Abstract
Using mappings as model problem we study the structure around the last invariant KAM torus for different values of the perturbing parameter. We used the standard map for the analysis of the hierarchical structure existing around invariant KAM tori applying two complementary methods: the Laskar's frequency map analysis and the sup-map analysis. We recover and extend the theoretical prediction recently given by Morbidelli and Giorgilli about the existence of a neighborhood almost completely full of slave tori around a chief torus. We then make tests about the diffusion in order to measure the barrier to diffusion which still remains after the break-up of the last KAM torus.
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References
Arnold, V.I. (1963): “Proof of a Theorem by A.N. Kolmogorov on the invariance of quasiperiodic motions under small perturbations of the Hamiltonian”, Russ. math. Surv., 18, 9.
Celletti, A. (1990): “Analysis of resonances in the spin-orbit problem in Celestial Mechanics: the synchronous resonance (Part I)”, J. of Applied mathematics and Physics (ZAMP), 41, 174.
Froeschlé, C. (1992), Mapping in Astrodynamics, in Chaos, Resonance and Collective dynamical phenomena in the solar system, S.Ferraz-Mello (ed.) Kluwer Academic Publishers, 375–390.
Froeschlé, C., Froeschlé, Ch. and Lohinger, E. (1993): “Generalized Lyapunov characteristic indicators and corresponding Kolmogorov like entropy of the standard mapping”, Celestial Mech., 56, 307–314.
Froeschlé, C. and Lega, E. (1995): “Polynomial approximation of Poincaré maps for Hamiltonian systems”, Earth, Moon and Planets, in press.
Hadjidemetriou, J.D. (1991): “Mapping models for Hamiltonian Systems with applications to resonant asteroid motion”, in Predictability, Stability and Chaos in N-body Dynamical Systems, A.E. Roy (ed.), 157–175, Kluwer Publ.
Hadjidemetriou, J.D. (1992): “The elliptic restricted problem at the 3:1 resonance”, Celest. Mech., 53, 153–181.
Hadjidemetriou, J.D. (1993a): “Asteroid motion near the 3:1 resonance”, Celest. Mech., 56, 563–599.
Hadjidemetriou, J.D. (1993b): “Resonant motion in the restricted three body problem”, Celest. Mech., 56, 201–219.
Henrard, J. and Morbidelli, A. (1993): “Slow crossing of a stochastic layer”, Physica D, 68, 187–200.
Hénon,M. (1969): “Numerical study of quadratic area-preserving mapping”, Quarterly of Applied Mathematics, 27, 292–306.
Kolmogorov, A.N (1954): “On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian”, Dokl. Akad. Nauk. SSSR, 98, 469.
Laskar, J., Froeschle, C. and Celletti, A. (1992): “The measure of chaos by the numerical analysis of the fundamental frequencies”, Physica D, 56, 253–269.
Lega, E. and Froeschlé, C. (1995): “Numerical measures of the structure around an invariant KAM torus using the frequency map analysis”, submitted to Physica D.
Lohinger,E. and Froeschlé C. (1993): “Fourier analysis of local Lyapunov characteristic exponents for satellite-type motions”, Celestial Mech., 57, 369–372.
Morbidelli, A. and Giorgilli, A. (1995): “Superexponential Stability of KAM tori”, J. Stat. Phys., 78, 1607–1616.
Petit, J.M. and Froeschlé, C. (1994): “Polynomial approximations of Poincare maps for Hamiltonian systems. II”, Astron. Astrophys., 282, 291–303.
Voglis, N. and Contopoulos, G.J (1994): “Invariant spectra of orbits in dynamical systems”, J. Phys. A: Math. Gen., 27, 4899–4909.
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Froeschlé, C., Lega, E. On the measure of the structure around the last kam torus before and after its break-up. Celestial Mech Dyn Astr 64, 21–31 (1996). https://doi.org/10.1007/BF00051602
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DOI: https://doi.org/10.1007/BF00051602