester, R.B. White, Calculation of turbulent diffusion for the Chirikov-Taylor model. Phys. Rev. Lett. 44, 1586–1589 (1980)MathSciNetADSGoogle Scholar
  281. 281.
    A. Rényi, On measures of information and entropy, in Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability 1960, University of California Press, Berkeley/Los Angeles, 1961, pp. 547–561Google Scholar
  282. 282.
    J.A. Rice, Mathematical Statistics and Data Analysis, Second edn. (Duxbury Press, Belmont, 1995)MATHGoogle Scholar
  283. 283.
    B. Rink, Symmetric invariant manifolds in the Fermi-Pasta-Ulam lattice. Phys. D 175, 31–42 (2003)MathSciNetMATHGoogle Scholar
  284. 284.
    G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, M. Inguscio, Anderson localization of a non-interacting Bose-Einstein condensate. Nature 453, 895–899 (2008)ADSGoogle Scholar
  285. 285.
    A. Rodríguez, V. Schwämmle, C. Tsallis, Strictly and asymptotically scale invariant probabilistic models of N correlated binary random variables having q-Gaussians as N →  limiting distributions. J. Stat. Mech.-Theory Exp. 2008, P09006 (2008)Google Scholar
  286. 286.
    R.M. Rosenberg, The normal modes of nonlinear n-degree-of-freedom systems. J. Appl. Mech. 29, 7–14 (1962)ADSMATHGoogle Scholar
  287. 287.
    V.M. Rothos, T. Bountis, Mel’nikov analysis of phase space transport in a N-degree-of-freedom Hamiltonian system. Nonlinear Anal. Theor. 30, 1365–1374 (1997)MathSciNetMATHGoogle Scholar
  288. 288.
    V.M. Rothos, T. Bountis, Mel’nikov’s vector and singularity analysis of periodically perturbed 2 d.o.f. Hamiltonian systems, in Hamiltonian Systems of Three or More Degrees of Freedom, ed. by C. Simó. NATO Advanced Study Institute, vol. 533 (Kluwer, Dordrecht, 1999), pp. 544–548Google Scholar
  289. 289.
    D. Ruelle, Ergodic theory of differentiable dynamical systems. Publ. Math. IHÉS 50, 27–58 (1979)MathSciNetMATHGoogle Scholar
  290. 290.
    D. Ruelle, Measures describing a turbulent flow. Ann. NY Acad.Sci. 357, 1–9 (1980)Google Scholar
  291. 291.
    D. Ruelle, Large volume limit of the distribution of characteristic exponents in turbulence. Commun. Math. Phys. 87, 287–302 (1982)MathSciNetADSMATHGoogle Scholar
  292. 292.
    G. Ruiz, C. Tsallis, Nonextensivity at the edge of chaos of a new universality class of one-dimensional unimodal dissipative maps. Eur. Phys. J. B 67, 577–584 (2009)MathSciNetADSMATHGoogle Scholar
  293. 293.
    G. Ruiz, T. Bountis, C. Tsallis, Time-evolving statistics of chaotic orbits of conservative maps in the context of the central limit theorem. Int. J. Bifurc. Chaos. (2012, In Press) arXiv:1106.6226Google Scholar
  294. 294.
    V.P. Sakhnenko, G.M. Chechin, Symmetrical selection rules in nonlinear dynamics of atomic systems. Sov. Phys. Dokl. 38, 219–221 (1993)Google Scholar
  295. 295.
    V.P. Sakhnenko, G.M. Chechin, Bushes of modes and normal modes for nonlinear dynamical systems with discrete symmetry. Sov. Phys. Dokl. 39, 625–628 (1994)MathSciNetGoogle Scholar
  296. 296.
    Zs. Sándor, B. Érdi, C. Efthymiopoulos, The phase space structure around L 4 in the restricted three-body problem. Celest. Mech. Dyn. Astron. 78, 113–123 (2000)Google Scholar
  297. 297.
    Zs. Sándor, B. Érdi, A. Széll, B. Funk, The relative Lyapunov indicator: an efficient method of chaos detection. Celest. Mech. Dyn. Astron. 90, 127–138 (2004)Google Scholar
  298. 298.
    K.W. Sandusky, 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)ADSGoogle Scholar
  299. 299.
    T. Schwartz, G. Bartal, S. Fishman, M. Segev, Transport and Anderson localization in disordered two-dimensional photonic lattices. Nature 446, 52–55 (2007)ADSGoogle Scholar
  300. 300.
    H. Segur, M.D. Kruskal, Nonexistence of small-amplitude breather solutions in ϕ4 theory. Phys. Rev. Lett. 58, 747–750 (1987)MathSciNetADSGoogle Scholar
  301. 301.
    V.D. Shapiro, R.Z. Sagdeev, Nonlinear wave-particle interaction and conditions for the applicability of quasilinear theory. Phys. Rep. 283, 49–71 (1997)ADSGoogle Scholar
  302. 302.
    H. Shiba, N. Ito, Anomalous heat conduction in three-dimensional nonlinear lattices. J. Phys. Soc. Jpn. 77, 05400 (2008)Google Scholar
  303. 303.
    S. Shinohara, Low-dimensional solutions in the quartic Fermi-Pasta-Ulam system. J. Phys. Soc. Jpn. 71, 1802–1804 (2002)ADSGoogle Scholar
  304. 304.
    S. Shinohara, Low-dimensional subsystems in anharmonic lattices. Prog. Theor. Phys. Suppl. 150, 423–434 (2003)ADSGoogle Scholar
  305. 305.
    I.V. Sideris, Measure of orbital stickiness and chaos strength. Phys. Rev. E 73, 066217 (2006)ADSGoogle Scholar
  306. 306.
    A.J. Sievers, S. Takeno, Intrinsic localized modes in anharmonic crystals. Phys. Rev. Lett. 61, 970–973 (1988)ADSGoogle Scholar
  307. 307.
    Y.G. Sinai, Dynamical systems with elastic reflections. Russ. Math. Surv. 25(2), 137–189 (1970)MathSciNetMATHGoogle Scholar
  308. 308.
    Ya.G. Sinai, Gibbs measures in ergodic theory. Russ. Math. Surv. 27(4), 21–69 (1972)Google Scholar
  309. 309.
    Ch. Skokos, Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits. J. Phys. A-Math. Gen. 34, 10029–10043 (2001)MathSciNetADSMATHGoogle Scholar
  310. 310.
    Ch. Skokos, The Lyapunov characteristic exponents and their computation. Lect. Notes Phys. 790, 63–135 (2010)ADSGoogle Scholar
  311. 311.
    Ch. Skokos, S. Flach, Spreading of wave packets in disordered systems with tunable nonlinearity. Phys. Rev. E 82, 016208 (2010)ADSGoogle Scholar
  312. 312.
    Ch. Skokos, E. Gerlach, Numerical integration of variational equations. Phys. Rev. E 82, 036704 (2010)MathSciNetADSGoogle Scholar
  313. 313.
    Ch. Skokos, Ch. Antonopoulos, T.C. Bountis, M.N. Vrahatis, How does the smaller alignment index (SALI) distinguish order from chaos? Prog. Theor. Phys. Suppl. 150, 439–443 (2003)ADSGoogle Scholar
  314. 314.
    Ch. Skokos, Ch. Antonopoulos, T.C. Bountis, M.N. Vrahatis, Detecting order and chaos in Hamiltonian systems by the SALI method. J. Phys. A-Math. Gen. 37, 6269–6284 (2004)MathSciNetADSGoogle Scholar
  315. 315.
    Ch. Skokos, T.C. Bountis, Ch. Antonopoulos, Geometrical properties of local dynamics in Hamiltonian systems: the generalized alignment index (GALI) method. Phys. D 231, 30–54 (2007)MathSciNetMATHGoogle Scholar
  316. 316.
    Ch. Skokos, T. Bountis, Ch. Antonopoulos, Detecting chaos, determining the dimensions of tori and predicting slow diffusion in Fermi-Pasta-Ulam lattices by the generalized alignment index method. Eur. Phys. J.-Spec. Top. 165, 5–14 (2008)Google Scholar
  317. 317.
    Ch. Skokos, D.O. Krimer, S. Komineas, S. Flach, Delocalization of wave packets in disordered nonlinear chains. Phys. Rev. E 79, 056211 (2009)MathSciNetADSGoogle Scholar
  318. 318.
    A. Smerzi, A. Trombettoni, Nonlinear tight-binding approximation for Bose-Einstein condensates in a lattice. Phys. Rev. A 68, 023613 (2003)MathSciNetADSGoogle Scholar
  319. 319.
    P. Soulis, T. Bountis, R. Dvorak, Stability of motion in the Sitnikov 3-body problem. Celest. Mech. Dyn. Astron. 99, 129–148 (2007)MathSciNetADSMATHGoogle Scholar
  320. 320.
    P.S. Soulis, K.E. Papadakis, T. Bountis, Periodic orbits and bifurcations in the Sitnikov four-body problem. Celest. Mech. Dyn. Astron. 100, 251–266 (2008)MathSciNetADSGoogle Scholar
  321. 321.
    M. Spivak, Comprehensive Introduction to Differential Geometry, vol. 1 (Perish Inc., Houston, 1999)MATHGoogle Scholar
  322. 322.
    P. Stránský, P. Hruška, P. Cejnar, Quantum chaos in the nuclear collective model: classical-quantum correspondence. Phys. Rev. E 79, 046202 (2009)ADSGoogle Scholar
  323. 323.
    M. Strözer, P. Gross, C.M. Aegerter, G. Maret, Observation of the critical regime near Anderson localization of light. Phys. Rev. Lett. 96, 063904 (2006)ADSGoogle Scholar
  324. 324.
    Á. Süli, Motion indicators in the 2D standard map. PADEU 17, 47–62 (2006)ADSGoogle Scholar
  325. 325.
    A. Széll, B. Érdi, Z. Sándor, B. Steves, Chaotic and stable behaviour in the Caledonian symmetric four-body problem. Mon. Not. R. Astron. Soc. 347, 380–388 (2004)ADSGoogle Scholar
  326. 326.
    M. Terraneo, M. Peyrard, G. Casati, Controlling the energy flow in nonlinear lattices: a model for a thermal rectifier. Phys. Rev. Lett. 88, 094302 (2002)ADSGoogle Scholar
  327. 327.
    U. Tirnakli, C. Beck, C. Tsallis, Central limit behavior of deterministic dynamical systems. Phys. Rev. E 75, 040106 (2007)ADSGoogle Scholar
  328. 328.
    U. Tirnakli, C. Tsallis, C. Beck, Closer look at time averages of the logistic map at the edge of chaos. Phys. Rev. E 79, 056209 (2009)MathSciNetADSGoogle Scholar
  329. 329.
    M. Toda, Theory of Nonlinear Lattices, (2nd edn.) (Springer, Berlin, 1989)MATHGoogle Scholar
  330. 330.
    S. Trillo, W. Torruellas (eds.), Spatial Solitons (Springer, Berlin, 2001)Google Scholar
  331. 331.
    A. Trombettoni, A. Smerzi, Discrete solitons and breathers with dilute Bose-Einstein condensates. Phys. Rev. Lett. 86, 2353–2356 (2001)ADSGoogle Scholar
  332. 332.
    C. Tsallis, Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World (Springer, New York, 2009)MATHGoogle Scholar
  333. 333.
    C. Tsallis, U. Tirnakli, Nonadditive entropy and nonextensive statistical mechanics – Some central concepts and recent applications. J. Phys. Conf. Ser. 201, 012001 (2010)ADSGoogle Scholar
  334. 334.
    G.P. Tsironis, An algebraic approach to discrete breather construction. J. Phys. A-Math. Theor. 35, 951–957 (2002)MathSciNetADSMATHGoogle Scholar
  335. 335.
    S. Umarov, C. Tsallis, S. Steinberg, On a q-central limit theorem consistent with nonextensive statistical mechanics. Milan J. Math. 76, 307–328 (2008)MathSciNetMATHGoogle Scholar
  336. 336.
    S. Umarov, C. Tsallis, M. Gell-Mann, S. Steinberg, Generalization of symmetric α-stable Lévy distributions for q > 1. J. Math. Phys. 51, 033502 (2010)MathSciNetADSGoogle Scholar
  337. 337.
    A.A. Vedenov, E.P. Velikhov, R.Z. Sagdeev, Nonlinear oscillations of rarified plasma. Nucl. Fusion 1, 82–100 (1961)Google Scholar
  338. 338.
    H. Veksler, Y. Krivolapov, S. Fishman, Spreading for the generalized nonlinear Schrödinger equation with disorder. Phys. Rev. E 80, 037201 (2009)ADSGoogle Scholar
  339. 339.
    H. Veksler, Y. Krivolapov, S. Fishman, Double-humped states in the nonlinear Schrödinger equation with a random potential. Phys. Rev. E 81, 017201 (2010)MathSciNetADSGoogle Scholar
  340. 340.
    N. Voglis, G. Contopoulos, Invariant spectra of orbits in dynamical systems. J. Phys. A-Math. Gen. 27, 4899–4909 (1994)MathSciNetADSMATHGoogle Scholar
  341. 341.
    N. Voglis, G. Contopoulos, C. Efthymiopoulos, Method for distinguishing between ordered and chaotic orbits in four-dimensional maps. Phys. Rev. E 57, 372–377 (1998)ADSGoogle Scholar
  342. 342.
    G. Voyatzis, S. Ichtiaroglou, On the spectral analysis of trajectories in near-integrable Hamiltonian systems. J. Phys. A-Math. Gen. 25, 5931–5943 (1992)MathSciNetADSMATHGoogle Scholar
  343. 343.
    J.-S. Wang, B. Li, Intriguing heat conduction of a chain with transverse motions. Phys. Rev. Lett. 92, 074302 (2004)ADSGoogle Scholar
  344. 344.
    E.T. Whittaker, G.N. Watson, A Course in Modern Analysis, 4th edn. (Cambridge University Press, Cambridge, 1927)/(Cambridge Mathematical Library, Cambridge, 2002)Google Scholar
  345. 345.
    D.S. Wiersma, P. Bartolini, A. Lagendijk, R. Righini, Localization of light in a disordered medium. Nature 390, 671–673 (1997)ADSGoogle Scholar
  346. 346.
    S. Wiggins, Applied Nonlinear Dynamical Systems and Chaos (Springer, New York, 1990)MATHGoogle Scholar
  347. 347.
    S. Wiggins, Chaotic Transport in Dynamical Systems (Springer, New York, 1992)MATHGoogle Scholar
  348. 348.
    N. Yang, G. Zhang, B. Li, Carbon nanocone: a promising thermal rectifier. Appl. Phys. Lett. 93, 243111 (2008)ADSGoogle Scholar
  349. 349.
    H. Yoshida, Construction of higher order symplectic integrators. Phys. Lett. A 150, 262–268 (1990)MathSciNetADSGoogle Scholar
  350. 350.
    H. Yoshida, Recent progress in the theory and application of symplectic integrators. Celest. Mech. Dyn. Astr. 56, 27–43 (1993)ADSMATHGoogle Scholar
  351. 351.
    K. Yoshimura, Modulational instability of zone boundary mode in nonlinear lattices: rigorous results. Phys. Rev. E 70, 016611 (2004)MathSciNetADSGoogle Scholar
  352. 352.
    N.J. Zabusky, M.D. Kruskal, Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)ADSMATHGoogle Scholar
  353. 353.
    G.M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461–580 (2002)MathSciNetADSMATHGoogle Scholar
  354. 354.
    Y. Zou, D. Pazó, M.C. Romano, M. Thiel, J. Kurths, Distinguishing quasiperiodic dynamics from chaos in short-time series. Phys. Rev. E 76, 016210 (2007)MathSciNetADSGoogle Scholar
  355. 355.
    Y. Zou, M. Thiel, M.C. Romano, J. Kurths, Characterization of stickiness by means of recurrence. Chaos 17, 043101 (2007)MathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Tassos Bountis
    • 1
  • Haris Skokos
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of PatrasPatrasGreece
  2. 2.MPI for the Physics of Complex SystemsDresdenGermany
  3. 3.Physics DepartmentAristotle University of ThessalonikiThessalonikiGreece

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