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Estimate of the Transition Value of Librational Invariant Curves

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Abstract

We investigate the break-down threshold of librational invariant curves. As a model problem, we consider a variant of a mapping introduced by M. Hénon, which well describes the dynamics of librational motions surrounding a stable invariant point. We verify in concrete examples the applicability of Greene's method, by computing the instability transition values of a sequence of periodic orbits approaching an invariant curve with fixed noble frequency. However, this method requires the knowledge of the location of the periodic orbits within a very good approximation. This task appears to be difficult to realize for a libration regime, due to the different topology of the phase space. To compute the break-down threshold, we tried an alternative method very easy to implement, based on the computation of the fast Lyapunov indicators and frequency analysis. Such technique does not require the knowledge of the periodic orbits, but again, it appears very difficult to have a precision better than Greene's method for the computation of the critical parameter.

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References

  • Arnold, V. I.: 1963, 'Proof of a theorem by A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian', Russ. Math. Surv. 18(9).

  • Benettin, G., Galgani, L., Giorgilli, A. and Strelcyn J. P.: 1980, 'Lyapunov characteristic exponents for smooth dynamical systems; a method for computing all of them', Part I: Theory, 9-20; Part II: Numerical applications, 21-30 Meccanica 15.

    Google Scholar 

  • Celletti, A.: 1995, 'Normal form for librational curves of the standard-map', Meccanica 3, 251-2604.

    Google Scholar 

  • Celletti, A. and Froeschlé, C.: 1995, 'On the determination of the stochasticity threshold of invariant curves', Int. J. Bifur. Chaos. 5(6), 1713.

    Google Scholar 

  • Falcolini, C. and de la Llave, R.: 1992, 'A rigorous partial justification of Greene's criterion', J. Stat Phys. 67(3/4), 609.

    Google Scholar 

  • Froeschlé, C.: 1970, 'A numerical study of the stochasticity of dynamical systems with two degrees of freedom', Astron. Astroph. 9, 15-23.

    Google Scholar 

  • Froeschlé, C.: 1984, 'The Lyapunov characteristic exponents and applications', J. de Mécanique théorique et appliquée, Numéro spécial 101-132.

  • Froeschlé, C., Gonczi, R. and Lega E.: 1997A, 'The fast Lyapunov indicator: a simple tool to detect weak chaos. Application to the structure of the main asteroidal belt', Planet. Space Sci. 45, 881-886.

  • Froeschlé, C., Lega, E. and Gonczi, R.: 1997B 'Fast Lyapunov indicators. Application to asteroidal motion', Celest. Mech. & Dyn. Astr. 67, 41-62.

    Google Scholar 

  • Froeschlé, C. and Lega, E.: 2000, 'On the structure of symplectic mappings. The fast Lyapunov indicator: a very sensitive tool', Preprint.

  • Froeschlé, C., Lega, E. and Guzzo, M.: 2000, 'Graphical evolution of the Arnold web: from order to chaos', Science 289.

  • Greene, J. M.: 1979, 'A method for determining a stochastic transition', J. Math. Phys. 20, 1183.

    Google Scholar 

  • Greene, J. M., MacKay, R. S. and Stark, J.: 1986, 'Boundary circles for area-preserving maps', Phys. D 21, 267-295.

    Google Scholar 

  • Hénon, M.: 1969, 'Numerical study of quadratic area-preserving mappings', Quart. Appl. Math. 27(3), 291-312.

    Google Scholar 

  • Hénon, M.: Private communication.

  • Kolmogorov, A. N.: 1954, 'On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian', Dokl. Akad. Nauk. SSR 98, 469.

    Google Scholar 

  • Laskar, J.: 1993, 'Frequency analysis for multi-dimensional systems. Global dynamics and diffusion', Physica D 67, 257.

    Google Scholar 

  • Lega, E. and Froeschlé, C.: 1996, 'Numerical investigations of the structure around an invariant KAM torus using the frequency map analysis', Physica D 95, 97-106.

    Google Scholar 

  • Lega, E. and Froeschlé, C.: 1997, 'Fast Lyapunov indicators. Comparison with other chaos indicators. Application to two and four-dimensional maps', In: Jacques Henrard and Rudolf Dvorak (eds), The Dynamical Behaviour of our Planetary System, Kluwer Academic Publishers.

  • Lega, E. and Froeschlé, C.: 2001, 'On the relation between fast Lyapunov indicator and periodic orbits for symplectic mappings', To appear in Celest. Mech. & Dyn. Astr. Preprint.

  • MacKay, R. S.: 1992, 'Greene's residue criterion', Nonlinearity 5, 161-187.

    Google Scholar 

  • Morbidelli, A. and Giorgilli, A.: 1995, 'Superexponential stability of KAM tori', J. Stat. Phy. 78, 1607.

    Google Scholar 

  • Moser, J.: 1962, 'On invariant curves of area-preserving mappings of an annulus', Nach. Akad. Wiss. Göttingen, Math. Phys. Kl. II 1(1).

  • Olvera, A. and Simó C.: 1987, 'An obstruction method for the destruction of invariant curves', Physica D 26, 181.

    Google Scholar 

  • Paladin, G. and Vulpiani, A.: 1987, 'Anomalous scaling laws in multifractal objects', Phys. Rep. 156, 147.

    Google Scholar 

  • Siegel, C. L. and Moser, J. K.: 1971, Lectures on Celestial Mechanics, Springer-Verlag, Berlin, Heidelberg, New York.

    Google Scholar 

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Celletti, A., Della Penna, G. & Froeschlé, C. Estimate of the Transition Value of Librational Invariant Curves. Celestial Mechanics and Dynamical Astronomy 83, 257–274 (2002). https://doi.org/10.1023/A:1020119922393

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