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Astron. Astrophys. 349, 70-76 (1999)

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4. Discussion

In this paper we have studied the regular and chaotic motion in the Hamiltonian system (2) using a map, derived from the averaged Hamiltonian and numerical integration. Note that Hamiltonian (2) is integrable when [FORMULA] and when [FORMULA]. Depending on the choice of the parameters [FORMULA] this Hamiltonian displays three different types of [FORMULA] phase plane named as types A,B,C. In the case of the two first types of phase plane, the map describes in a very satisfactory way the real properties of orbits up to [FORMULA]. Extensive numerical calculations have shown that the map does not produce any chaotic regions in the cases A and B. Therefore, one concludes that the map fails to describe the small chaotic regions near the separatrix found by numerical integration. Here we must note that at the beginning we had started with smaller values of [FORMULA], but we observed that the map persisted not to produce chaos up to [FORMULA]. This explains why [FORMULA] was chosen.

The situation is quite different in the case of the phase plane of type C. Here the system has large chaotic regions, increasing when [FORMULA] approaches [FORMULA] and the map describes them satisfactorily. Moreover, the comparison of the spectra and LCNs of orbits found using the map and numerical integration shows that one can trust the map. This is very important because the map is at least 10-20 times faster than numerical integration and one can make faster all the time consuming calculations. Note that in all cases the number of periods in the calculation of the spectra using the map or numerical integration was the same in order to be able to compare the corresponding results.

We have also investigated the symmetry of the spectra of ordered orbits. It was found that the quantity Q increases, with a second order polynomial dependence, as we are moving far from the stable periodic orbit.

Finally the authors would like to make clear that (i) the proposed map has some qualitative similarities with the Poincaré map of the original Hamiltonian system (2), although the numerical differences may be important and (ii) the results of this work correspond to the particular Hamiltonian system (2) and for the resonance case 1:1. Interesting results, on the spectra of orbits, in galactic Hamiltonians made up of harmonic oscillators in the 4:3 resonance, have been given by Contopoulos et al. (1995). Also spectra of orbits have been studied, in the standard map, by Voglis & Contopoulos (1994), Contopoulos et al. (1997). This paper was focused on the comparison of the results (structure of the phase plane and spectra of orbits) given by numerical integration and the map.

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© European Southern Observatory (ESO) 1999

Online publication: August 25, 1999