Abstract
Frequency map analysis is a numerical method which provides a clear representation of the global dynamics of many multi-dimensional systems, and which is particularly useful for systems of 3 degrees of freedom and more. The frequency map dependence with time also allows refined estimates of the diffusion of the orbits in the frequency domain. Here are presented some applications of frequency map analysis to the special case of a quasi-periodic perturbation, and to the study of the global dynamics of a 4 dimensional symplectic mapping with definite or indefinite torsion. The convergence of the frequency analysis algorithm on KAM solutions is demonstrated, and the asymptotic value of the error is provided.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Arnold V., Proof of Kolmogorov’s theorem on the preservation of quasi-periodic motions under small perturbations of the Hamiltonian, Rus. Math. Surv. 18, N6 (1963) 9–36.
Arnold V.: Dynamical Systems IIISpringer-Verlag, Berlin ,(1988).
Chirikov, B.V., A universal instability of many dimensional oscillator systems, Physics Reports 52 (1979) 263–379.
Dumas, H. S., Laskar, J.: Global Dynamics and Long-Time Stability in Hamiltonian Systems via Numerical Frequency Analysis, Phys. Rev. Lett. 70, (1993), 2975–2979.
Froeschlé, C., On the number of isolating integrals in systems with three degrees of freedom, Astrophys. and Space Sciences 14 (1971) 110–117.
Giorgilli A., Delshams A., Fontich E., Galgani L., Simó C., Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem. J. Diff. Equa. 77 (1989) 167–198.
Greene, J.M., A method for determining stochastic transitions, J. Math. Phys. 20 (1979) 1183.
Herman, M.R., Sur les courbes invariantes par les difféomorphismes de l’anneau. Volume 1, Astérisque 103–104 (1983).
Herman, M.R.: Dynamics connected with indefinite normal torsion, preprint, (1990).
Knezevic, Z., and Milani, A: Asteroid proper elements: the big picture, in Symposium IAU 160, A. Milani, M. Di Martino, A. Cellino, eds, (1994), 31–44, Kluwer, Dordrecht.
Kolmogorov, A.N.: 1954, On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian Dokl. Akad. Nauk. SSSR, 98, (1954), 469.
Laskar, J., Secular evolution of the solar system over 10 million years, Astron. Astrophys. 198 (1988) 341–362.
Laskar, J.: The chaotic motion of the solar system. A numerical estimate of the size of the chaotic zones, Icarus, 88, (1990), 266–291.
Laskar, J., Frequency analysis for multi-dimensional systems. Global dynamics and diffusion, Physica D 67 (1993) 257–281.
Laskar, J., Frequency map analysis of an Hamiltonian system, Workshop on Non-Linear dynamics in Particle accelerators, Arcidosso sept. 1994, (1995) AIP Conf. Proc., 344, 130–159.
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, (1992) 253–269.
Laskar, J. Robutel, P.: The chaotic obliquity of the planets, Nature, 361, (1993), 608–612.
Laskar, J., Joutel, F., Robutel, P., Stabilization of the Earth’s obliquity by the Moon, Nature 361 (1993) 615–617.
McGill C., Binney, J.: Torus construction in general gravitational potentials, Mon. Not. R. astr. Soc., 244, (1990) 634–645.
morbidelli, A.: Asteroid secular resonant proper elements, Icarus, 105, (1993), 48–66.
Morbidelli, A., Giorgilli, A.: Superexponential stability of KAM tori, J. Stat. Phys. 78, (1995) 1607–1617.
J.K. Moser, Nachr. Akad. Wiss. Göttmgen, Math. Phys. Kl. II (1962), 1–20.
Nekhoroshev, N.N.: An exponential estimates for the time of stability of nearly iutegrable Hamiltonian systems, Russian Math. Surveys, 32, (1977), 1–65.
Nekhoroshev, N.N.: An exponential estimates for the time of stability of nearly iutegrable Hamiltonian systems, Russian Math. Surveys, 32, (1977), 1–65.
Poincaré, H.: Les Méthodes Nouvelles de la Mécanique Céleste, tomes I-III, Gauthier Villard, Paris, (1892–1899), reprinted by Blanchard, 1987.
J. Pöschel: Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math. 25 (1982). 653–695.
Schwartz, L.: Analyse, t. II. Calcul différentiel et équations différentielles, Hermann, Paris, 1992.
R.L. Warnock and R.. D. Ruth, Long-term hounds on nonlinear Hamiltonian motion, Physica D 56 (1992) 188–215.
Wisdom, J., Chaotic behaviour and the origine of the 3/1 Kirkwood gap, carus 56 (1983) 51–74.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Laskar, J. (1999). Introduction to Frequency Map Analysis. In: Simó, C. (eds) Hamiltonian Systems with Three or More Degrees of Freedom. NATO ASI Series, vol 533. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4673-9_13
Download citation
DOI: https://doi.org/10.1007/978-94-011-4673-9_13
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-5968-8
Online ISBN: 978-94-011-4673-9
eBook Packages: Springer Book Archive