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

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3. Dynamical spectra

In this section we shall study the spectra of the Hamiltonian system (2) using numerical integration and the map (4).Before doing this, it is necessary to remember some useful notions and definitions. The "stretching number" [FORMULA] (see Voglis & Contopoulos 1994, Contopoulos at al. 1995, Contopoulos & Voglis 1997) is defined as

[EQUATION]

where [FORMULA] is the next image on the Poincare phase plane of an infinitesimal deviation [FORMULA] between two nearby orbits. The spectrum of the stretching numbers is their distribution function

[EQUATION]

where [FORMULA] is the number of stretching numbers in the interval [FORMULA] after N iterations.The maximal Lyapunov characteristic number can be written as

[EQUATION]

in other words, the LCN is the average value of the stretching number [FORMULA].

Today we know that the distribution of the successive stretching numbers forms a spectrum which is invariant with respect to (i) the initial conditions along an orbit and the direction of the deviation from this orbit and (ii) the initial conditions of the orbits belonging to the same chaotic region.

In what follows we shall give the spectra [FORMULA] of orbits of the Hamiltonian system (2) derived using the map (4) as well as numerical integration. In the case of the Hamiltonian where t is a continuous quantity (that is in the numerical integration of the equations of motion) we use for the derivation of the stretching numbers and spectra the method used by Contopoulos et al. (1995). All calculations correspond to the case C because the results are much more interesting.

Fig. 8 shows the spectra of two orbits (the first with solid line the other with dots) derived using numerical integration. The orbits were started in the chaotic region with different initial conditions. It is evident that the two spectra are close to each other. Fig. 9 shows the spectra of the same two orbits found using the map. Again the two spectra are close to each other. The spectra in both cases were calculated for [FORMULA] periods. As one observes the pattern shown in both Figs. 8 and 9 have the characteristics of the spectrum of orbits belonging to a chaotic region. Fig. 10 shows the LCNs for two chaotic orbits. Number 1 was derived using the map, while number 2 was found using numerical integration. As one can see the mean exponential divergence of the two nearby chaotic trajectories, described by the map, is larger than that given by numerical integration. Nevertheless the two curves are qualitatively similar.

[FIGURE] Fig. 8. The spectrum [FORMULA] of two chaotic orbits derived using numerical integration. The values of the parameters are as in Fig. 6. Initial conditions [FORMULA], [FORMULA] and [FORMULA], [FORMULA], [FORMULA]. The value of [FORMULA] is found from the energy integral.

[FIGURE] Fig. 9. Same as Fig. 8 derived using the map.

[FIGURE] Fig. 10. LCNs for the same chaotic orbit given by the map 1 and numerical integration 2. Initial conditions [FORMULA], [FORMULA].

In Figs. 11 and 12 we give the spectra of the same regular orbit derived by numerical integration and the map respectively. This is an orbit with initial conditions near the periodic orbit [FORMULA]. The spectra were calculated for [FORMULA] periods. As one can see the agreement between the two spectra is good. From other cases we know that the agreement is much better if we calculate an orbit for [FORMULA], or [FORMULA] periods. Both are "U-shaped", with two large and two small peaks. Such spectra are characteristic of quasi-periodic orbits, starting close to a stable periodic orbit (see Patsis et al. 1997). Fig. 13 shows the LCNs for the orbit of Fig. 11 derived using a map (dots) and numerical integration (solid line). Again one can see that the map describes well the qualitative properties of regular orbits.

[FIGURE] Fig. 11. The spectrum [FORMULA] of a regular orbit derived using numerical integration. The values of the parameters are as in Fig. 6. Initial conditions [FORMULA], [FORMULA]. The value of [FORMULA] is found from the energy integral.

[FIGURE] Fig. 12. Same as Fig. 11 derived using the map.

[FIGURE] Fig. 13. LCNs for the same regular orbit given by the map (dots) and numerical integration (solid line). Initial conditions as in Fig 11.

Let us now come to the symmetry of the spectra. It was observed that the spectra of ordered orbits, starting close to a stable periodic orbit, are almost symmetric with respect to the [FORMULA] axis, while, when we go far from the stable periodic orbit they become asymmetric. In order to give an estimate between closeness and symmetry we have made extensive numerical experiments near the exact periodic orbit [FORMULA], [FORMULA].

We define as "symmetry factor" the quantity

[EQUATION]

As one can see, [FORMULA] corresponds to a perfectly symmetric spectrum, while [FORMULA] to a totally asymmetric with all non zero values of [FORMULA] to belong to either positive or negative [FORMULA]. The results Q vs [FORMULA] for the map, in the case C, when [FORMULA], are shown in Fig. 14. Dots correspond to values given by the numerical experiments, while the solid line corresponds to the best fit

[EQUATION]

where [FORMULA], [FORMULA] and [FORMULA]. Fig. 15 is the same as Fig. 14, for the Hamiltonian in the case C when [FORMULA]. The best fit is now with [FORMULA], [FORMULA] and [FORMULA]. As we can see we have a second order polynomial growth of the asymmetry of the spectrum as we move far from the stable periodic orbit. The value of N in all cases was [FORMULA] iterations.

[FIGURE] Fig. 14. Q vs [FORMULA] for the map in case C, when [FORMULA]

[FIGURE] Fig. 15. Same as Fig. 14 for the Hamiltonian in case C when [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: August 25, 1999
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