Astronomy and Astrophysics – Astronomy
Scientific paper
Oct 1994
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994cemda..60..249c&link_type=abstract
Celestial Mechanics & Dynamical Astronomy (ISSN 0923-2958), vol. 60, no. 2, p. 249-271
Astronomy and Astrophysics
Astronomy
7
Celestial Mechanics, Chaos, Orbit Perturbation, Orbital Mechanics, Orbits, Potential Energy, Asymptotic Properties, Galaxies, Hamiltonian Functions, Perturbation Theory, Systems Stability
Scientific paper
We study the structure of chaos in a simple Hamiltonian system that does not have an escape energy. This system has 5 main periodic orbits that are presented on the surface of section(y, first derivative of y) by the points (1) O(0,0), (2) C1 C2(+/- yc, 0), (3) B1 B2(O, +/- 1) and (4) the boundary y squared + (first derivative of y)squared = 1. The periodic orbits (1) and (4) have infinite transitions from stability (S) to instability (U) and vice-versa; the transition values of epsilon are given by simple approximate formulae. At every transition S goes to U a set of 4 asymptotic curves is formed at O. For larger epsilon the size and the oscillations of these curves grow until they destroy the closed invariant curves that surround O, and they intersect the asymptotic curves of the orbits C1, C2 at infinite heteroclinic points. At every transition U goes to S these asymptotic curves are duplicated and they start at two unstable invariant points bifurcating from O. At the transition itself the asymptotic curves from O are tangent to each other. The areas of the lobes from O increase with epsilon; these lobes increase even after O becomes stable again. The asymptotic curves of the unstable periodic orbits follow certain rules. Whenever there are heteroclinic points the asymptotic curves of one unstable orbit approach the asymptotic curves of another orbit in a definite way. Finally we study the tangencies and the spirals formed by the asymptotic curves of the orbits B1, B2. We find indications that the number of spiral rotations tends to infinity as epsilon approaches infinity. Therefore new tangencies between the asymptotic curves appear for arbitrarily large epsilon. As a consequence there are infinite new families of stable periodic orbits that appear for arbitrarily large epsilon.
Contopoulos George
Papadaki H.
Polymilis C.
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