Journal of Statistical Physics

, Volume 145, Issue 5, pp 1256–1274 | Cite as

Strong and Weak Chaos in Weakly Nonintegrable Many-Body Hamiltonian Systems

  • M. Mulansky
  • K. Ahnert
  • A. Pikovsky
  • D. L. Shepelyansky
Article

Abstract

We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology.

Keywords

Lyapunov exponent Arnold diffusion Chaos spreading 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Poincaré, H.: Acta Math. 13, 1 (1890) MATHGoogle Scholar
  2. 2.
    Chirikov, B.V.: Phys. Rep. 52, 265 (1979) CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics. Springer, New York (1992) MATHGoogle Scholar
  4. 4.
    Arnold, V.I.: Dokl. Akad. Nauk SSSR 156, 9 (1964) MathSciNetGoogle Scholar
  5. 5.
    Chirikov, B.V.: Research concerning the theory of non-linear resonance and stochasticity. Report 267, Inst. of Nuclear Phys., Novosibirsk (1969) [English CERN Trans. 71-40, Geneva (1971)] Google Scholar
  6. 6.
    Nekhoroshev, N.N.: Usp. Mat. Nauk 32(6), 5 (1977) MATHGoogle Scholar
  7. 7.
    Lochak, P.: Usp. Mat. Nauk (Russ. Math. Surv.) 47(6), 57 (1992) ADSMathSciNetGoogle Scholar
  8. 8.
    Kaloshin, V., Levi, M.: SIAM Rev. 50(4), 702 (2008) CrossRefMATHADSMathSciNetGoogle Scholar
  9. 9.
    Chirikov, B.V., Vecheslavov, V.V.: KAM integrability. In: Rabinowitz, P.H., Zehnder, E. (eds.) Analysis, et cetera. Research Papers Published in honor of Jurgen Moser’s 60th Birthday, p. 219. Academic Press, New York (1990) Google Scholar
  10. 10.
    Chirikov, B.V., Vecheslavov, V.V.: J. Stat. Phys. 71, 243 (1993) CrossRefMATHADSGoogle Scholar
  11. 11.
    Chirikov, B.V., Vecheslavov, V.V.: Sov. Phys. JETP 85(3), 616 (1997) [Zh. Eksp. Teor. Fiz. 112, 1132 (1997)] CrossRefADSGoogle Scholar
  12. 12.
    Chirikov, B.V., Lieberman, M.A., Shepelyansky, D.L., Vivaldi, F.: Physica D 14, 289 (1985) CrossRefMATHADSMathSciNetGoogle Scholar
  13. 13.
    Fermi, E., Pasta, J., Ulam, S., Tsingou, M.: Los Alamos Report No. LA-1940, 1955 (unpublished) Google Scholar
  14. 14.
    Fermi, E.: Collected Papers, vol. 2. University of Chicago Press, Chicago (1965). 978 pages Google Scholar
  15. 15.
    Campbell, D.K., Rosenau, P., Zaslavsky, G. (eds.): A focus issue on “The “Fermi-Pasta-Ulam” Problem—The First 50 Years”. Chaos 15(1) (2005) Google Scholar
  16. 16.
    Gallavotti, G. (ed.): The Fermi-Pasta-Ulam Problem. Springer Lecture Notes in Physics, vol. 728 (2008) MATHGoogle Scholar
  17. 17.
    Benettin, G., Livi, R., Ponno, A.: J. Stat. Phys. 135(5–6), 873 (2009) CrossRefMATHADSMathSciNetGoogle Scholar
  18. 18.
    Pettini, M., Landolfi, M.: Phys. Rev. A 41(2), 768–783 (1990) CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Pettini, M., Cerruti-Sola, M.: Phys. Rev. A 44(2), 975–987 (1991) CrossRefADSGoogle Scholar
  20. 20.
    Casetti, L., Cerruti-Sola, M., Pettini, M., Cohen, E.G.D.: Phys. Rev. E 55(6), 6566–6574 (1997) CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Pettini, M., Casetti, L., Cerruti-Sola, M., Franzosi, R., Cohen, E.G.D.: CHAOS 15, 015106 (2005) CrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Shepelyansky, D.L.: Phys. Rev. Lett. 70, 1787 (1993) CrossRefADSGoogle Scholar
  23. 23.
    Molina, M.I.: Phys. Rev. B 58(19), 12547 (1998) CrossRefADSGoogle Scholar
  24. 24.
    Pikovsky, A.S., Shepelyansky, D.L.: Phys. Rev. Lett. 100(9), 094101 (2008) CrossRefADSGoogle Scholar
  25. 25.
    Garcia-Mata, I., Shepelyansky, D.L.: Eur. Phys. J. B 71(1), 121 (2009) CrossRefADSGoogle Scholar
  26. 26.
    Flach, S., Krimer, D.O., Skokos, C.: Phys. Rev. Lett. 102(2), 024101 (2009) CrossRefADSGoogle Scholar
  27. 27.
    Skokos, C., Krimer, D.O., Komineas, S., Flach, S.: Phys. Rev. E 79(5, Part 2), 056211 (2009) CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Mulansky, M., Ahnert, K., Pikovsky, A., Shepelyansky, D.L.: Phys. Rev. E 80, 056212 (2009) CrossRefADSGoogle Scholar
  29. 29.
    Skokos, Ch., Flach, S.: Phys. Rev. E 82(1), 016208 (2010) CrossRefADSGoogle Scholar
  30. 30.
    Flach, S.: Chem. Phys. 375(2–3), 548 (2010) CrossRefADSGoogle Scholar
  31. 31.
    Laptyeva, T.V., Bodyfelt, J.D., Krimer, D.O., Skokos, Ch., Flach, S.: Europhys. Lett. 91(3), 30001 (2010) CrossRefADSGoogle Scholar
  32. 32.
    Mulansky, M., Pikovsky, A.: Europhys. Lett. 90, 10015 (2010) CrossRefADSGoogle Scholar
  33. 33.
    Johansson, M., Kopidakis, G., Aubry, S.: Europhys. Lett. 91(5), 50001 (2010) CrossRefADSGoogle Scholar
  34. 34.
    Basko, D.M.: Weak chaos in the disordered nonlinear Schroedinger chain: destruction of Anderson localization by Arnold diffusion. Ann. Phys. 326(7), 1577–1655 (2011). Spec. Iss. CrossRefMATHADSMathSciNetGoogle Scholar
  35. 35.
    Krimer, D.O., Flach, S.: Phys. Rev. E 82(4, Part 2), 046221 (2010) CrossRefADSGoogle Scholar
  36. 36.
    Pikovsky, A., Fishman, S.: Phys. Rev. E 83, 025201 (2011) CrossRefADSGoogle Scholar
  37. 37.
    Wang, W.-M., Zhang, Z.: e-print arXiv:0805.3520 (2008)
  38. 38.
    Bourgain, J., Wang, W.-M.: J. Eur. Math. Soc. 10, 1 (2008) CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Kaneko, K., Konishi, T.: Phys. Rev. A 40(10), 40 (1989) CrossRefGoogle Scholar
  40. 40.
    Konishi, T., Kaneko, K.: J. Phys. A 32, L715 (1990) CrossRefMathSciNetGoogle Scholar
  41. 41.
    Falcioni, M., Paladin, G., Vulpiani, A.: Europhys. Lett. 10(3), 201 (1989) CrossRefADSGoogle Scholar
  42. 42.
    Falcioni, M., Marconi, U.M.B., Vulpiani, A.: Phys. Rev. A 44, 2263 (1991) CrossRefADSGoogle Scholar
  43. 43.
    Lichtenberg, A.J., Aswani, A.M.: Phys. Rev. E 57(5), 5325 (1998) CrossRefADSGoogle Scholar
  44. 44.
    Ahnert, K., Pikovsky, A.: Phys. Rev. E 79, 026209 (2009) CrossRefADSMathSciNetGoogle Scholar
  45. 45.
    Mulansky, M., Ahnert, K., Pikovsky, A.: Phys. Rev. E 83, 026205 (2011) CrossRefADSGoogle Scholar
  46. 46.
    Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (1992) Google Scholar
  47. 47.
    Chirikov, B.V., Shepelyansky, D.L.: Sov. J. Nucl. Phys. 36, 908 (1982) MATHGoogle Scholar
  48. 48.
    Shepelyansky, D.L.: Phys. Rev. E 82, 055202(R) (2010) CrossRefADSMathSciNetGoogle Scholar
  49. 49.
    Reiner, M.: The Deborah number. Phys. Today 17(1), 62 (1964) CrossRefGoogle Scholar
  50. 50.
    Malkin, A.Ya., Isayev, A.I.: Rheology: Concepts, Methods, & Applications. ChemTech Publ., Toronto (2006) Google Scholar
  51. 51.
    Rao, M.A.: Rheology of Fluid and Semisolid Foods: Principles and Applications. Springer, Berlin (2007) CrossRefGoogle Scholar
  52. 52.
    Barenblatt, G.I.: Scaling. Cambridge Univ. Press, Cambridge (2003) MATHGoogle Scholar
  53. 53.
    Brambilla, G., Buzzaccaro, S., Piazza, R., Berthier, L., Cilelleti, L.: Phys. Rev. Lett. 106, 118302 (2011) CrossRefADSGoogle Scholar
  54. 54.
    Sollich, P., Lequeux, E., Hébraud, P., Cates, M.E.: Phys. Rev. Lett. 78, 2020 (1997) CrossRefADSGoogle Scholar
  55. 55.
    Sollich, P.: Soft glassy rheology. In: Weiss, R.G., Terech, P. (eds.) Molecular Gels: Materials with Self-assembled Fibrillar Networks, p. 161. Springer, Berlin (2006) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • M. Mulansky
    • 1
  • K. Ahnert
    • 1
  • A. Pikovsky
    • 1
  • D. L. Shepelyansky
    • 2
  1. 1.Department of Physics and AstronomyPotsdam UniversityPotsdam-GolmGermany
  2. 2.Laboratoire de Physique Théorique du CNRS (IRSAMC), UPSUniversité de ToulouseToulouseFrance

Personalised recommendations