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

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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.

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References

  1. Poincaré, H.: Acta Math. 13, 1 (1890)

    MATH  Google Scholar 

  2. Chirikov, B.V.: Phys. Rep. 52, 265 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  3. Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics. Springer, New York (1992)

    MATH  Google Scholar 

  4. Arnold, V.I.: Dokl. Akad. Nauk SSSR 156, 9 (1964)

    MathSciNet  Google Scholar 

  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)]

  6. Nekhoroshev, N.N.: Usp. Mat. Nauk 32(6), 5 (1977)

    MATH  Google Scholar 

  7. Lochak, P.: Usp. Mat. Nauk (Russ. Math. Surv.) 47(6), 57 (1992)

    ADS  MathSciNet  Google Scholar 

  8. Kaloshin, V., Levi, M.: SIAM Rev. 50(4), 702 (2008)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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. Chirikov, B.V., Vecheslavov, V.V.: J. Stat. Phys. 71, 243 (1993)

    Article  MATH  ADS  Google Scholar 

  11. Chirikov, B.V., Vecheslavov, V.V.: Sov. Phys. JETP 85(3), 616 (1997) [Zh. Eksp. Teor. Fiz. 112, 1132 (1997)]

    Article  ADS  Google Scholar 

  12. Chirikov, B.V., Lieberman, M.A., Shepelyansky, D.L., Vivaldi, F.: Physica D 14, 289 (1985)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Fermi, E., Pasta, J., Ulam, S., Tsingou, M.: Los Alamos Report No. LA-1940, 1955 (unpublished)

  14. Fermi, E.: Collected Papers, vol. 2. University of Chicago Press, Chicago (1965). 978 pages

    Google Scholar 

  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)

  16. Gallavotti, G. (ed.): The Fermi-Pasta-Ulam Problem. Springer Lecture Notes in Physics, vol. 728 (2008)

    MATH  Google Scholar 

  17. Benettin, G., Livi, R., Ponno, A.: J. Stat. Phys. 135(5–6), 873 (2009)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Pettini, M., Landolfi, M.: Phys. Rev. A 41(2), 768–783 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  19. Pettini, M., Cerruti-Sola, M.: Phys. Rev. A 44(2), 975–987 (1991)

    Article  ADS  Google Scholar 

  20. Casetti, L., Cerruti-Sola, M., Pettini, M., Cohen, E.G.D.: Phys. Rev. E 55(6), 6566–6574 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  21. Pettini, M., Casetti, L., Cerruti-Sola, M., Franzosi, R., Cohen, E.G.D.: CHAOS 15, 015106 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  22. Shepelyansky, D.L.: Phys. Rev. Lett. 70, 1787 (1993)

    Article  ADS  Google Scholar 

  23. Molina, M.I.: Phys. Rev. B 58(19), 12547 (1998)

    Article  ADS  Google Scholar 

  24. Pikovsky, A.S., Shepelyansky, D.L.: Phys. Rev. Lett. 100(9), 094101 (2008)

    Article  ADS  Google Scholar 

  25. Garcia-Mata, I., Shepelyansky, D.L.: Eur. Phys. J. B 71(1), 121 (2009)

    Article  ADS  Google Scholar 

  26. Flach, S., Krimer, D.O., Skokos, C.: Phys. Rev. Lett. 102(2), 024101 (2009)

    Article  ADS  Google Scholar 

  27. Skokos, C., Krimer, D.O., Komineas, S., Flach, S.: Phys. Rev. E 79(5, Part 2), 056211 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  28. Mulansky, M., Ahnert, K., Pikovsky, A., Shepelyansky, D.L.: Phys. Rev. E 80, 056212 (2009)

    Article  ADS  Google Scholar 

  29. Skokos, Ch., Flach, S.: Phys. Rev. E 82(1), 016208 (2010)

    Article  ADS  Google Scholar 

  30. Flach, S.: Chem. Phys. 375(2–3), 548 (2010)

    Article  ADS  Google Scholar 

  31. Laptyeva, T.V., Bodyfelt, J.D., Krimer, D.O., Skokos, Ch., Flach, S.: Europhys. Lett. 91(3), 30001 (2010)

    Article  ADS  Google Scholar 

  32. Mulansky, M., Pikovsky, A.: Europhys. Lett. 90, 10015 (2010)

    Article  ADS  Google Scholar 

  33. Johansson, M., Kopidakis, G., Aubry, S.: Europhys. Lett. 91(5), 50001 (2010)

    Article  ADS  Google Scholar 

  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.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  35. Krimer, D.O., Flach, S.: Phys. Rev. E 82(4, Part 2), 046221 (2010)

    Article  ADS  Google Scholar 

  36. Pikovsky, A., Fishman, S.: Phys. Rev. E 83, 025201 (2011)

    Article  ADS  Google Scholar 

  37. Wang, W.-M., Zhang, Z.: e-print arXiv:0805.3520 (2008)

  38. Bourgain, J., Wang, W.-M.: J. Eur. Math. Soc. 10, 1 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  39. Kaneko, K., Konishi, T.: Phys. Rev. A 40(10), 40 (1989)

    Article  Google Scholar 

  40. Konishi, T., Kaneko, K.: J. Phys. A 32, L715 (1990)

    Article  MathSciNet  Google Scholar 

  41. Falcioni, M., Paladin, G., Vulpiani, A.: Europhys. Lett. 10(3), 201 (1989)

    Article  ADS  Google Scholar 

  42. Falcioni, M., Marconi, U.M.B., Vulpiani, A.: Phys. Rev. A 44, 2263 (1991)

    Article  ADS  Google Scholar 

  43. Lichtenberg, A.J., Aswani, A.M.: Phys. Rev. E 57(5), 5325 (1998)

    Article  ADS  Google Scholar 

  44. Ahnert, K., Pikovsky, A.: Phys. Rev. E 79, 026209 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  45. Mulansky, M., Ahnert, K., Pikovsky, A.: Phys. Rev. E 83, 026205 (2011)

    Article  ADS  Google Scholar 

  46. Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  47. Chirikov, B.V., Shepelyansky, D.L.: Sov. J. Nucl. Phys. 36, 908 (1982)

    MATH  Google Scholar 

  48. Shepelyansky, D.L.: Phys. Rev. E 82, 055202(R) (2010)

    Article  ADS  MathSciNet  Google Scholar 

  49. Reiner, M.: The Deborah number. Phys. Today 17(1), 62 (1964)

    Article  Google Scholar 

  50. Malkin, A.Ya., Isayev, A.I.: Rheology: Concepts, Methods, & Applications. ChemTech Publ., Toronto (2006)

    Google Scholar 

  51. Rao, M.A.: Rheology of Fluid and Semisolid Foods: Principles and Applications. Springer, Berlin (2007)

    Book  Google Scholar 

  52. Barenblatt, G.I.: Scaling. Cambridge Univ. Press, Cambridge (2003)

    MATH  Google Scholar 

  53. Brambilla, G., Buzzaccaro, S., Piazza, R., Berthier, L., Cilelleti, L.: Phys. Rev. Lett. 106, 118302 (2011)

    Article  ADS  Google Scholar 

  54. Sollich, P., Lequeux, E., Hébraud, P., Cates, M.E.: Phys. Rev. Lett. 78, 2020 (1997)

    Article  ADS  Google Scholar 

  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 

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Authors and Affiliations

  1. Department of Physics and Astronomy, Potsdam University, Karl-Liebknecht-Str. 24, 14476, Potsdam-Golm, Germany

    M. Mulansky, K. Ahnert & A. Pikovsky

  2. Laboratoire de Physique Théorique du CNRS (IRSAMC), UPS, Université de Toulouse, 31062, Toulouse, France

    D. L. Shepelyansky

Authors
  1. M. Mulansky
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  2. K. Ahnert
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  3. A. Pikovsky
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  4. D. L. Shepelyansky
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Correspondence to A. Pikovsky.

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Mulansky, M., Ahnert, K., Pikovsky, A. et al. Strong and Weak Chaos in Weakly Nonintegrable Many-Body Hamiltonian Systems. J Stat Phys 145, 1256–1274 (2011). https://doi.org/10.1007/s10955-011-0335-3

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  • DOI: https://doi.org/10.1007/s10955-011-0335-3

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