Abstract
We present a very simple and fast method to separate chaotic from regular orbits for non-integrable Hamiltonian systems. We use the standard map and the Hénon and Heiles potential as model problems and show that this method appears to be at least as sensitive as the frequency-analysis method. We also study the chaoticity of asteroidal motion.
Similar content being viewed by others
References
Benettin, G., Galgani, L. and Strelcyn, J. M.: 1976, Phys. Rev. A, 14, 2338.
Benettin, G. and Galgani, L.: 1979, Lyapunov characteristic exponents and Stochasticity, Intrinsic Stochasticity in plasmas. Laval G. and Gresillon, D. (eds), Les éditions de Physique Coutaboeuf Orsay-France.
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 II: Numerical applications, 21-30.
Celletti, A. and Froeschlé, C.: 1995, On the determination of the stochasticity threshold of invariant curves. Int. J. of Bifurcation and Chaos, 5, n.6.
Contopoulos, G., Galgani, L. and Giorgilli, A.: 1978, Rhys. Rev. A., 18, 1183.
Contopoulos, G., Voglis, N., Efthymiopoulos, C., Froeschlé, C., Gonczi, R., Lega, E., Dvorak, R. and Lohinger, E.: 1996, Transition spectra of dynamical systems. Celest. Mech. and Dynamical Astron., submitted.
Contopoulos, G. and Voglis, N.: 1996a, A fast method for distinguishing between ordered and chaotic motion. Astron. Astrophys., in press.
Franklin, F.: 1994, AJ, 107: 1890.
Froeschlé, C.: 1970, A numerical study of the stochasticity of dynamical systems with two degrees of freedom. Astron. and Astrophys., 9:15–23.
Froeschlé, C. and Gonczi, R.: 1980, Lyapunov characteristic numbers and Kolmogorov entropy of a four-dimensional mapping. Il Nouvo Cimento, 55B: 59–69.
Froeschlé, C.: 1984, The Lyapunov characteristic exponents and applications. Journal de Méc. théor. et apll., Numero spécial: 101–132.
Froeschlé, C., Froeschlé, Ch. and Lohinger, E.: 1993, Generalized Lyapunov characteristic indicators and corresponding Kolmogorov like entropy of the standard mapping. Celest. Mech. and Dynamical Astron., 56: 307–315.
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. and Dynamical Astron., 64: 21–31.
Froeschlé, C. and Scheidecker, J. P.: 1973, Numerical study of a four dimensional mapping II. Astron. Astrophys., 22: 431–436.
Froeschlé, C. and Scheidecker, J. P.: 1973, On the disappearance of isolating integrals in systems with more than two degrees of freedom. Astrophys. and Space Sc., 25: 373–386.
Henon, M.: 1981, Cours des Houches XXXVI. North Holland, Amsterdam 57.
Hénon, M. and Heiles, C.: 1964, The applicability of the third integral of motion: Some numerical experiments. A. J., 1:73–79.
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.
Lecar, M., Franklin, F. and Murison, M.: 1992, AJ, 104: 1230.
Lyapunov, A. M.: 1949, Problème General de la Stabilité du Muvement. Ann. Math. Studies. Princeton University Press., 17.
Milani, A., Nobili, A. and Knezevic, Z.: 1995, Icarus, submitted.
Milani, A. and Groupe E.U.R.O.P.A.: 1996, Celest. Mech. and Dynam. Astr.,The Trojan asteroid belt: proper elements, stability, chaos and families. Celest. Mech. and Dynam. Astr., 57: 59–94.
Milani, A. and Nobili, A.: 1992, An example of stable chaos in the Solar System. Nature, 357: 569–571.
Morbidelli, A. and Froeschlé. C.: 1996, On the relationship between Lyapunov times and macroscopic instability times. Celest. Mech. and Dynam. Astr., 63: 227–239.
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.
Oseledec, V. I.: 1968, A multiplicative ergodic theorem. The Lyapunov characteristic numbers of dynamical systems. Mosc. Math. Soc., 19: 197–231.
Soper, M., Franklin, F. and Lecar, M.: 1990, On the original distribution of the asteroids. Icarus, 87: 265–284.
Voglis, N. and Contopoulos, G. J.: 1994, J. Phys. A: Math. Gen., 27: 4899–4909.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
FROESCHLÉ, C., LEGA, E. & GONCZI, R. FAST LYAPUNOV INDICATORS. APPLICATION TO ASTEROIDAL MOTION. Celestial Mechanics and Dynamical Astronomy 67, 41–62 (1997). https://doi.org/10.1023/A:1008276418601
Issue Date:
DOI: https://doi.org/10.1023/A:1008276418601