4. Frequency map analysis of the triaxial logarithmic system: quasi-periodic approximations
In this section, we will proceed as in Paper I, in order to verify our conjectures regarding the principal categories of orbits. In effect, we will endeavour to understand their complicated structure through the quasi-periodic approximations given by the frequency map analysis.
4.1. Box orbits
We apply the NAFF algorithm to the representative box orbit (Fig. 2a), generated by the initial conditions . The orbit is integrated for about 20 orbital periods represented by the number of crossings of the surface . This time span assures a precision in the determination of the frequency vector for quasi-periodic KAM solutions of the order of (Sect. 2).
We present in Table 1 the quasi-periodic approximations given by the frequency analysis when applied to each pair of conjugate variables, considered as complex functions of the form , and , respectively (long, middle and short axis variables for this choice of the perturbation parameters). Here, the number of terms N is typically 40. For reasons of economy in space we display only the ten leading terms of the solutions, which will provide, though, enough information about the dynamics of the orbits in question. Let us first remark that the solutions displayed in Table 1 are indeed quasi-periodic, as every can be identified by linear combinations of the three fundamental frequencies , and , corresponding to the three terms of maximum amplitude of each quasi-periodic approximation.
Table 1. Frequency analysis of the Box orbit (Fig. 2a). The quasi-periodic solution has the form , where , with a real amplitude and the phase and the should be linear combinations of three fundamental frequencies and , while is a measure of the precision of the quasi-periodic decomposition.
The three fundamental frequencies are non resonant, since the boxes, as their 2-dimensional analogues (Paper I), are orbits circulating in the phase space parameterized by the Cartesian coordinates and their conjugate momenta. They can be generated by the coupling of the 1 degree of freedom integrable systems studied in detail in Sect. 3.1. To be more precise, let us first note that the only axial periodic orbit which is linearly elliptic for values of q close to 1 is the long axial orbit, i.e. the one generated by the -system (see Fig. 1). This is reflected also in the quasi-periodic approximation representing the function (Table 1) which is the only one where we can distinguish uncoupled harmonics of the form (see also Paper I). Thus, it is most probable that a perturbation of the "long axis system" by adding two non-axially symmetric dimensions in the logarithm of Eq. (16) will generate quasi-periodic orbits. For and , we may write the Hamiltonian of the complete system, as follows: 4
The second part of the Hamiltonian (24) can be considered as a perturbation if x and z are small enough comparing to y. Nevertheless, a perturbative study would not be possible (Paper I) as the variables used are very far from the actions and angles of the integrable system, a fact which is also apparent in the complexity and the slow convergence of the quasi-periodic approximations.
4.2. Tube orbits
4.2.1. Short axis tubes
We generate a typical small axis tube orbit , whose projection in the configuration space is plotted in Fig. 2b, by integrating the equations of motion of the system with initial conditions for the same time span as for the box orbit. The quasi-periodic approximations extracted by the NAFF algorithm for the complex functions , and are represented in Table 2.
Table 2. Frequency analysis of the short axis tube orbit .
In Sect. 3.4, we inferred that the short axis tubes should be resonant in the employed variables. This is indeed true as we may observe from the frequencies of maximum amplitude of the quasi-periodic approximations involving the long and middle axis variables. By considering , and as fundamental frequencies, we manage to determine with a good precision all the harmonics of the quasi-periodic approximations. On the other hand, the solutions corresponding to the resonant variables are very similar to the ones of an axisymmetric system (see Paper I). Thus, we may suspect that the small axis tubes are generated by the perturbation of the axisymmetric system restricted on the long-middle variables () hypersurface by setting the perturbation parameter ( in our case) different from 1 and a second perturbation by adding in the argument of the logarithm of the potential a third non-axisymmetric dimension. This perturbation may generate quasi-periodic orbits, considering the fact that the periodic orbit on the plane formed by the middle and long axis spatial variables (x and y in our case) is linearly elliptic. For the used values of the axial ratios, we may write:
where . The part is a perturbation if is close to 1 and z is sufficiently small comparing to .
The part of the phase space covered by these resonant orbits can be thought as an extension of a librating island (or mathematically speaking of a central manifold of dimension 4). In that part of the phase space, the ratios between the frequencies of maximum amplitude is constant, due to the fact that the variables associated with these frequencies label the tori of the circulating part of the phase space. Consequently, in order to explore the librating part corresponding to the short axis tube orbits, we should find a proper system of variables where the resonance is eliminated. The approach the most direct is the construction of some suitable normal forms which would transform the motion into products of distorted circles. However, we have observed in Paper I, that the transformation to some adapted action-angle variables is rather cumbersome for such a complex system.
Alternatively, the resonant long and middle axis variables may be transformed in polar coordinates, as in Eq. 25(see also Paper I), which is a more adapted set of variables for the study of the short axis tubes. A third solution can be given by using a trick inspired by the averaging procedure in dynamical systems. In fact, as we have determined the resonant frequency given by the frequency analysis, we may integrate the equations of motion of the system using as step the associated period . Thus, we "average" in a certain way the system, as it is performed, for example, in planetary dynamics, in order to eliminate the fast motions of the orbit and recover the "real" frequencies, which are "hidden" by the resonance. Principally due to lack of space, we will not present here any example of these "averaged" quasi-periodic solutions (for more details see Papaphilippou 1997).
4.2.2. Outer long axis tubes
The third important family of orbits of the logarithmic potential is the outer long axis tubes. In the case of the chosen axial ratios, these are the orbits which are moving around the y -axis in the configuration space. We proceed in the same way as in the two previous subsections by applying the NAFF algorithm to the representative of this family (Fig. 2c). As before, we integrate the equations of motion with initial conditions , for approximately 20 orbital periods. The quasi-periodic approximations for the 3 pairs of conjugate variables are represented in Table 3.
Table 3. Frequency analysis of the outer long axis tube orbit .
The conjecture that these orbits present a 1:1 resonance between the middle and the short axis variables is verified by the computed quasi-periodic approximations. All the harmonics can be considered as linear combinations of three fundamental frequencies , and and this particular orbit seems quasi-periodic.
We remark the prominent resemblance of these quasi-periodic approximations with the ones issued by the application of the method in the short axis tubes (see Table 2), especially in the quasi-periodic decomposition of the harmonics. More specifically, the resonant variables present approximately the same frequencies in their quasi-periodic approximations. Indeed, these orbits may be considered as the perturbation of an axisymmetric 2 degrees of freedom system by setting the axial ratio different from one, plus a second perturbation by adding a third dimension which is not rotationally symmetric with the two previous ones, as in the case of the short axis tubes. Considering the fact that the periodic orbit in the middle-short axis plane (the -plane in the present choice of the axial ratios) is linearly elliptic, the Hamiltonian of the system may be written:
where . The second part of the Hamiltonian is a perturbation if is close to 1 and y is sufficiently small comparing to . The same formula may be written for any range of the axial ratios with a proper reorganisation of the variables.
Formally, the Hamiltonian of the system can be also written as a perturbation of the axisymmetric system restricted in the surface of the long-short axis variables (here the ) with a perturbation on the middle (x) axis, as for the short axis and outer long axis tubes. However, this kind of perturbative approach is not applicable whenever the variable x labels the middle axis of the system. In that case, the periodic orbit lying on the long-short axis plane is partially hyperbolic and a perturbation of this orbit by adding a third degree of freedom should create irregular orbits.
4.2.3. Inner long axis tubes
The last principal quasi-periodic family of orbits of the logarithmic system are the inner long axis tubes. The behaviour of these orbits in the configuration space is very similar with the outer long axis tubes. In order to study more closely their dynamics, we display the quasi-periodic approximations given by the frequency analysis (Table 4) when applied to the orbit of Fig. 2d. In fact, we integrate the equations of motion of the system, with initial conditions , which are slightly different from the ones of the previously studied outer long axis tube.
Table 4. Frequency analysis of the inner long axis tube orbit .
It is straightforward to check by the quasi-periodic approximations of the functions and that there exists a 1:1 resonance between the middle and short axis variables, as for the outer long axis tubes. The difference though is apparent in the quasi-periodic approximation of the function (first part of Table 4). First, for a small change of the initial conditions comparing to the ones of the outer long axis tube (see Sect. 4.2.2), the leading frequency has changed dramatically from -2.5947176 to -2.1157969. Furthermore, we observe the appearance of many uncoupled harmonics, as for the corresponding approximation of the box orbits (see Table 1), which testifies that the system is circulating in the long axis variables. In consequence, it is reasonable to consider that the inner long axis tubes can be generated by a perturbation of the long axial integrable system, as in the case of box orbits but with the fundamental difference that this perturbation is performed by a slightly perturbed axisymmetric system. We may thus write:
is indeed a perturbation, if is close to 1 and is sufficiently small comparing to y. This perturbative approach can explain why we cannot have inner small axis tubes of that type: the stability diagrams of the three rectilinear orbits (see Fig. 1) show clearly that for all the values of the axial ratios which are relatively close to 1, the corresponding short axial periodic orbit is always hyperbolic.
© European Southern Observatory (ESO) 1998
Online publication: December 8, 1997