Astron. Astrophys. 329, 451-481 (1998)

5. The frequency map

The knowledge gained by the quasi-periodic approximations of the major types of orbits can be directly exploited for the construction of frequency maps through which we will be able to study the global dynamics of the system.

5.1. Choice of parameters

As mentioned above, we will choose a variety of representative values for the parameters and . On the other hand, the energy is kept constant to an arbitrary value () and the core radius is fixed to (approximately 1/8th of the maximum value on the x -axis). In that way, we can cover the possible dynamical behaviour of the logarithmic system and study its evolution through the change of shape of the isopotentials and the isodensity surfaces.

For this, it is essential that the chosen parameters correspond to physical density values. Moreover, they should approximate the ratios of the observed configurations of elliptical galaxies. In effect, as we argued in Sect. 3, the isodensities are two to three times flatter than the equipotentials. The isodensities may be labeled by two ratios and , with the convention that , a rough estimation of which was given in Eqs. (13) and (14). These ratios determine the triaxiality parameter (Franx, Illingworth & de Zeeuw 1991)

which qualifies the flattening of the model: when tends to 0 the model is oblate, whereas when it tends to 1 the model becomes prolate. In the case of the logarithmic system the axial ratios are parameterized by the value of the density (see Appendix A). In order to have a better estimation of the three shape parameters , and we may calculate the density value for which the variable labeling the long axis attains its maximum value, determined through Eq. (11), which corresponds to a minimal possible value taken by an isodensity line. We may next define the points where the given density surface cuts the two remaining axes and compute the three shape parameters, as performed by Lees & Schwarzschild (1992).

In Table 5, we present the 12 selected couples of parameters , in addition with the corresponding ratios of the equipotentials and shape parameters. This choice enables us to have an overview of models, covering a variety of density configurations, with a short to long axis ratio varying from 1/6 to 1/2, approximately, and a triaxiality parameter going from 0.94 (very close to a prolate model) to 0.25 (approaching the oblate case). However, as the isodensity surfaces are flatter than the isopotentials, the chosen models are quite elongated.

Table 5. Perturbation parameters , , equipotential axial ratios , and shape parameters , , .

5.2. Construction of the frequency map

A very useful way to view the phase space of a 2 degrees of freedom Hamiltonian system is by simply producing its Poincaré map (Poincaré 1892). Nevertheless, this map is rather unexploitable for systems with 3 or more degrees of freedom, as it has at least 4 dimensions. To our knowledge, the only satisfactory method designed to display the dynamics of multidimensional systems is the frequency map analysis. Indeed, the frequency map is a direct representation of the system's complicated network of resonances, known as the Arnol'd web (Arnol'd 1964; Arnol'd & Avez 1967, p.93).

In order to construct the frequency map, we will carry out a more drastic reduction of the dimension of the system than the one performed by a simple surface of section, by fixing all the variables considered as the angles to arbitrary values. For a 3 degrees of freedom system this implies to fix three angle-like variables and take initial conditions in two action-like variables, with the last variable determined by the restriction of the system in a constant energy manifold. We essentially require that this section cuts transversally the tori under consideration, taking into account the symmetries of the system (for details see Laskar 1993, 1996) .

5.2.1. Box orbits

The phase space of the logarithmic system can be roughly divided in four regions, corresponding to a different type of orbit. First, the boxes are orbits which are generated by the coupling of the 1 degree of freedom systems (Eq. 16). The phase space of these integrable systems is parameterized by the constant value of the energy. If we set now all the spatial variables equal to 0, the initial conditions in the conjugate momenta , and are directly associated with the energy values of each integrable system. Hence, in the general system, by taking initial conditions on one of the moment planes, e.g. the , and calculate the Z variable by Eq. (11), we are able to produce box orbits. Due to the mirror symmetries in all the system's variables we may consider only the positive subspaces of initial conditions. More precisely, we construct the map:

which maps the moments and labeling the long and middle axis to the rotation numbers and , defined as the ratios of the frequencies and associated with these variables, respectively divided by the frequency , which is conjugate to the variables labeling the short axis. The zone of circulation can be also generated by fixing all momenta to 0 and taking initial conditions in the spatial variables. This was conjectured by Schwarzschild (1993), for the scale free logarithmic potential, who called this plane "stationary" start space. Indeed, the sufficient condition in order to produce a box is to give to the integrated orbit a zero initial angular momentum. We prefer to take initial conditions in the momenta due to the fact that these variables seem to be more adapted as action-like variables. Moreover, the chosen plane of initial conditions is a straightforward generalisation of the section selected for the study of the 2D box orbits (Paper I).

We construct 12 representative frequency maps (Eq. 29) for the part of the phase space filled with box orbits by taking approximately 10000 initial conditions on the plane formed by the long and middle axis momenta. The results are displayed in Figs. 3, 5 and 7. The orbits are integrated for about 100 periods of the periodic orbit moving along the long axis of the system. This integration time assures a precision in the determination of the fundamental frequencies of the order of for quasi-periodic KAM solutions (Laskar 1996). Each orbit is represented as a dot on the plane permitting a very detailed study of the boxes dynamics. In order to have a more comprehensive view, we should point out that the image displayed on the frequency plane may be regarded as a superposition of frequency lines (Paper I, Figs. 8-12) produced by following a line of initial conditions, and making successive steps in another direction (Laskar 1993).

Fig. 3. Frequency maps for the part of the phase space covered with box orbits for different values of the perturbation parameters a , , b , , c , and d , (see Table 5 for the corresponding shape parameters).

Let us first comment on the characteristic features of the frequency map by concentrating on Fig. 3a. The left part of the plot contains orderly arranged points. The map seems to be regular in that area and, thus, we may conclude that there exist many KAM tori corresponding to these initial conditions. Yet, the actual KAM tori do not separate the phase space of a system with 3 or more degrees of freedom. Therefore, it is possible that the orbits corresponding to initial conditions in the complementary space, in between the tori, may diffuse (Arnol'd 1964). However, recent rigorous results (Morbidelli & Giorgilli 1995) using arguments of the KAM (Kolmogorov 1954; Arnol'd; 1963; Moser 1962) and the Nekhorochev theory (Nekhorochev 1977) prove that orbits in the neighbourhood of KAM tori will remain there for superexponentially long times. Hence, in the areas of the frequency maps where the number of KAM tori is large, we should be reassured that the orbits will remain there for very long times.

On the other hand, the regular regions are interrupted by lines of the form , with (at least two of them non-zero), which delineate resonant tori of the system. This complicated network of resonant lines represented on the maps is a snapshot of the Arnol'd web of the system. The resonant lines traced by empty gaps are related with whiskered tori, i.e. tori with a hyperbolic character (see Arnold & Avez 1967, p.93). On the other hand, when the plane of initial conditions cuts elliptic resonant tori (the generalisation of elliptic islands of stability in 3 degrees of freedom systems), we obtain the bold resonant lines on the map (Laskar 1993). The most important resonant lines are prolonged by fictive dashed lines and labeled with the three corresponding parameters .

The horizontal, vertical and parallel to the diagonal resonant lines (when , or is 0) are related with resonances of the 2D systems formed by the short-middle, short-long or middle-long axes variables, respectively. The principal resonant lines of the first two 2D systems, which correspond to zero values in one of the rotation numbers, are left out of the diagram's frame. The third 2D system cannot be viewed on this map, as, for , the rotation numbers and are singular. All the other lines are due to the coupling of these systems. The intersections between the resonant lines indicate periodic orbits, which can be directly identified by the two rotation numbers. We are able to have an idea on the linear stability of these orbits, by checking the character of the intersecting resonant lines, as described in the previous paragraph. All the important periodic orbits can be found in Table 6, with the related values of the rotation numbers , the resonant lines whose intersection indicates the existence of each one of them and the labels of the corresponding figures. Let us note that the axial periodic orbits do not belong to the map, as their rotation numbers are singular or zero. Nevertheless, the left, bottom and right part of the map, respectively, is covered with orbits which are very close to them.

Table 6. Periodic box orbits corresponding to the frequency maps of Figs. 3, 5 and 7, with their associated values of the rotation numbers and , the intersecting resonant lines defining them and the labels of the related figures.

On the right part of the diagram the frequency map is irregular. This fact is testified by the area of dispersed points. For these initial conditions, the quasi-periodic approximation does no longer hold and we can conclude that the scattered points correspond to chaotic orbits. In this part of the map we perceive two principal lines of accumulation of points. The one formed in the upper part of the diagram is connected with a 1:1 resonance between the middle and the short axis variables and the other, which is the diagonal of the frequency plane, is related to a 1:1 resonance between the middle and long axis variables. These lines are the result of the perturbation of the short axis periodic orbit. We have remarked in the linear stability analysis of the rectilinear periodic orbits (Fig. 1) that, apart from the integrable spherical case, the short axis rectilinear orbit (whichever is the variable representing it) is hyperbolic-hyperbolic, for practically all physical values of the perturbation parameters. Thus, when the parameters correspond to values for which the system is non-integrable (for and ), we may surmise that this orbit creates a stable and an unstable manifold of dimension 3 which intersect transversally. The homoclinic intersections of these manifolds, and, as the perturbation raises, the overlapping of nearby resonances generate the chaotic zone. This chaotic region traces the boundaries of the large libration zones containing the long and short axis tubes. The two lines are formed by orbits which are chaotic and turn from libration (1:1 resonance in between the short/middle and the long/middle axis variables as the long and short axis tubes, respectively) to circulation (as the boxes) and vice-versa. The large gap between these two lines and the major body of the map is due to the strong hyperbolicity of the periodic orbit, which is related with a singularity in the rotation numbers (Laskar 1993). Moreover, as the middle axial periodic orbit is partially hyperbolic for the values of the perturbation parameters corresponding to all the studied maps, the perturbed 2-dimensional stable and unstable manifolds of this orbit enlarge further the chaotic region (bottom part of the maps).

The dynamical behaviour of the system is proved to be quite complicated, as it was predicted in some extent by Schwarzschild (1993) for the singular logarithmic potential and in recent studies of 3-dimensional galactic potentials with a cusp (Merritt & Fridman 1996). In these studies some resonances of the 2D systems, (the so called "boxlets" - see Miralda-Escudé & Schwarzschild 1989; Lees & Schwarzschild 1992) and periodic orbits were identified. However, the objects playing the fundamental role for the dynamics of a 3 degrees of freedom system are the 3-dimensional resonant tori (the coupling between the "boxlets"), which are directly identified in the frequency maps with their associated resonant conditions and their actual strength. Indeed, all these features, which are intrinsically connected with the addition of a third dimension in the system, are generally very hard to be discovered and visualized with any classical method.

The dynamical evolution of the system through the change of the perturbation parameters can be followed by comparing the frequency maps. In the first 8 maps (Figs. 3 and 5) the short to middle axis ratio of the equipotentials is kept constant and equal to 0.9 which corresponds to an isodensity ratio for the related axes of 0.6-0.7, approximately. On the other hand, by varying the long axis, we are able to cover values of the shape parameter extending from 1/2 to 1/5, with a triaxiality parameter going from a moderate value () to a prolate case ( - see Table 5).

The most important resonant line (with the lowest value of ) is the (1,-2,1). It is the most visible one in the middle of the maps and has a strongly hyperbolic character, for the chosen initial condition plane. Most of the resonant lines in all the diagrams are connected with the perturbation of the short axial periodic orbit (radial lines the prolongation of which passes from the point of the map). As we may observe in Fig. 3, when the perturbation increases, the resonant lines overlap and orbits may diffuse through heteroclinic intersections across the resonances. This should be an extension of the so called Chirikov diffusion ⁵ for systems with 3 degrees of freedom (see Laskar 1993, 1996 for numerical evidence). Indeed, in Fig. 3d which corresponds to the higher value of the perturbation for this group of maps, all tori up to the resonant line (3,-1,-1) have been destroyed producing a large chaotic region. Besides, by varying the axial ratios of the equipotentials, the length of the compact phase space following the long axis direction increases, thereby permitting the appearance of new resonant lines. The last comment for this group of maps regards the long axial periodic orbit. In the first three maps this orbit is elliptic-elliptic for the corresponding values of and - a fact which is also reflected by the regular area in the left part of the diagram. On the other hand, according to the linear stability analysis effectuated in Sect. 3.1, this orbit becomes partially hyperbolic for the perturbation parameters corresponding to the map of Fig. 3d ( and ). Consequently, a small irregular region appears in the vicinity of this orbit characterized by some dispersed points in the left part of the map.

In addition to the frequency maps, a very useful representation of the dynamics of box orbits is obtained through Figs. 4, 6 and 8, where we map the initial condition space (the axes represent squares of diminishing initial momenta) with a certain diffusion parameter. Actually, we obtain the rotation numbers corresponding to each orbit for a second quasi-periodic approximation by continuing the integration for another 100 periods. Then, we compute the maximum between the temporal derivatives of the two consecutive rotation number couples (Laskar 1993, 1996; Dumas & Laskar 1993). Each point on this graph is labeled through the diffusion parameter, by increasing the grey colour weight of each point proportionally to the logarithm of the diffusion rate. Only the fine points represent quasi-periodic orbits up to the numerical error on the estimation of the frequencies. Attempts to construct diagrams of this kind for galactic models were also performed by Schwarzschild (1993) and by Merritt and his collaborators (Merritt and Fridman 1996; Merritt and Valluri 1996) by using the classical Lyapunov exponents in order to distinguish regular from chaotic orbits. The advantage of our representation, apart from the very quick and detailed scanning of the zone, is the fact that it is based on the calculation of the rotation numbers which can be considered as numerical integrals of motion. Thus, their first derivative can be regarded as a quasi-linear diffusion coefficient (see Lichtenberg & Liebermann 1992, p. 328).

Fig. 4. Initial conditions' space (momenta) corresponding to the boxes' frequency maps of Fig. 3.

Fig. 5. Frequency maps for the part of the phase space covered with box orbits for different values of the perturbation parameters a , , b , , c , and d , (see Table 5 for the corresponding shape parameters).

Fig. 6. Initial conditions' space (momenta) corresponding to the boxes' frequency maps of Fig. 5.

Fig. 7. Frequency maps for the part of the phase space covered with box orbits for different values of the perturbation parameters a , , b , , c , and d , (see Table 5 for the corresponding shape parameters).

Fig. 8. Initial conditions' space (momenta) corresponding to the boxes' frequency maps of Fig. 7.

For the first group of maps (Fig. 4), we may observe that the most irregular region corresponds to orbits produced for moderate to small values of the long axis moment. These initial conditions generate boxes laying close to a plane of the configuration space formed by the short and middle axes. This fact stresses once more the importance of the hyperbolicity of the two rectilinear orbits, moving along these axes. In these figures, we may also view the resonant lines, which appear as bold curves. If the square of the moments were exactly equal to the actions of a system of coupled rotators, all the latter should have appeared as straight lines. When the perturbation magnifies, a larger proportion of these diagrams becomes irregular. As we may see in Fig. 4d, the chaotic region is now apparent in the bottom of the diagram near the long axis system, as the corresponding orbit is partially hyperbolic.

In the four maps presented in Fig. 5, where the triaxiality parameter takes values approaching the prolate case, the irregular part of the phase space is enlarging. Indeed, as previously mentioned, this is also partly due to the hyperbolicity of the long axial periodic orbit, which is related with the appearance of the (2,0,-1) resonant line, in the left part of Fig. 5a. Due to resonance overlapping, most of the region depicted on the frequency map is covered with irregular orbits. The irregular orbits are multiplied in Fig. 5b, where almost every box orbit is chaotic. The principal lines of accumulation of points in the main body of the map are broken resonant tori, and the principal attracting point in the middle of this diagram is in the neighbourhood of the 4:6:7 elliptic periodic orbit. This phenomenon is common in the chaotic regions where the orbits are "trapped" in the vicinity of resonances for quite some time, due to the effect of long time correlations (Contopoulos 1971; Shirts & Reinhardt 1982). This is one of the principal reasons for which we cannot consider that a chaotic orbit fills ergodically the permitted space even if it has a high rate of diffusion.

The dramatic change in the behaviour of the circulating part of the phase space can be easily visualized in the "action space". Indeed, a large proportion of the diagrams presented in Figs. 6a and b, are entirely black, due to the high values of the diffusion parameter. Only some small space in the middle of the Fig. 6b is covered with fine points, where we can locate the initial conditions of orbits which seem regular for this time span.

The situation changes slightly in the case of Fig. 5c, for and . For these values of the perturbation parameters, the rectilinear long axis periodic orbit becomes once more linearly elliptic, and the left part of the diagram is covered with regular orbits. This regular region is also depicted in the left part of the "action" space in Fig. 6c, close to the long axial periodic orbit, corresponding to a maximum value of the long axis moment. Nonetheless, some new resonant lines do appear, and their traces can be also identified in the left part of the "action" diagram.

For values of the perturbation parameter between 1.80 and 2.25 (Figs. 5c and d) the long axial periodic orbit becomes partially hyperbolic bifurcating the 3:0:1 periodic orbit, whose perturbation (by increasing the dimension of the system) gives rise to the (-3,0,1) resonant lines and to many others, due to the coupling. Then it becomes once again elliptic. In the last map of this group, all these resonances have overlapped, up to the (3,-1,0) line and, moreover, the long axial periodic orbit is becoming again hyperbolic, giving rise to the 4:0:1 periodic orbit. This instability can be observed through the space of initial conditions, where the upper part of the diagram, close to this orbit, seems quite irregular.

This highly irregular behaviour of orbits in these models for small values of the shape parameters and (approaching the prolate case) is not surprising. As we showed in Paper I (see Paper I, Fig. 10), in the case of the 2-dimensional logarithmic system, and for values of the perturbation parameter q which are similar to the short to long axis ratio of the studied triaxial models, we were able to discern the existence of a multitude of resonances surrounded by chaotic zones. Thus, by adding a third degree of freedom in the system, the interaction between these resonances due to the coupling, even for axial ratios which are fairly close, produces large chaotic regions. The effect of the addition of one dimension is also apparent in the first group of maps (Fig. 3). In contrast to the their associated 2D systems which are mostly regular (Paper I, Fig. 8), the maps of the triaxial system present many resonances and quite significant chaotic zones.

The last group of maps correspond to models which are somewhat flatter then the group of Fig. 3 (), but with a similar middle to long axis isodensity ratio , letting us cover values of the triaxiality parameter which are close to the oblate case () and beyond (Table 5). The behaviour of the short axis rectilinear orbit is similar as before, and thus we observe the existence of a chaotic zone in its vicinity. Furthermore, many resonant lines are springing as a result of the interaction of this orbit with resonances of the 2D systems. On the other hand, the middle axis periodic orbit is partially hyperbolic for all the values of these perturbation parameters. Thus, it contributes to the chaotic zone (bottom edge of the maps). These aspects are clearly visible on the "action plane" where, as previously, most of the initial conditions corresponding to chaotic orbits are located in the vicinity of the 2D system formed by the variables labeling the short and middle axes. We may finally observe in the first two maps the resonant lines originating from the short-middle axes system (vertical lines (3,0,-2) and (5,0,-3)) and many others which arise as a result of the coupling. These lines are also visible in the "action" space appearing as a spider web penetrating inside the regular regions.

In Fig. 7c, we have the apparition of the (2,0,-1) resonant line, which is connected with the hyperbolicity of the long axis rectilinear orbit and its bifurcation to a 2:0:1 periodic orbit on the restricted 2D system. Thus, the chaotic zone covers that part of the map represented by the grey region depicted in the vicinity of the orbit. This zone extends near the long-short axes 2D system (Fig. 8c). The effect of overlapping of resonances is mostly apparent in the neighbourhood of the short axial periodic orbit on the right side of the map. At last, with the magnification of the perturbation, the overlapping of all these resonant tori gives rise to a large chaotic region in the central body of the map. However, many chaotic orbits seem to be trapped in the vicinity of two families of resonant tori corresponding to the (2,1,-2) and (3,-1,-1) resonant lines, which intersect on the 3:4:5 periodic orbit. These regions correspond to the center of the action diagram and have smaller diffusion rates (Fig. 8d). On the other hand, the vicinity of the long axial orbit is regular, as shown also by the linear stability analysis of Sect. 3.1. This last map, for which we have the appearance of chaotic regions in a rather large scale, corresponds to a strongly perturbed system, with a triaxiality parameter equal to 0.5 and a short to long axis ratio approximately equal to 1/5.

5.2.2. Tube orbits

Whereas the circulation zone can be displayed by taking initial conditions in one of the surfaces with vanishing initial angular momentum components (setting all initial positions or moments equal to zero), the situation is quite complex in the case of the libration zones, where the initial angular momentum components vary within a wide range of values. Thus, we do not expect to map only one kind of orbit on each of the remaining initial condition planes which are produced by keeping three non-conjugate variables equal to 0 and by evolving in the other three on the constant energy manifold. In general, it is possible to distinguish the type of orbit generated by each initial condition through the leading frequencies of their quasi-periodic approximation given by the frequency map analysis. The regions occupied by different kind of orbits on the 6 remaining initial condition planes are delineated on Fig. 9, for values of the axial ratios close to the spherical case ( and ). The areas covered by boxes are represented by the empty spaces, the fine points label the inner long axis tubes, whereas the light and deep grey regions depict the outer long axis and short axis tubes, respectively. The bold points represent chaotic orbits, which bound the different kind of quasi-periodic motion. Let us remind that for the chosen axial ratios the position variables x, y and z correspond to the middle, long and short axis, respectively.

Fig. 9a-f. Planes of initial conditions labelled by the regions of different kind of orbits, for and . They are constructed by fixing three variables to 0 and evolving in the remaining three: (a) middle, short axis positions and long axis moment , (b) middle, long axis positions and short axis moment , (c) long, short axis positions and middle axis moment , (d) long axis position, short and middle axis moments , (e) short axis position, long and middle axis moments and (f) middle axis position, long and middle axis moments .

As mentioned above, the distribution of the different types of orbits on these images can be elucidated through the initial value of their angular momentum components (Schwarzschild 1993). For the chosen couple of perturbation parameters, the components are , and , parallel to the middle, long and short axis, respectively. The short and long axes components play a fundamental role in the Hamiltonian structure of the system: they correspond to the integrals of the integrable cases, the perturbations of which generate tube orbits, as shown in Eqs. (25) - (27). This dependence is also apparent in the physical characteristics of these orbits (e.g. their shape in the configuration space).

The diagrams of Fig. 9 can be divided in three groups depending on which initial angular momentum components are non-zero. The initial conditions portrayed in Figs. 9a and 9d correspond generally to non-vanishing initial values of the angular momentum components parallel to the middle and short axes. The middle axis angular momentum, which is the integral of the short/long axis axisymmetric system, is not related with quasi-periodic motions due to the partial hyperbolicity of the corresponding periodic orbit in the complete 3D system (see Sect. 4.2.2). On the contrary, the initial non-vanishing value of the angular momentum component following the short axis can generate short axis tubes. Nevertheless, these planes will also contain some boxes, for vanishing values of the initial angular momentum corresponding to the bottom left and right corners of the figures, i.e. when the initial conditions of the middle axis variables ( and ) are small.

The planes displayed in Figs. 9b and 9e correspond to non-zero initial values of the angular momentum component following the long axis. Hence, naturally, outer long axis tubes are produced until the initial long axis angular momentum reaches a critical value, for which the system begins the circulation in the long axis variables. These regions which correspond to an important value of the long axis spatial variable (here y) are covered with inner long axis tubes. As before, the initial conditions for which the angular momentum vanishes produce boxes.

At last, in the case of the planes and , the situation is more involved, as both the long and short axis angular momenta accept non-vanishing initial values. Therefore, we expect to have all possible kind of orbits (see Figs. 9c and f), depending on which initial angular momentum component is dominant. Indeed, short axis tubes cover the area of high initial short axis angular momentum, that is when the products (Fig. 9c) or (Fig. 9f) attain high values. The outer long axis tubes appear when the long axis angular momentum is large (corresponding to the products and for each plane, respectively). When the variables labelling the long axis reach values which permit the circulation following the latter direction, while keeping a high value of the long axis angular momentum, the orbits turn to inner long axis tubes. At last, boxes are confined in the regions where the initial components of the angular momenta are close to 0.

An important dynamical aspect of the system which was mentioned in recent studies (see Merritt & Fridman 1996) is the existence of chaotic regions separating the different types of quasi-periodic motion in the libration zones of the system. In fact, as we commented in the case of box orbits (see Sect. 5.2.1), the stability of the principal rectilinear periodic orbits is crucial for the existence of these regions. For the values of the axial ratios employed here, the long axial orbit is linearly elliptic, while the middle and short axial ones are simply and doubly unstable, respectively (Fig. 1). As stressed in Sect. 5.2.1, by adding two non-symmetric dimensions, these latter orbits create a stable and an unstable manifold of dimension 2 in the case of the partially hyperbolic orbit, and of dimension 3 for the hyperbolic-hyperbolic orbit. In each case, the stable manifold intersects transversally with its corresponding unstable manifold thereby producing chaotic zones which can be enlarged as the perturbation increases. Indeed, the traces of these regions, and especially of the one produced by the small axial periodic system, are very apparent in the initial condition planes of Fig. 9.

More specifically, in Figs. 9a and d, the chaotic zone generated by the small axial periodic orbit separates the box orbits from the small axis tubes. To this chaotic region, at least for initial conditions very close to the middle/long axes system, contributes the small irregular zone produced due to the partial hyperbolicity of the middle axial periodic orbit. In Figs. 9b and e, the chaotic region produced due to short axial periodic orbit appears in three areas of the initial condition planes. In the extreme right and bottom corners of each diagram, respectively, the small chaotic region corresponding to the middle axial periodic orbit is superposed with the chaotic zones produced by the short axial orbit. It is interesting to observe that the perturbed manifolds of the short axial periodic orbit split in two, bounding the three different types of orbits and also separate the inner long axis tubes in two parts. This phenomenon is also apparent in the last two figures (Figs. 9c and f), where the chaotic zone produced due to the hyperbolicity of the short axial periodic orbit separates all four types of quasi-periodic motion.

On the other hand, in the case of the middle/short and middle/long axes initial condition planes (Fig.9a and b), the chaotic region expands along the short and long axes of the system, respectively, due to the partial hyperbolicity of the periodic orbit around which loops librate, in the restricted 2D system. It is already known that around the circular orbit of the 4-dimensional restriction of the axisymmetric system (setting the middle axis variables equal to 0) there exist a family of quasi-periodic orbits parameterized by the angular momentum. In the complete system, (setting the middle axis variables slightly different from zero) this is a one parameter family of partially hyperbolic tori, the so called whiskered tori. Cresson (1997) gave a mathematically rigorous result regarding the existence of these objects and the persistence of a large number of them in the triaxial system, provided that the axial ratio between the small and the middle axis is close enough to 1, in the sense of the hypotheses of KAM theorem. Moreover, he proved that under these circumstances, and as the second integral of motion (the angular momentum) is destroyed, there exists a neighbourhood for each whiskered torus in which the associated unstable manifold of the later object intersects transversally with the stable manifold of every partially hyperbolic torus inside this neighbourhood. Hence, it is formed a transition chain, through which orbits may diffuse, a process known under the name of Arnol'd diffusion (Arnol'd 1964; Arnol'd & Avez 1967). The logarithmic system provides one of the rare examples discovered up to this moment for which the possibility of the existence of instability orbits can be rigorously demonstrated, even if this type of diffusion should be extremely slow in order to be physically significant in the case of galaxies. However, when the perturbation increases, the existence of this instability is responsible for the chaotic regions viewed in Figs. 9a and b, for initial conditions very close to the short/long axis system (with vanishing values of the middle axis spatial variable).

Let us now turn back to the distribution of quasi-periodic orbits on the initial condition planes. In fact, these figures are similar to the orbit classification diagrams in the action space of a Stäckel system, as projected on a surface of constant energy (de Zeeuw 1985; Statler 1987; Schwarzschild 1993). In that case, though, the initial condition plane formed by the long and short axes, covers the possible dynamical behaviour of the system. On the contrary, in the logarithmic system (and in its scale free limit), this is not true. Schwarzschild (1993) restricted his study for the tubes on that plane by conjecturing that most of the tube orbits should cross it. In fact, this plane gives the necessary information for the construction of a self-consistent model, as it contains all type of tubes. Nonetheless, by checking the range of the rotation numbers values parameterizing the motion of the regular orbits of the system for each initial condition plane, we can show that, in our case, almost all possible orbits may be covered by taking two planes of initial conditions: one for the long axis tubes formed by the middle and long axis spatial variables (Fig. 9b) and a second one on the middle and short axis spatial variables (Fig. 9a), for the study of the short axis tubes. This choice seems much more appropriate, also dictated by the perturbative approach followed in order to explain the existence of these orbits (Sect. 4.2).

The complexity of the initial condition planes is a direct indication that the Cartesian coordinates provide a bad parametrization for the study of the libration zones. Indeed, due to symmetries, initial conditions producing the same orbit may be found in two different points of these planes. For the short and outer long axis tubes, this can be understood by checking their initial angular momenta (Schwarzschild 1993). These quantities can take similar values in two parts of the planes, separated by the arc of maximum initial total angular momentum connecting the two periodic orbits of the restricted 2D systems. This line designates orbits which are occasionally named as thin tubes due to the fact that they are produced by a very slight perturbation of the corresponding planar periodic orbit. Thus, for the short and outer long axis tubes a better parametrization would have been furnished by taking initial conditions following the angular momentum components after transformation of the system in the corresponding polar coordinates, as performed in the case of the loops of the 2D system (Paper I). However, this procedure cannot be applied in the case of inner long axis tubes, which are separated by the chaotic zone produced by the doubly hyperbolic small axial orbit. What is more troublesome is that even in the case of the short and outer long axis tubes the polar coordinates do not give a good parametrization of the libration zones. The latter was very apparent in the numerical experiments conducted in order to produce frequency maps for these zones as the ones constructed previously for box orbits. Indeed, for some orbits, it is very difficult to determine the fundamental frequencies of motion even in these variables (they do not even appear in the 20 first terms of the quasi-periodic approximation). There is no simple solution to this problem. The construction of numerical normal forms, apart from being cumbersome, it is doubtful whether it can improve the convergence of the series (Paper I). On the other hand the averaging procedure (see Sect. 4) needs long computer integration times. Nevertheless, the construction of the frequency maps for the libration zones is not essential in order to understand the global dynamics of the system. Indeed, the important resonances inside these regions are quite limited in number and occupy a small fraction of the phase space. In addition to this fact, when the perturbation increases, the libration zones corresponding to quasi-periodic motion shrink, as the chaotic regions cover a larger part of the phase space.

What seems crucial for the comprehension of the tube orbits' dynamics is the distinction between the initial conditions generating quasi-periodic orbits from the ones triggering chaotic motions. As we referred in Sect. 5.2.1, in previous studies, this was attempted by the use of Lyapounov exponents (see Schwarzschild 1993; Merritt & Fridman 1996). The frequency map analysis provides an elegant and much more efficient alternative solution, by mapping the maximum diffusion rates on the initial conditions planes, as performed in the case of the boxes, for the same group of axial ratios (see Table 5). The diffusion rates for the tubes are calculated by the temporal derivative of the leading frequencies ratios between two successive integrations and application of the frequency map analysis. We should point out that these ratios do not represent the real rotation numbers parameterizing the motion of this orbits, as the analysed Cartesian coordinates are resonant in that case. In spite of this fact, the information of whether the motion is regular or not is transparent even in these frequencies.

In Figs. 10 and 11, we present the diffusion rates mapped on the middle/long axes and middle/short axes planes, for the first group of perturbation parameters (Table 5 - see also Fig. 4 for the "action" space related with box orbits). Let us first turn our attention to Fig. 10 representing long axis tubes for the majority of initial conditions and some boxes limited in an extreme exterior arc of initial conditions near the limit of the compact phase space. The main chaotic regions are spread near the small axial periodic orbit, in the lower right and left and the upper parts of the diagrams. The extension of the chaotic region near the middle axis maximum value follows from the partial hyperbolicity of the middle axial periodic orbit. In addition, the partial hyperbolic tori produced by small perturbations of the short/long axes 2D system give rise to the chaotic region in the area of small initial values of the middle axis variable. In Fig. 10a the traces of some resonant lines appear in the middle part of the diagram, which are symmetric with respect to an imaginary line defining initial conditions with maximal initial total angular momentum. In all four diagrams, the perturbed manifolds of the doubly unstable small axial periodic orbit, in the upper and lower right part of the diagram split in two the initial conditions producing inner long axis tubes. Further, two lines corresponding to irregular orbits separate the inner long axis tubes from the boxes and the outer long axis tubes. We should emphasise the fact that apart from the strongly chaotic motion due to the instabilities of the principal periodic orbits, at least for the last three cases of the axial ratios (Figs. 12b, c and d) there exist large areas corresponding to inner and outer long axis tubes for which the motion is nor quasi-periodic but neither strongly chaotic. The latter proves that the chaotic motion inside the libration zones was generally underestimated in previous studies. With the increase of the perturbation the fraction of orbits which seem to be quasi-periodic diminishes. Especially in the last figure (Fig. 12d), the chaotic regions are extending even in the part of the diagram near the long axial periodic orbit (maximum of the corresponding variable), as it becomes partially hyperbolic for these values of the perturbation parameters. This behaviour is quite different from the one observed for loops in the 2D system where the chaotic regions were quite limited near the unstable small axis orbit (see Paper I).

Fig. 10. Initial conditions space (middle/long axis positions) covered with the two families of long axis tubes and boxes for different values of the perturbation parameters a , , b , , c , and d , .

Fig. 11. Initial conditions space (middle/short axis positions) covered by short axis tubes and boxes for different values of the perturbation parameters a , , b , , c , and d , .

Fig. 12. Initial conditions space (middle/long axis positions) covered with the two families of long axis tubes and boxes for different values of the perturbation parameters a , , b , , c , and d , .

In Fig. 11, the majority of orbits are small axis tubes, apart from the ones corresponding to the zone near the rectilinear periodic orbit moving along the long axis of the system (for vanishing values of the middle/short axis variables). Boxes are always separated from the small axis tubes by a line of chaotic orbits springing from the perturbation of the small axial periodic orbit. The chaotic regions produced by this orbit, in addition with the ones of the partially hyperbolic tori generated by a perturbation of the small/long axial system extend along the small axis for small values of the middle axis variables. In these figures (especially in Fig. 11a) we may observe many dark lines corresponding to resonances. These lines seem to originate from the chaotic zone produced around the small axial periodic orbit and have a certain symmetry due to the before-mentioned fact, i.e. equal values of the angular momentum produce the same orbits in two parts of the initial condition planes. On the other hand the increase of the perturbation raises the proportion of the initial conditions for which the motion is chaotic. Besides, as the long axis length is magnified with respect to the other two axes which remain unchanged, the boxes occupy a larger zone. As before, for and , (Fig. 12d), we may descry the small chaotic region created near the long axis orbit due to its partial hyperbolicity.

In Figs. 12 and 13, we display the initial condition planes for the second group of axial ratios. In these cases, most of the initial conditions lead to chaotic motions in both planes, which is in agreement with the behaviour of box orbits in their corresponding "action" planes (Fig. 6). Indeed, the proportion of initial conditions producing chaotic motions is large for and (Figs. 12c and 13c), where all the principal rectilinear periodic orbits are hyperbolic. In the figures representing the middle/long axis initial condition plane, there exists a central part that seems more regular and separates the inner long axis tubes by the other two types of orbits. On the other hand, the region corresponding to small axis tubes shrinks as the long axis increases its length with respect to the other two axes. Indeed, in the last diagram, for and , the small axis tubes represent a very small fraction of initial conditions near the maximum values of the middle axis variables.

Fig. 13. Initial conditions space (middle/short axis positions) covered by short axis tubes and boxes for different values of the perturbation parameters a , , b , , c , and d , .

Finally, the initial condition planes depicted in Figs. 14 and 15 correspond to the last group of the chosen perturbation parameters. In these figures, the situation is quite similar as in the first groups of perturbation parameters displayed in Figs. 10 and 11. For small perturbations, the chaotic regions are concentrated near the unstable periodic orbits and are magnified considerably in the last diagram, for and , at least in the case of long axis tubes (Fig. 14d). The initial condition planes covered with small axis tubes and some boxes seem more regular. A very interesting aspect depicted in these diagrams are the traces of the resonant lines forming a kind of triangle in the bottom part of the diagrams, which overlap with the main chaotic region generated by the hyperbolic small axial orbit, for the last couple of perturbation parameters (Fig. 15d). In this last diagram, we may observe some resonant lines springing from the region near the middle axial periodic orbit, which is partially hyperbolic for the considered axial ratios.

Fig. 14. Initial conditions space (middle/long axis positions) covered with the two families of long axis tubes and boxes for different values of the perturbation parameters a , , b , , c , and d , .

Fig. 15. Initial conditions space (middle/short axis positions) covered by short axis tubes and boxes for different values of the perturbation parameters a , , b , , c , and d , .

© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997
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