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On the connection between the Nekhoroshev theorem and Arnold diffusion

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Abstract

The analytical techniques of the Nekhoroshev theorem are used to provide estimates on the coefficient of Arnold diffusion along a particular resonance in the Hamiltonian model of Froeschlé et al. (Science 289:2108–2110, 2000). A resonant normal form is constructed by a computer program and the size of its remainder ||R opt || at the optimal order of normalization is calculated as a function of the small parameter \({\epsilon}\) . We find that the diffusion coefficient scales as \({D \propto ||R_{opt}||^3}\) , while the size of the optimal remainder scales as \({||R_{opt}|| \propto {\rm exp}(1/\epsilon^{0.21})}\) in the range \({10^{-4} \leq \epsilon \leq 10^{-2}}\) . A comparison is made with the numerical results of Lega et al. (Physica D 182:179–187, 2003) in the same model.

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References

  • Arnold V.I.: Instability of dynamical systems with several degrees of freedom. Sov. Math. Dokl. 6, 581–585 (1964)

    Google Scholar 

  • Benettin G., Galgani L., Giorgilli A.: A proof of Nekhoroshev’s theorem for the stability times in nearly integrable Hamiltoniar systems. Cel. Mech. 37, 1–25 (1985)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Boccaletti D., Pucacco G.: Theory of Orbits. Springer, Berlin (1996)

    Google Scholar 

  • Chirikov B.V., Lieberman M.A., Shepelyansky D.L., Vivaldi F.M.: A theory of modulational diffusion. Physica 14, 289–304 (1985)

    MATH  MathSciNet  Google Scholar 

  • Contopoulos G.: Order and Chaos in Dynamical Astronomy. Springer, Berlin (2002)

    MATH  Google Scholar 

  • Dumas H.S., Laskar J.: Global dynamics and long-time stability in Hamiltonian systems via numerical frequency analysis. Phys. Rev. Lett. 70, 2975–2979 (1993)

    Article  ADS  Google Scholar 

  • Ferraz-Mello S.: Canonical Perturbation Theories: Degenerate Systems and Resonance. Springer, New York (2007)

    MATH  Google Scholar 

  • Froeschlé C., Guzzo M., Lega E.: Graphical evolution of the Arnold web: from order to chaos. Science 289(5487), 2108–2110 (2000)

    Article  ADS  Google Scholar 

  • Froeschlé C., Guzzo M., Lega E.: Local and global diffusion along resonant lines in discrete quasi-integrable dynamical systems. Cel. Mech. Dyn. Astron. 92, 243–255 (2005)

    Article  MATH  ADS  Google Scholar 

  • Giordano C.M., Cincotta P.M.: Chaotic diffusion of orbits in systems with divided phase space. Astron. Astrophys. 423, 745–753 (2004)

    Article  ADS  Google Scholar 

  • Giorgilli, A.: Notes on exponential stability of Hamiltonian systems, in Dynamical Systems. Part I: Hamiltonian Systems and Celestial Mechanics, Pubblicazioni della Classe di Scienze, Scuola Normale Superiore, Pisa (2002)

  • Guzzo M., Lega E., Froeschlé C.: First numerical evidence of global Arnold diffusion in discrete quasi-integrable systems. Dis. Con. Dyn. Sys. B 5, 687–698 (2005)

    MATH  Google Scholar 

  • Kaneko K., Konishi T.: Diffusion in Hamiltonian dynamical systems. Phys. Rev. A 40, 6130 (1989)

    Article  ADS  Google Scholar 

  • Laskar J.: Global dynamics and diffusion. Physica D 67, 257–281 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Lega E., Guzzo M., Froeschlé C.: Detection of Arnold diffusion in Hamiltonian systems. Physica D 182, 179–187 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Lochak P.: Canonical perturbation theory via simultaneous approximation. Russ. Math. Surv. 47, 57–133 (1992)

    Article  MathSciNet  Google Scholar 

  • Morbidelli A.: Modern Celestial Mechanics. Aspects of Solar System Dynamics. Taylor and Francis, London (2002)

    Google Scholar 

  • Morbidelli A., Guzzo M.: The Nekhoroshev theorem and the asteroid belt dynamical system. Cel. Mech. Dyn. Astron. 65, 107–136 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Morbidelli A., Giorgilli A.: On the role of high order resonances in normal forms and in separatrix splitting. Physica D 102, 195–207 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Nekhoroshev N.N.: Exponential estimates of the stability time of near-integrable in Hamiltonian systems’. Russ. Math. Surv. 32(6), 1–65 (1977)

    Article  MATH  Google Scholar 

  • Pöshel J.: Math. Z. 213, 187 (1993)

    Article  MathSciNet  Google Scholar 

  • Skokos C., Contopoulo G., Polymilis C.: Structures in the phase space of a four dimensional symplectic map. Cel. Mech. Dyn. Astr. 65, 223–251 (1997)

    Article  MATH  ADS  Google Scholar 

  • Wood B.P., Lichtenberg A.J., Lieberman M.A.: Arnold diffusion in weakly coupled standard maps. Phys. Rev. A 42, 5885–5893 (1990)

    Article  ADS  MathSciNet  Google Scholar 

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  1. Research Center for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efessiou 4, Athens, 115 27, Greece

    Christos Efthymiopoulos

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  1. Christos Efthymiopoulos
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Correspondence to Christos Efthymiopoulos.

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Efthymiopoulos, C. On the connection between the Nekhoroshev theorem and Arnold diffusion. Celest Mech Dyn Astr 102, 49–68 (2008). https://doi.org/10.1007/s10569-008-9151-8

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  • DOI: https://doi.org/10.1007/s10569-008-9151-8

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