Abstract
It is already known (Froeschlé, Lega and Gonczi, 1997) that the Fast Lyapunov Indicator (FLI), that is the computation on a relatively short time of the largest Lyapunov indicator, allows to discriminate between ordered and weak chaotic motion. We have found that, under certain conditions, the FLI also discriminates between resonant and non-resonant orbits, not only for two-dimensional symplectic mappings but also for higher dimensional ones. Using this indicator, we present an example of the Arnold web detection for four and six-dimensional symplectic maps. We show that this method allows to detect the global transition of the system from an exponentially stable Nekhoroshev’s like regime to the diffusive Chirikov’s one.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Benettin, G., Galgani, L., Giorgilli, A. and Strelcyn, J. M.: 1980, ‘Lyapunov characteristic exponents for smooth dynamical systems; a method for computing all of them’, Meccanica 15, Part I, theory, 9–20 — Part 2: Numerical applications, —21–30.
Contopoulos, G. and Voglis, N.: 1997, ‘A fast method for distinguishing between order and chaotic orbits’, Astr. Astrophys. 317, 73–82.
Froeschlé, C.: 1970, ‘A numerical study of the stochasticity of dynamical systems with two degrees of freedom’, Astr. Astrophys. 9, 15–23.
Froeschlé, C.: 1971, ‘On the number of isolating integrals in systems with three degrees of freedom’, Astrophys. Space Sci. 14, 110.
Froeschlé, C.: 1972, ‘Numerical study of a four-dimensional mapping’, Aste. Astrophys. 16, 172.
Froeschlé, C.: 1984, ‘The Lyapunov characteristic exponents and applications’, J. de Méc. théor et appl. Numero spécial, 101–132.
Froeschlé, C. and Gonczi, R.: 1980, ‘Lyapunov characteristic numbers and Kolmogorov entropy of a four-dimensional mapping’, Il Nuovo Cimento 55B, 59–69.
Froeschlé, C. and Lega, E.: 1998, ‘Twist angles: a fast method for distinguishing islands, tori and weak chaotic orbits. Comparison with other methods of analysis’, Astr. Astrophys. 334, 355–362.
Froeschlé, C. and Lega, E.: 1999 ‘Weak chaos and diffusion in Hamiltonian systems. From Nekhoroshev to Kirkwood’, in: A. E. Roy (ed.), The Dynamics of Small Bodies in the Solar System: a Major Key to Solar System Studies, the NATO/ASI series Vol. 522.
Froeschlé, C. and Lega, E.: 1996, ‘On the measure of the structure around the last KAM torus before and after its break-up’, Celest. Mech. & Dyn. Aste 64, 21–31.
Froeschlé, C., Lega, E. and Gonczi, R.: 1997, ‘Fast Lyapunov indicators. Application to asteroidal motion’, Celest. Mech. & Dyn. Aste 67, 41–62.
Guzzo, M., Lega, E. and Froeschlé, C.: 2001, ‘On the numerical detection of the stability of chaotic motions in quasi-integrable systems’, Physica D (submitted).
Laskar, J.: 1993, ‘Frequency analysis for multi-dimensional systems. Global dynamics and diffusion’, Physica D 67, 257–281.
Laskar, J.: 1994, ‘Large scale chaos in the solar system’, Astr. Astrophys. 287, L9 — L12.
Laskar, J., Froeschlé, C. and Celletti, A.: 1992, ‘The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping’, Physica D 56, 253.
Lega, E. and Froeschlé, C.: 1997, ‘Fast Lyapunov Indicators. Comparison with other chaos indicators. Application to two and four dimensional maps’, in: J. Henrard and R. Dvorak (eds), The Dynamical Behaviour of our Planetary System., Kluwer Academic publisher.
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.
Lichtenberg, A. J. and Lieberman, M. A.: 1983, Regular and Stochastic Motion, Springer, Berlin, Heidelberg, New York.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Froeschlé, C., Lega, E. (2001). On the Structure of Symplectic Mappings. The Fast Lyapunov Indicator: A Very Sensitive Tool. In: Dvorak, R., Henrard, J. (eds) New Developments in the Dynamics of Planetary Systems. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2414-2_12
Download citation
DOI: https://doi.org/10.1007/978-94-017-2414-2_12
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5702-0
Online ISBN: 978-94-017-2414-2
eBook Packages: Springer Book Archive