Analysis of the chaotic behaviour of orbits diffusing along the Arnold web

  • Claude Froeschlé
  • Elena Lega
  • Massimiliano Guzzo
Conference paper


In a previous work [Guzzo et al. DCDS B 5, 687–698 (2005)] we have provided numerical evidence of global diffusion occurring in slightly perturbed integrable Hamiltonian systems and symplectic maps. We have shown that even if a system is sufficiently close to be integrable, global diffusion occurs on a set with peculiar topology, the so-called Arnold web, and is qualitatively different from Chirikov diffusion, occurring in more perturbed systems. In the present work we study in more detail the chaotic behaviour of a set of 90 orbits which diffuse on the Arnold web. We find that the largest Lyapunov exponent does not seem to converge for the individual orbits while the mean Lyapunov exponent on the set of 90 orbits does converge. In other words, a kind of average mixing characterizes the diffusion. Moreover, the Local Lyapunov Characteristic Numbers (LLCNs), on individual orbits appear to reflect the different zones of the Arnold web revealed by the Fast Lyapunov Indicator. Finally, using the LLCNs we study the ergodicity of the chaotic part of the Arnold web.


KAM and Nekhoroshev theorem Arnold’s diffusion Chaos detection 


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  1. Arnold, V.I.: Proof of a theorem by A.N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surveys 18, 9–36 (1963)CrossRefADSGoogle Scholar
  2. Arnold, V.I.: Instability of dynamical systems with several degrees of freedom. Sov. Math. Dokl 6, 581–585 (1964)Google Scholar
  3. Contopoulos, G.: Order and Chaos in Dynamical Astronomy. Springer-Verlag, Berlin, Heidelberg (2002)zbMATHGoogle Scholar
  4. Chirikov, B.V.: An universal instability of many dimensional oscillator system. Phys. Rep. 52, 263–379 (1979)CrossRefADSMathSciNetGoogle Scholar
  5. Froeschlé, C. Froeschlé, Ch., Lohinger, E.: Generalized Lyapunov characteristics indicators and corresponding like entropy of the standard mapping, Celest. Mech. Dynam. Astron. 56, 307–315 (1993)CrossRefADSzbMATHGoogle Scholar
  6. Froeschlé, C., Guzzo, M., Lega, E.: Graphical evolution of the Arnold web: from order to chaos. Science 289, 2108–2110 (2000)CrossRefADSGoogle Scholar
  7. Froeschlé, C., Lega, E., Gonczi, R.: Fast Lyapunov indicators. Application to asteroidal motion. Celest. Mech. Dynam. Astron. 67, 41–62 (1997)CrossRefADSzbMATHGoogle Scholar
  8. Froeschlé, C., Guzzo, M., Lega, E.: Local and global diffusion along resonant lines in discrete quasi-integrable dynamical systems. Celest. Mech. Dynam. Astron. 92, 243–255 (2005)CrossRefADSzbMATHGoogle Scholar
  9. Guzzo, M.: A direct proof of the Nekhoroshev theorem for nearly integrable symplectic maps. Ann. Henry Poincaré 5, 1013–1039 (2004)CrossRefMathSciNetzbMATHADSGoogle Scholar
  10. Guzzo, M., Lega, E., Froeschlé, C.: On the numerical detection of the effective stability of chaotic motions in quasi-integrable systems. Physica D 163, 1–25 (2002)CrossRefADSMathSciNetzbMATHGoogle Scholar
  11. Guzzo, M., Lega, E., Froeschlé, C.: First numerical evidence of global Arnold diffusion in quasi-integrable systems. DCDS-B, 5, 687–698 (2005)zbMATHCrossRefGoogle Scholar
  12. Kolmogorov, A.N.: On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian. Dokl. Akad. Nauk SSSR 98, 527 (1954)MathSciNetzbMATHGoogle Scholar
  13. Kuksin, S. B.: On the inclusion of an almost integrable analytic symplectomorphism into a Hamiltonian flow. Russ. J. Math. Phys. 1, 191–207 (1993)MathSciNetzbMATHGoogle Scholar
  14. Kuksin S.B., Pöschel, J.: On the inclusion of analytic symplectic maps in analytic Hamiltonian flows and its applications. Nonlinear Diff. Eq. Appl. 12, 96–116 (1994)Google Scholar
  15. Lega, E., Guzzo, M., Froeschlé, C.: Detection ofArnold diffusion in Hamiltonian systems. Physica D 182:179–187 (2003)CrossRefADSMathSciNetzbMATHGoogle Scholar
  16. Morbidelli, A., Giorgilli, A.: Super-exponential stability of KAM tori. J. Stat. Phys. 78, 1607 (1995a)CrossRefADSMathSciNetzbMATHGoogle Scholar
  17. Morbidelli, A.: Modern Celestial Mechanics. Aspects of Solar System Dynamics. Taylor and Francis (2002).Google Scholar
  18. Morbidelli, A., Giorgilli, A.: On a connection between KAM and Nekhoroshev’s theorems. Physica D 86, 514–516 (1995b)CrossRefADSMathSciNetzbMATHGoogle Scholar
  19. Moser, J.: On invariant curves of area-preserving maps of an annulus. Comm. Pure Appl. Math. 11, 81–114 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  20. Nekhoroshev, N.N.: Exponential estimates of the stability time of near-integrable Hamiltonian systems. Russ. Math. Surveys 32, 1–65 (1977)CrossRefzbMATHADSGoogle Scholar
  21. Poschel, J.: Nekhoroshev estimates for quasi-convex Hamiltonian systems. Math. Z. 213, 187 (1993)MathSciNetCrossRefGoogle Scholar
  22. Vergassola, M.: Standard and anomalous diffusion in dynamical systems. In: Benest, D., Froeschlé, C. (eds.), Analysis and Modeling of Discrete Dynamical Systems. Gordon and Breach Science Publishers (1998)Google Scholar
  23. Voglis, N., Contopoulos, G.: Invariant spectra of orbits in dynamical systems. J. Phys. A: Math. Gen. 27, 4899–4909 (1994)CrossRefADSMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • Claude Froeschlé
    • 1
  • Elena Lega
    • 1
  • Massimiliano Guzzo
    • 2
  1. 1.Observatoire de la Côte d’AzurNice Cedex 4France
  2. 2.Dipartimento di Matematica pura e Applicata Università di PadovaPadovaItaly

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