Astronomy and Astrophysics – Astronomy
Scientific paper
Feb 2003
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2003cemda..85..105p&link_type=abstract
Celestial Mechanics and Dynamical Astronomy, v. 85, Issue 2, p. 105-144 (2003).
Astronomy and Astrophysics
Astronomy
Periodic Orbits, Asymptotic Curves, Homoclinic Tangle
Scientific paper
We study in great detail the geometry of the homoclinic tangle, with respect to the energy, corresponding to an unstable periodic orbit of type 1:2, on a surface of section representing a 2-D Hamiltonian system. The tangle consists of two resonance areas, in contrast with the tangles of type-l or -{l, m, k, x = 0} considered in previous studies, that consist of only one resonance area. We study the intersections of the inner and outer lobes of the same resonance area and of the two resonance areas. The intersections of the lobes follow certain rules. The detailed study of these rules allows us to derive quantitative relations about the number of intersections and to understand the complex behavior of the higher order lobes by studying the lower order lobes. We find 1st, 2nd, 3rd, etc. order intersections formed by lobes making 1, 2, 3, etc. turns around an island. After a sufficiently high order of iterations a lobe may intersect its image and thus produce a Poincaré recurrence. Numerical results for a wide interval of energies are presented. The number of intersections changes through tangencies. In any finite interval of the energy between two tangencies of 1st order, an infinite number of higher order tangencies occur and thus, according to the Newhouse theorem, there exist nearby islands of stability.
Contopoulos George
Dokoumetzidis A.
Polymilis C.
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