Foundations of Computational Mathematics

, Volume 15, Issue 1, pp 89–123 | Cite as

Effect of Islands in Diffusive Properties of the Standard Map for Large Parameter Values

  • Narcís Miguel
  • Carles SimóEmail author
  • Arturo Vieiro


In this paper we review, based on massive, long-term, numerical simulations, the effect of islands on the statistical properties of the standard map for large parameter values. Different sources of discrepancy with respect to typical diffusion are identified, and their individual roles are compared and explained in terms of available limit models.


Standard map Accelerator modes Diffusion Stickiness 

Mathematics Subject Classification

Primary: 37A50 Secondary: 37D45 37J99 37M99 



The authors were supported by Grants MTM2010-16425 (Spain) and 2009 SGR 67 (Catalonia). We thank Jaume Timoneda for maintaining the computing facilities of the Dynamical Systems Group of the Universitat de Barcelona, which were the main facilities used in this work, up to a total of floating point operations that exceeds 5 exaflop. The authors also want to thank Robert MacKay and Vered Rom-Kedar for helpful discussions on related topics.


  1. 1.
    B. V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep. 52 (1979), 264–379.Google Scholar
  2. 2.
    B- V. Chirikov, Chaotic dynamics in Hamiltonian systems with divided phase space, in Proceed. Sitges Conference on Dynamical Systems and Chaos (L. Garrido, ed.) Lecture Notes in Physics 179, Springer, 1983.Google Scholar
  3. 3.
    B. V. Chirikov and D. L. Shepelyansky, Statistics of Poincaré recurrences and the structure of the stochastic layer of a nonlinear resonance, Ninth international conference on nonlinear oscillations, Vol. 2 Kiev, 1981. Translation to English: Plasma physics laboratory, Princeton University, 1983.Google Scholar
  4. 4.
    B. V. Chirikov and D. L. Shepelyansky, Correlation properties of dynamical chaos in Hamiltonian systems, Physica 13 D (1984), 395–400.Google Scholar
  5. 5.
    G. Contopoulos and M. Harsoula, Stickiness effects in conservative systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 (2010), 2005–2043.Google Scholar
  6. 6.
    P. Duarte, Plenty of elliptic islands for the standard family of area preserving maps, Ann. Inst. H. Poincar Anal. Non Linaire 11 (1994), 359–409.Google Scholar
  7. 7.
    F. Dumortier, S. Ibáñez, H. Kokubu and C. Simó, About the unfolding of a Hopf-zero singularity, Discrete Contin. Dyn. Syst. Ser. A 33 (2013), 4435–4471.Google Scholar
  8. 8.
    K. M. Frahm and D. L. Shepelyansky, Ulam method for the Chirikov standard map, Eur. Phys. J. B. 76 (2010), 57–68.Google Scholar
  9. 9.
    K. M. Frahm and D. L. Shepelyansky, Poincaré recurrences and Ulam method for the Chirikov standard map. Eur. Phys. J. B. 86 (2013), 322–333.Google Scholar
  10. 10.
    A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem. J. Diff. Eq. 77 (1989), 167–198.Google Scholar
  11. 11.
    M. Gonchenko, Homoclinic phenomena in conservative systems, Ph.D. Thesis, Universitat Politècnica de Catalunya, 2013.Google Scholar
  12. 12.
    S. V. Gonchenko and L. P. Shilnikov, On two-dimensional area-preserving mappings with homoclinic tangencies, Russian Math. Dokl. 63 (2001), 395–399.Google Scholar
  13. 13.
    J. M. Greene, A method for determining stochastic transition, J. Math. Phys. 6 (1979), 1183–1201.Google Scholar
  14. 14.
    M. Hénon, Numerical study of quadratic area-preserving mappings, Quart. Appl. Math. 27 (1969), 291–312.Google Scholar
  15. 15.
    I. Jungreis, A method for proving that monotone twist maps have no invariant circles, Ergod. Th. & Dynam. Sys. 11 (1991), 79–84.Google Scholar
  16. 16.
    C. F. F. Karney, Long time correlations in the stochastic regime, Physica 3 D (1983), 360–380.Google Scholar
  17. 17.
    C. F. F. Karney, A. Rechester and B. White, Effect of noise on the standard mapping, Physica 4 D (1982), 425–438.Google Scholar
  18. 18.
    F. Ledrappier, M. Shub, C. Simó and A. Wilkinson, Random versus deterministic exponents in a rich family of diffeomorphisms, J. Stat Phys. 113 (2003), 85–149.Google Scholar
  19. 19.
    A. J. Lichtenberg and M. A. Lieberman, Regular And Chaotic Dynamics, Applied Mathematical Sciences, 2nd edition, Springer, New York, 1992.Google Scholar
  20. 20.
    R. S. MacKay, A renormalisation approach to invariant circles in area-preserving maps, Physica 7 D (1983), 283–300. Order in chaos (Los Alamos, N.M., 1982).Google Scholar
  21. 21.
    R. S. MacKay, Renormalisation in area-preserving maps, Advanced Series in Nonlinear Dynamics, 6. World Scientific. 1992.Google Scholar
  22. 22.
    R. S. MacKay, J. D. Meiss and I. C. Percival, Transport in Hamiltonian systems, Physica 13 D (1984), 55–81.Google Scholar
  23. 23.
    R. S. MacKay and I. C. Percival, Converse KAM: theory and practice, Comm. Math. Phys. 98 (1985), 469–512.Google Scholar
  24. 24.
    R. S. MacKay and J. Stark, Locally most robust circles and boundary circles for area-preserving maps, Nonlinearity 5 (1992), 867–888.Google Scholar
  25. 25.
    S. Marmi and J. Stark, On the standard map critical function, Nonlinearity 5 (1992), 743–761.Google Scholar
  26. 26.
    J. N. Mather, Nonexistence of invariant circles, Ergod. Th. & Dynam. Sys. 4 (1984), 301–309.Google Scholar
  27. 27.
    J. D. Meiss, Class renormalization: Islands around islands, Phys. Rev. A 34 (1986), 2375–2383.Google Scholar
  28. 28.
    J. D. Meiss, Average exit time for volume-preserving maps, Chaos 7 (1997), 139–147.Google Scholar
  29. 29.
    J. D. Meiss, J. R. Cary, C. Grebogi, J. D. Crawford, A. N. Kaufman and H. D. I. Abarbanel, Correlations of Periodic Area-Preserving Maps, Physica 6 D (1983), 375–384.Google Scholar
  30. 30.
    J. D. Meiss and E. Ott, Markov tree model of transport in area-preserving maps, Physica 20 D (1986), 387–402.Google Scholar
  31. 31.
    N. Miguel, C. Simó, and A. Vieiro, From the Hénon conservative map to the Chirikov standard map for large parameter values, Regular and Chaotic Dynamics 20 (2013), 469–489.Google Scholar
  32. 32.
    N. N. Nekhorosev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Russian Mathematical Surveys 32 (1977), 1–65.Google Scholar
  33. 33.
    A. Olvera and C. Simó, An obstruction method for the destruction of invariant curves, Physica 26 D (1987), 181–192.Google Scholar
  34. 34.
    Ya. Pesin, Characteristic exponents and smooth ergodic theory, Russian Math. Surveys 32 (1977), 55–114.Google Scholar
  35. 35.
    V. Rom-Kedar and G. Zaslavsky, Islands of accelerator modes and homoclinic tangles, Chaos 9 (1999), 697–705.Google Scholar
  36. 36.
    J. Sánchez, M. Net and C. Simó, Computation of invariant tori by Newton-Krylov methods in large-scale dissipative systems, Physica 239 D (2010), 123–133.Google Scholar
  37. 37.
    C. Simó, Analytical and numerical computation of invariant manifolds, in Modern methods in celestial mechanics (D. Benest and C. Froeschlé, eds), Editions Frontières, Paris, 1990, pp. 285–330.Google Scholar
  38. 38.
    C. Simó, Some properties of the global behaviour of conservative low dimensional systems, in Foundations of Computational Mathematics: Hong Kong 2008 (F. Cucker et al. eds), London Math. Soc. Lecture Notes Series 363, 2009, New York, Cambridge University Press, pp. 163–189.Google Scholar
  39. 39.
    C. Simó, P. Sousa-Silva and M. Terra, Practical Stability Domains near \(L_{4,5}\) in the Restricted Three-Body Problem: Some preliminary facts, in Progress and Challenges in Dynamical Systems, Vol. 54, Springer, 2013, pp. 367–382.Google Scholar
  40. 40.
    C. Simó and A. Vieiro, Resonant zones, inner and outer splittings in generic and low order resonances of Area Preserving Maps, Nonlinearity 22 (2009), 1191–1245.Google Scholar
  41. 41.
    C. Simó and A. Vieiro, Dynamics in chaotic zones of area preserving maps: close to separatrix and global instability zones, Physica 240 D (2011), 732–753.Google Scholar
  42. 42.
    R. Venegeroles, Calculation of superdiffusion for the Chirikov-Taylor model, Physical Review Letters 101 (2008): 054102.Google Scholar
  43. 43.
    R. Venegeroles, Universality of Algebraic Laws in Hamiltonian Systems, Physical Review Letters 102 (2009): 064101.Google Scholar
  44. 44.
    G. M. Zaslavsky, Dynamical traps, Physica 168–169 D (2002), 292–304.Google Scholar
  45. 45.
    G. M. Zaslavsky, M. Edelman and B. A. Niyazov, Self-similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics, Chaos 7 (1997), 159–181.Google Scholar

Copyright information

© SFoCM 2014

Authors and Affiliations

  1. 1.Dept. de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelona, CatalunyaSpain

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