Abstract
We applied the method of the short time Lyapunov indicators to the planar circular and to the planar elliptic restricted three-body problem in order to study the structure of the phase space in some selected regions. In the circular case we computed the short-time averages of the stretching numbers to distinguish between regular and chaotic domains of the phase space. The results obtained in this way are in good agreement with the corresponding Poincaré's surface of sections. Using the short time Lyapunov indicators we determined the detailed structure of the phase space in the semi-major axis-eccentricity plane of the test particle both in the circular and in the elliptic restricted problem (in the latter case for some values of the eccentricities of the primaries) and we studied the main features of the phase space.
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References
Contopoulos, G.: 1971, ‘Orbits in highly perturbed dynamical systems. III. Nonperiodic orbits’, Astron. J. 76, 147.
Contopoulos, G. and Voglis, N.: 1996, ‘Spectra of stretching numbers and helicity angles in dynamical systems’, Celest. Mech. & Dyn. Astron. 64, 1.
Contopoulos, G. and Voglis, N.: 1997, ‘A fast method for distinguishing between ordered and chaotic orbits’, Astron. Astrophys. 317, 73.
Contopoulos, G., Voglis, N., Efthymiopoulos, C., Froeschlé, C., Gonczi, R., Lega, E., Dvorak, R. and Lohinger, E.: 1997, ‘Transition spectra of dynamical systems’ Celest. Mech. & Dyn. Astron. 67, 293.
Dermott, S. and Murray, C. D.: 1983, ‘Nature of the Kirwood gaps in the asteroidal belt’, Nature 301, 201.
Froeschlé, C., Froeschlé, Ch. and Lohinger, E.: 1993, ‘Generalized Lyapunov characteristic indicators and corresponding Kolmogorov like entropy of the standard mapping’, Celest. Mech. & Dyn. Astr. 56, 307.
Froeschlé, C., Lega, E. and Gonczi, R.: 1997a, ‘Fast Lyapunov indicators. Application to asteroidal motion’, Celest. Mech. & Dyn. Astr. 67, 41.
Froeschlé, C., Gonczi, R. and Lega, E.: 1997b, ‘The fast Lyapunov indicator: a simple tool to detect weak chaos. Application to the structure of the main asteroidal belt’, Planet. Space Sci. 45, 881.
Lohinger, E.: 1997, in: R. Dvorak and J. Henrard (eds.), The Dynamical Behaviour of Our Planetary System, Kluwer Academic Publishers, Dordrecht, Holland, pp. 385-392.
Lohinger, E. and Froeschlé, C.: 1995, in: J. Hagel, M. Cunha and R. Dvorak (eds.), Perturbation Theory and Chaos in Nonlinear Dynamics with Emphasis to Celestial Mechanics, University of Vienna and Universidade da Madeira, pp. 119-131.
Patsis, P. A., Efthymiopoulos, C., Voglis, N. and Contopoulos, G.: 1997, ‘Dynamical spectra of barred galaxies’, Astron. Astrophys. 326, 493.
Shirts, R. B. and Reinhardt, W. P.: 1982, ‘Approximate constants of motion for classically chaotic vibrational dynamics-vague tori, semiclassical quantization, and classical intramolecular energy flow’, J. Chem. Phys. 77, 5204.
Szebehely, V. G.: 1967, Theory of Orbits, Academic Press, New York.
Voglis, N. and Contopoulos, G.: 1994, ‘Invariant spectra of orbits in dynamical systems’, J. Phys. A. 27, 4899.
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Sàndor, Z., Balla, R., Téger, F. et al. Short Time Lyapunov Indicators in the Restricted Three-Body Problem. Celestial Mechanics and Dynamical Astronomy 79, 29–40 (2001). https://doi.org/10.1023/A:1011113928311
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DOI: https://doi.org/10.1023/A:1011113928311