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

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2. Map and stability conditions

The averaged Hamiltonian corresponding to the Hamiltonian (2) reads

[EQUATION]

Following the steps described in Caranicolas (1990) we find the map

[EQUATION]

where [FORMULA]. The map describes the motion in the [FORMULA] plane and we return to [FORMULA] variables through [FORMULA]. The fixed points of (4) are at

[EQUATION]

There are three distinguished case of the [FORMULA] phase plane portrait in the Hamiltonian (2). Type A, when both fixed points (i) and (ii) are stable. Type B when points (i) are stable while fixed points (ii) are unstable. In type C phase plane, fixed points (i) and (ii) are unstable and stable respectively. Applying the stability conditions (see Lichtenberg & Lieberman 1983) we find after some straightforward calculations: Type A phase plane appears when [FORMULA] [FORMULA]. Type B appears when [FORMULA] [FORMULA]. We observe the phase plane of type C when [FORMULA] [FORMULA] or if [FORMULA].

In all three cases there is, for a fixed value of the energy h, a value of the perturbation strenght [FORMULA], such as for [FORMULA] curves of zero velocity [FORMULA] open and the test particle is free to escape. We do not consider cases where the curves of zero velocity are always closed that is cases where the Hamiltonian (1) has no [FORMULA]. The value of [FORMULA] can be found using the method described in Caranicolas & Varvoglis (1984). For the type A phase plane we find

[EQUATION]

while in the cases B and C [FORMULA] is given by the formula

[EQUATION]

Let us now go to see the three different types of the phase plane produced by the Hamiltonian system (2). In all numerical calculation we use the value [FORMULA]. Fig. 1 shows the type A [FORMULA] phase plane derived by numerical integration. The values of [FORMULA] are 1.2, 0.8 respectively while [FORMULA]. The motion is everywhere regular except near the hyperbolic point in the center and in a thin strip along the separatrix.(see Fig. 3). Fig. 2 is the corresponding figure produced by the map. As one can see the agreement is good. One significant difference is that the map is inadequate to produce the chaotic layer seen in Fig. 3. Also note that [FORMULA] in Fig. 1 is greater than 2 while in Fig. 2 is smaller.

[FIGURE] Fig. 1. The [FORMULA] (type A) phase plane derived by numerical integration. The values of the parameters are [FORMULA], [FORMULA], [FORMULA], [FORMULA].

[FIGURE] Fig. 2. Same as Fig. 1 derived by the map.

[FIGURE] Fig. 3. The area near the center of Fig. 1

Fig. 4 and Fig. 5 show the type B phase plane derived using numerical integration and the map respectively. The values of the parameters are [FORMULA], [FORMULA], while [FORMULA]. The results are similar to those observed in Figs. 1 and 2. Again the map describes well the real phase plane except in a small chaotic region near the two hyperbolic fixed points. Thus, our numerical calculations suggest that, the map (4) describes well the properties of motion in the Hamiltonian (2) up to the largest perturbation, that is [FORMULA] but it is insufficient to describe the small chaotic region observed when using numerical integration.

[FIGURE] Fig. 4. The [FORMULA] (type B) phase plane derived by numerical integration. The values of the parameters are [FORMULA], [FORMULA], [FORMULA], [FORMULA].

[FIGURE] Fig. 5. Same as Fig. 4 derived by the map.

Type C phase plane is shown in Figs. 6 and 7. Fig. 6 comes from numerical integration while Fig. 7 was derived using the map. The values of the parameters are [FORMULA], while [FORMULA]. This value of [FORMULA] was chosen to maximize the chaotic effects. The most important characteristic, observed in both Figs. 6 and 7 is the large unified chaotic region. In contrast with the two previous cases A and B, here the map reproduces satisfactorily the chaotic sea found by numerical integration. On the other hand, it is evident that the map describes qualitatively in a satisfactory way the areas of regular motion around the elliptic points in the [FORMULA] axis. Some differences are observed in the area near the center between the two patterns. Another important characteristic, observed in case C, is that the chaotic areas are large even when [FORMULA]. Results, not shown here, suggest that considerable chaotic areas are observed in the phase plane derived using the map, when [FORMULA]. Therefore we must admit that our numerical experiments show that, in the case when the Hamiltonian system (2) has small chaotic regions, the map is inadequate to produce them but it describes them satisfactorily when they are large.

[FIGURE] Fig. 6. The [FORMULA] (type C) phase plane derived by numerical integration. The values of the parameters are [FORMULA], [FORMULA], [FORMULA], [FORMULA].

[FIGURE] Fig. 7. Same as Fig. 6 derived by the map.

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© European Southern Observatory (ESO) 1999

Online publication: August 25, 1999
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