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Theory and Applications of the Fast Lyapunov Indicator (FLI) Method

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Chaos Detection and Predictability

Part of the book series: Lecture Notes in Physics ((LNP,volume 915))

Abstract

In the last 20 years numerical experiments have allowed to study dynamical systems in a new way providing interesting results. The development of tools for the detection of regular and chaotic orbits has been one of the key points to access the global properties of dynamical systems. In many cases the visualization of suitably chosen sections of the phase space has been determinant for the comprehension of the fascinating and complex interplay between order and chaos. The Fast Lyapunov Indicator introduced in Froeschlé et al. (Celest Mech Dyn Astron 67:41–62, 1997) and further developed in Guzzo et al. (Physica D 163(1–2):1–25, 2002), is an easy to implement and sensitive tool for the detection of order and chaos in dynamical systems. Closely related to the computation of the Largest Lyapunov Exponent, the Fast Lyapunov Indicator relies on the idea that the computation of tangent vectors contains a lot of information even on short integration times, while for the Largest Lyapunov Indicator large integration times are required in order to accurately approximate a limit value. The aim of this Chapter is to provide the definition of the Fast Lyapunov indicator and some simple examples of applications for readers that would like to implement and use the indicator for the first time. We associate to each example of application the references to more specific papers that we have published during these years.

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Notes

  1. 1.

    H ε is real analytic and H 0 is isoenergetically non-degenerate.

References

  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. Surv. 18, 9 (1963)

    Google Scholar 

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

    Google Scholar 

  3. Barrio, R.: Sensitivity tools vs. Poincaré sections. Chaos Solitons Fractals 25, 711–726 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Barrio, R., Borczyk, W., Breiter, S.: Spurious structures in chaos indicators maps. Chaos Solitons Fractals 40, 1697–1714 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Celletti, A., Froeschlé, C., Lega, E.: Dissipative and weakly-dissipative regimes in nearly-integrable mappings. Discrete Contin. Dyn. Syst. Ser. A 16(4), 757–781 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Celletti, A., Froeschlé, C., Lega, E.: Dynamics of the conservative and dissipative spin–orbit problem. Planet. Space Sci. 55, 889–899 (2007)

    Article  ADS  Google Scholar 

  7. Celletti, A., Stefanelli, L., Lega, E., Froeschlé, C.: Global dynamics of the regularized restricted three-body problem with dissipation. Celest. Mech. Dyn. Astron. 109, 265–284 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Cincotta, P.M., Simó, C.: Simple tools to study global dynamics in non-axisymmetric galactic potentials - I. Astron. Astrophys. Suppl. Ser. 147, 205 (2000)

    Article  ADS  Google Scholar 

  9. Cincotta, P.M., Giordano, C.M., Simó, C.: Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits. Physica D 182(3–4), 151–178 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Froeschlé, C., Lega, E.: Weak chaos and diffusion in Hamiltonian systems. From Nekhoroshev to Kirkwood. In: Roy, A.E. (eds.) The Dynamics of Small Bodies in the Solar System: A Major Key to Solar System Studies. NATO/ASI. Kluwer Academic, Boston (1998)

    Google Scholar 

  11. Froeschlé, C., Lega, E.: On the structure of symplectic mappings. The fast Lyapunov indicator: a very sensitive tool. Celest. Mech. Dyn. Astron. 78(1/4), 167–195 (2000)

    ADS  MATH  Google Scholar 

  12. Froeschlé, C., Lega, E., Gonczi, R.: Fast Lyapunov indicators. Application to asteroidal motion. Celest. Mech. Dyn. Astron. 67, 41–62 (1997)

    Article  ADS  MATH  Google Scholar 

  13. 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 

  14. Froeschlé, C., Guzzo, M., Lega, E.: Local and global diffusion along resonant lines in discrete quasi-integrable dynamical systems. Celest. Mech. Dyn. Astron. 92(1–3), 243–255 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. Guzzo, M.: The web of three-planets resonances in the outer solar system. Icarus 174(1), 273–284 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  16. Guzzo, M.: The web of three-planet resonances in the outer solar system II: a source of orbital instability for Uranus and Neptune. Icarus 181, 475–485 (2006)

    Article  ADS  Google Scholar 

  17. Guzzo, M.: Chaos and diffusion in dynamical systems through stable–unstable manifolds. In: Perozzi, Mello, F. (eds.) Space Manifolds Dynamics: Novel Spaceways for Science and Exploration. Novel Spaceways for scientific and exploration missions, a dynamical systems approach to affordable and sustainable space applications held in Fucino Space Centre (Avezzano), 15–17 October 2007. Springer, New York/Dordrecht/Heidelberg/London (2010)

    Google Scholar 

  18. Guzzo, M., Lega, E.: The numerical detection of the Arnold web and its use for long-term diffusion studies in conservative and weakly dissipative systems. Chaos 23, 23124 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Guzzo, M., Lega, E.: On the identification of multiple close encounters in the planar circular restricted three-body problem. Mon. Not. R. Astron. Soc. Lett. 428, 2688–2694 (2013)

    Article  ADS  Google Scholar 

  20. Guzzo, M., Lega, E.: Evolution of the tangent vectors and localization of the stable and unstable manifolds of hyperbolic orbits by Fast Lyapunov Indicators. SIAM J. Appl. Math. 74(4), 1058–1086 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  21. Guzzo, M., Lega, E., Froeschlé, C.: On the numerical detection of the effective stability of chaotic motions in quasi-integrable systems. Physica D 163(1–2), 1–25 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Guzzo, M., Lega, E., Froeschlé, C.: First numerical evidence of Arnold diffusion in quasi-integrable systems. Discrete Continuous Dyn. Syst. Ser. B 5(3), 687–698 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Guzzo, M., Lega, E., Froeschlé, C.: Diffusion and stability in perturbed non-convex integrable systems. Nonlinearity 19, 1049–1067 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Guzzo, M., Lega, E., Froeschlé, C.: A numerical study of the topology of normally hyperbolic invariant manifolds supporting Arnold diffusion in quasi-integrable systems. Physica D 238, 1797–1807 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  25. Guzzo, M., Lega, E., Froeschlé, C.: A numerical study of Arnold diffusion in a priori unstable systems. Commun. Math. Phys. 290, 557–576 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. Guzzo, M., Lega, E., Froeschlé, C.: First numerical investigation of a conjecture by N.N. Nekhoroshev about stability in quasi-integrable systems. Chaos 21(3), 033101-1–033101-12 (2011)

    Google Scholar 

  27. Hénon, M., Heiles, C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73–79 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  28. Kolmogorov, A.N.: On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian. Dokl. Akad. Nauk. SSSR 98, 527–530 (1954)

    MATH  MathSciNet  Google Scholar 

  29. Laskar, J.: The chaotic motion of the Solar system. A numerical estimate of the size of the chaotic zones. Icarus 88, 266–291 (1990)

    Google Scholar 

  30. Laskar, J.: Frequency analysis for multi-dimensional systems. Global dynamics and diffusion. Physica D 67, 257–281 (1993)

    MATH  MathSciNet  Google Scholar 

  31. Laskar, J., Froeschlé, C., Celletti, A.: The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping. Physica D 56, 253 (1992)

    Google Scholar 

  32. Lega, E., Froeschlé, C.: Fast Lyapunov Indicators. Comparison with other chaos indicators. Application to two and four dimensional maps. In: Henrard, J., Dvorak, R. (eds.) The Dynamical Behaviour of our Planetary System. Springer, The Netherlands (1997)

    Google Scholar 

  33. Lega, E., Froeschlé, C.: Comparison of convergence towards invariant distributions for rotation angles, twist angles and local Lyapunov characteristic numbers. Planet. Space Sci. 46, 1525–1534 (1998)

    Article  ADS  Google Scholar 

  34. Lega, E., Froeschlé, C.: On the relationship between fast Lyapunov indicator and periodic orbits for symplectic mappings. Celest. Mech. Dyn. Astron. 81, 129–147 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  35. Lega, E., Guzzo, M., Froeschlé, C.: Physica D 182, 179–187 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  36. Lega, E., Froeschlé, C., Guzzo, M.: Diffusion in Hamiltonian quasi-integrable systems. In: Benest, D., Froeschlé, C., Lega, E. (eds.) Topics in Gravitational Dynamics. Lecture Notes in Physics, vol. 729. Springer, Berlin (2007)

    Google Scholar 

  37. Lega, E., Guzzo, M., Froeschlé, C.: Measure of the exponential splitting of the homoclinic tangle in four dimensional symplectic mappings. Celest. Mech. Dyn. Astron. 104, 191–204 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  38. Lega, E., Guzzo, M., Froeschlé, C.: A numerical study of the size of the homoclinic tangle of hyperbolic tori and its correlation with Arnold diffusion in Hamiltonian systems. Celest. Mech. Dyn. Astron. 107, 129–144 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  39. Lega, E., Guzzo, M., Froeschlé, C.: A numerical study of the hyperbolic manifolds in a priori unstable systems. A comparison with Melnikov approximations. Celest. Mech. Dyn. Astron. 107, 115–127 (2010)

    Article  ADS  MATH  Google Scholar 

  40. Lega, E., Guzzo, M., Froeschlé, C.: Numerical Studies of hyperbolic manifolds supporting diffusion in symplectic mappings. Eur. Phys. J. Spec. Top. 186, 3–31 (2010)

    Article  Google Scholar 

  41. Lega, E., Guzzo, M., Froeschlé, C.: Detection of Close encounters and resonances in three body problems through Levi-Civita regularization. Mon. Not. R. Astron. Soc. Lett. 418, 107–113 (2011)

    Article  ADS  Google Scholar 

  42. Mitchenko, T.A., Ferraz–Mello, S.: Astron. J. 122, 474–481 (2001)

    Google Scholar 

  43. Moser, J.: On invariant curves of area-preserving maps of an annulus. Commun. Pure Appl. Math. 11, 81–114 (1958)

    Article  Google Scholar 

  44. Namouni, F.: Astron. J. 130, 280 (2005)

    Article  ADS  Google Scholar 

  45. Namouni, F.: LNP 729, 233 (2007)

    ADS  MathSciNet  Google Scholar 

  46. Namouni, F., Guzzo, M.: Celest. Mech. Dyn. Astron. 99, 31 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  47. Namouni, F., Guzzo, M., Lega, E.: On the integrability of stellar motion in an accelerated logarithmic potential. Astron. Astrophys. 489, 1363 (2008)

    Article  ADS  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  49. Robutel, P.: Frequency map analysis and quasiperiodic decompositions. In: Benest et al. (eds.) Hamiltonian Systems and Fourier Analysis, pp. 179–198. Taylor and Francis. Adv. Astron. Astrophys., Cambridge Sci. Publ., Cambridge (2005)

    Google Scholar 

  50. Robutel, P., Galern, F.: The resonant structure of Jupiter’s Trojan asteroids I. Long term stability and diffusion. Mon. Not. R. Astron. Soc. 372, 1463–1482 (2006)

    Article  ADS  Google Scholar 

  51. Robutel, P., Laskar, J.: Frequency map and global dynamics in the Solar System I. Icarus 52(1), 4–28 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  52. Tang, X.Z., Boozer, A.H.: Finite time Lyapunov exponent and advection-diffusion equation. Physica D 95(3–4), 283–305 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  53. Todorović, N., Guzzo, M., Lega, E., Froeschlé, C.: A numerical study of the stabilization effect of steepness. Celest. Mech. Dyn. Astron. 110(4), 389–398 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  54. Villac, B.F.: Using FLI maps for preliminary spacecraft trajectory design in multi-body environments. Celest. Mech. Dyn. Astron. 102, 29–48 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  55. Wayne, B.H., Malykh, A.V., Danforth, C.M.: The interplay of chaos between the terrestrial and giant planets. Mon. Not. R. Astron. Soc. 407(3), 1859–1865 (2010)

    Article  ADS  Google Scholar 

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Author information

Authors and Affiliations

  1. Université de Nice Sophia Antipolis, CNRS UMR 7293, Observatoire de la Côte d’Azur, Bv. de l’Observatoire, CS 34229, 06304, Nice cedex 4, France

    Elena Lega

  2. Dipartimento di Matematica, Università degli studi di Padova, Via Trieste, 63, 35121, Padova, Italy

    Massimiliano Guzzo

  3. Université de Nice Sophia Antipolis, CNRS UMR 7293, Observatoire de la Côte d’Azur, Bv. de l’Observatoire, 4229, 06304, Nice cedex 4, France

    Claude Froeschlé

Authors
  1. Elena Lega
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  2. Massimiliano Guzzo
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  3. Claude Froeschlé
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Corresponding author

Correspondence to Elena Lega .

Editor information

Editors and Affiliations

  1. University of Cape Town, Department of Mathematics and Applied Ma, Rondebosch, South Africa

    Charalampos (Haris) Skokos

  2. School of Mathematics and Statistics, University of Sydney, Sydney, Australia

    Georg A. Gottwald

  3. Observatoire de Paris, IMCCE, Paris, France

    Jacques Laskar

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Lega, E., Guzzo, M., Froeschlé, C. (2016). Theory and Applications of the Fast Lyapunov Indicator (FLI) Method. In: Skokos, C., Gottwald, G., Laskar, J. (eds) Chaos Detection and Predictability. Lecture Notes in Physics, vol 915. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48410-4_2

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