Abstract
On the basis of the general theory of Hamiltonian systems, we consider the relationship between Lyapunov times and macroscopic diffusion times. We find out that there are two regimes: the Nekhoroshev regime and the resonant overlapping regime. In the first case the diffusion time is exponentially long with respect to Lyapunov times. In the second case, the relationship is polynomial although we do not find any theoretical reason for the existence of a ‘universal’ power law. We show numerical evidences which confirm our theoretical considerations.
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Arnold, V. I., 1963b: ‘Proof of A. N. Kolmogorov's theorem on the conservation of conditionally periodic motions with a small variation in the Hamiltonian’,Russian Math. Surv.,18, 9–36.
Benettin G. and Gallavotti G., 1986: ‘Stability of motions near resonances in quasi-integrable Hamiltonian systems’,J. Stat. Phys.,44, 293–338.
Chirikov, B. V., 1960: ‘Resonance processes in magnetic traps’,Plasma Phys., 1, 253–260.
Chirikov, B. V., 1979: ‘A universal instability of many dimensional oscillator systems’,Physics reports,52, 265–379.
Ferraz-Mello, S., 1994: ‘Dynamics of the asteroidal 2/1 resonance’,AJ,108, 2330–2337.
Franklin, F., 1994: ‘An examination of the relation between chaotic orbits and the Kirkwood gap at the 2/1 resonance’,AJ,107, 1890–1899.
Kolmogorov, A. N., 1954: ‘Preservation of conditionally periodic movements with small change in the Hamiltonian function’,Dokl. Akad. Nauk SSSR,98, 527–530; english translation inLecture notes in Physics, N.93, 51–56, Casati, G. and Ford, J.
Lecar, M., Franklin, F., and Murison, M., 1992: ‘On predicting long-term orbital instability: a relation between Lyapunov time and sudden orbital transitions’,AJ,104, 1230–1236.
Levison, H. F. and Duncan, M. J., 1993: ‘The gravitational scultping of the Kuiper belt’,Astroph. J. Lett.,406, L35-L38.
Lichtemberg, A. J. and Lieberman, M. A., 1983: ‘Regular and stochastic motion’, Springer-Verlag, New York.
Lochak, P., 1993: ‘Hamiltonian perturbation theory: periodic orbits, resonances and intermittency’,Nonlinearity,6, 885–904.
Milani, A. and Nobili, A., 1992: ‘An example of stable chaos in the Solar System’,Nature,357, 569–571.
Milani, A., Nobili A., and Knezevic, Z., 1995: ‘Stable chaos in the asteroid belt’, Icarus, submitted.
Morbidelli, A. and Giorgilli, A., 1995a: ‘Superexponential stability of KAM tori’,J. Stat. Phys.,78, 1607–1617.
Morbidelli, A. and Giorgilli, A., 1995b: ‘On a connection between KAm and Nekhoroshev's theorems’,Phys. D.,86, 514–516.
Morbidelli, A., Zappala, E., Moons, M., Cellino, A. and Gonczi, R., 1995: ‘Asteroid families close to mean motion resonances: dynamical effects and physical implications’,Icarus, in press.
Moser, J., 1962: ‘On invariant curves of area-preserving mappings of an annulus’,Nachr. Akad. Wiss. Göttingen Math. Phys. Kl. 2, 1.
Murison, M., Lecar, M., and Franklin, F., 1994: ‘Chaotic motion in the outer asteroid belt and its relation to the age of the solar system’,AJ,108, 2323–2329.
Neishtadt, A. I., 1984: ‘The separation of motions in systems with rapidly rotating phase’,J. Appl. Math. Mech.,48, 133–139.
Nekhoroshev, N. N., 1977: ‘Exponential estimates of the stability time of near-integrable Hamiltonian systems’,Russ. Math. Surveys,32, 1–65.
Nekhoroshev, N. N., 1979: ‘Exponential estimates of the stability time of near-integrable Hamiltonian systems, 2’,Trudy Sem. Petrovs.,5, 5–50.
Niederman, L., 1993: ‘Resonance et stabilité dans le problème planetaire: solutions de seconde éspece’,Ph. D. dissertation, Université Paris 6.
Shlesinger, M. F., Zaslavsky, G. M., and Klafter, J., 1993: ‘Strange Kinetics’,Nature,363, 31–37.
Soper, M., Franklin, F., and Lecar, M., 1990: ‘On the original distribution of the asteroids’,Icarus,87, 265–284.
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Morbidelli, A., Froeschlé, C. On the relationship between Lyapunov times and macroscopic instability times. Celestial Mech Dyn Astr 63, 227–239 (1995). https://doi.org/10.1007/BF00693416
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DOI: https://doi.org/10.1007/BF00693416