6 Liapunov exponents

The next step after the periodic orbit search is to determine the phase space extent of the regular orbits trapped around the stable closed orbits and of the chaotic orbits, which have no other integral than the Jacobi integral. The Poincaré surface of section method is well suited to highlight the regular and chaotic regions in phase space at constant value of the Hamiltonian, but not at constant position in real space. A better tool for this purpose are the Liapunov exponents, which also allow to quantify the degree of stochasticity of the orbits. These exponents describe the mean exponential rate of divergence of two trajectories initially close to each other in phase space and are defined as:

$$\lambda(\vec{x}_{\circ},\Delta \vec{x}_{\circ})= \lim_{\scrip... ...ert\Delta \vec{x}(t)\vert}{\vert\Delta \vec{x}_{\circ}\vert}},$$ (9)

where $\vec{x}_{\circ}$ and $\Delta \vec{x}_{\circ}$ are the initial (t=0) position and deviation, and $\Delta \vec{x}(t)$ is the deviation at time t. Such limit is proven to exist and is finite for all bound orbits (Oseledec 1968). The value of $\lambda$ generally depends on the direction of the deviation $\Delta \vec{x}_{\circ}$ and, if Nis the dimension of phase space, one can show (Oseledec 1968; Benettin et al. 1976) that there exists in fact N discrete exponents $\lambda_1\geq \lambda_2\geq \ldots \geq \lambda_N$. However, the largest exponent $\lambda_1$ is the most important because it results from almost all deviations in $\Delta \vec{x}_{\circ}$-space (e.g. Udry & Pfenniger 1988), and therefore is the one found in practice when the initial deviation is chosen randomly. Moreover, this exponent exclusively determines the orbital stability: if $\lambda_1=0$, all exponents vanish and the orbit is regular, and if $\lambda_1>0$, the orbit is chaotic and the amount of chaos increases with $\lambda_1$.

The numerical computation of the Liapunov exponents faces some problems related to the limits in Eq. (9). First, the finite initial deviation $\Delta \vec{x}_{\circ}$ may rapidly grow as large as the size of the orbits themselves, especially for chaotic orbits, and thus $\Delta \vec{x}(t)$ must be occasionally scaled down by a large factor. Noting $\Delta \vec{x}^0(t)$the deviation before the first rescaling, this is done in a way similar to Contopoulos & Barbanis (1989), by normalising $\Delta \vec{x}(t)$ to $\vert\Delta \vec{x}_{\circ}\vert$:

$$\Delta \vec{x}^i(t_i)=\frac{\vert\Delta \vec{x}_{\circ}\vert} {\vert\Delta \vec{x}^{i-1}(t_i)\vert}\Delta \vec{x}^{i-1}(t_i)$$ (10)

every time t_i when $\vert\Delta \vec{x}^{i-1}\vert>1.3\times 10^{-3}R_{\hbox{\tiny OLR}}$, so that after n iterations:

$$\ln{\frac{\vert\Delta \vec{x}(t)\vert}{\vert\Delta \vec{x}_{\... ...ec{x}^{i-1}(t_i)\vert}{\vert\Delta \vec{x}_{\circ}\vert}}\cdot$$ (11)

Another problem is that one cannot integrate orbits over an infinite time and thus the time limit in Eq. (9) must be replaced by the value at some finite time, which has been set to about 3 Hubble times when the models are scaled to realistic physical units.

The Liapunov exponent $\lambda_1$ has been calculated on a Cartesian grid of planar velocities for different positions of the observer, using the 2D potential of Eq. (5) and with an initial deviation of $+1.3\times 10^{-11} R_{\hbox{\tiny OLR}}$ in the R-coordinate (Figs. 7, 8 and 10). The results are presented as a divergence timescale $t_{\lambda}\equiv 1/\lambda_1$ in units of local circular period in the axisymmetric part of the potential, which provides a more obvious and useful measure of stochasticity. Diagrams have been computed for every $10^{\circ}$ in azimuth for $R/R_{\hbox{\tiny OLR}}=0.9$, 1.0, 1.05, 1.1, 1.2 and also at $\varphi =25^{\circ }$ over a larger radial range, but those diagrams between our $30^{\circ}$ sampling and at $R/R_{\hbox{\tiny OLR}}=1.05$ will not be shown here (and the same is true for the velocity distributions of the next section). We shall refer to such diagrams as "Liapunov'' diagrams.

The first thing to notice in these diagrams is that the stable and unstable periodic orbits fall within regular and chaotic regions respectively, as expected. There are some apparent exceptions like the 1/1 asym orbit at $\varphi=0$ and large R which must owe to the limited velocity resolution of the diagrams (=0.04$v_{\circ }$; see Fux 2001b for some higher resolution Liapunov diagrams). The fraction of chaotic orbits also obviously increases with bar strength. Furthermore, as a consequence of the four-fold symmetric barred potential, the diagrams at $\varphi=0$ and $90^{\circ}$ are symmetric with respect to u=0 and, more generally, diagrams at supplementary angles are anti-symmetric to each other in u, i.e. $t_{\lambda}(R,180^{\circ}-\varphi,v,u)=t_{\lambda}(R,\varphi,v,-u)$.

At $\varphi=90^{\circ}$ and for the radial range explored in Figs. 7 and 8, the 2/1 and 1/1 resonance curves, as well as all other not highlighted resonances of the form $2/\vert n_{\phi}\vert$ with integer $\vert n_{\phi}\vert$, are embedded in the middle of broad regular orbit arcs separated by mainly chaotic regions which come closer to u=0 as the bar strength increases. For (and $\varphi=90^{\circ}$), the regularity of the OLR arc gets destroyed near u=0, leaving the place to an unstable x₁^*(2) orbit. Between the dominant regular orbit arcs, secondary regular arcs associated with resonances of the form $4/\vert n_{\phi}\vert$ with odd integer $\vert n_{\phi}\vert$ can also be identified, especially for F=0.10. This includes in particular the 4/1 resonance arc visible for . In fact, the chaotic regions between the broad resonance arcs are probably densely filled by tiny arcs of higher order regular resonant orbits, but the filling factor must be very low.

At $\varphi=0$, the situation is reversed: the $2/\vert n_{\phi}\vert$ resonance curves lie in chaotic regions at large H which are spaced by regular orbit arcs right between the resonances. At intermediate angles, the regular and chaotic regions become offset from the resonance curves and the u-symmetry breaks. In particular, for $\varphi \sim 30^{\circ}$ and , i.e. realistic positions for the Sun, an extreme case of asymmetry arises near the OLR: a prominent region of regular orbits extends down to negative u, bounded roughly by the OLR curve on the right and penetrating well inside the hot orbit zone, whereas the positive upart of the OLR curve is surrounded by a wide chaotic region extending somewhat inside the H₁₂ contour and coinciding very well with the u-vlocation of the Hercules stream.

Figure 9: Location in the u-v plane of the regular orbits trapped around the stable x1(1) periodic orbits for a bar strength F=0.20 (dark points) and around the stable x1(2) orbits for F=0.10 (grey points). The computational details are described in the text. The horizontal and vertical axes of the frames and the different curves are as in Fig. 7. The H-contours are given for F=0.10.

Figure 9 shows the u-v region occupied by the regular orbits trapped around the stable x₁(1) and x₁(2) periodic orbits. To construct this figure, we have first derived many surfaces of section at mainly constant Hamiltonian interval $\Delta H=0.054v_{\circ}^2$ to locate the islands of invariant curves around these periodic orbits. Then the orbits within each islands of these maps have been sampled by 50 regularly spaced points along a straight line across the central periodic orbit and within the outermost invariant curve of the island. Finally, all the resulting initial conditions are integrated for 20 rotations in the rotating frame and the velocities are plotted when the orbits pass within a distance of $d_{\rm max}=R_{\circ}/80$of the considered space positions. The striped appearance of the regular x₁(1) and x₁(2) regions in the diagrams are due to the discrete H-sampling and the broadening of the stripes to the finite value of $d_{\circ}$ (not to an inaccurate orbital integration). The islands in the surfaces of section generally contain sub-resonances which have been included. The x₁(1) islands however have been truncated at the 1/1 resonance, so that the part of the x₁(1) region on the high-v side of the 1/1resonance curve is not represented in the u-v diagrams. The x₁(1) regions are derived for F=0.20 because the high level of chaos at this bar strength makes it easier to distinguish the boundaries of these regions in the surfaces of section, and the x₁(2) regions for F=0.10 in order to emphasise the more regular case where these regions are larger.

From this figure, it is obvious that the regular orbit arcs near the OLR, and in particular the prominent regular region at $\varphi \sim 30^{\circ}$and discussed previously, are produced by the regular orbits around the stable periodic orbits of the x₁(1) family, with an eccentricity increasing towards larger Hamiltonian values. The regular regions associated with the stable x₁(2) orbits can be viewed as divided into two parts, one involving only low-eccentricity orbits and the other one the higher eccentricity orbits falling close to the 4/1 resonance curve. The low-eccentricity orbit part exists for and over an angle range around $\varphi=90^{\circ}$ increasing as R decreases, and is generally enclosed between the H₁₂ contour and the OLR curve. Elsewhere, it is dissolved and only an unstable x₁^*(2) orbit remains (Fig. 7). The higher eccentricity part connects the low-eccentricity part at $R/R_{\hbox{\tiny OLR}}\approx 0.9$ and then progressively detach from it as the 4/1 resonance curve moves away from the H₁₂ contour at larger R. In particular, for and for F=0.10, there are no regular quasi-x₁(2) orbits between the H₁₂ contour and azimuthal velocities less than $v/v_{\circ}=-0.35$, whatever the angle $\varphi$, and no low-eccentricity such orbits at all for . Hence for this bar strength, the Hercules-like mode found in D2000's simulations at $R/R_{\hbox{\tiny OLR}}\approx 1.1$ and $\varphi \sim 25$ cannot be related to such quasi-x₁(2) orbits.

Figure 10: As Fig. 7, but for $\varphi =25^{\circ }$ and $0.6\leq R/R_{\hbox{\tiny OLR}}\leq 1.7$.

Figure 10 provides Liapunov diagrams over a larger radial range at $\varphi =25^{\circ }$ and for F=0.10. These diagrams clearly show that the H₁₂ contour marks the average transition between regular and chaotic motion. Hence disc orbits are essentially regular and hot orbits essentially chaotic. The diagrams in Fig. 10 also nicely illustrate the dependence of the H₁₂ and H₄₅ contours with galactocentric distance discussed in Sect. 3, and show how the velocity spacing between these contours decreases with increasing R.

Martinet & Raboud (1999) have computed a diagram $\Delta(v,u)$representing the relative pericentre deviation between planar orbits integrated in a barred N-body model and the corresponding orbits in the underlying axisymmetric potential, starting from a realistic space position of the Sun. Their diagram correlates well with our Liapunov diagrams in the sense that the larger values of $\Delta$ coincide with shorter Liapunov timescales. In particular, a clear jump of $\Delta$ occurs at $H\approx H_{\rm 12}$, with the high-H side displaying much larger pericentre deviations on the average, and there is also a tail of small $\Delta$-values at negative u extending inside the hot orbit region.

It is worth mentioning that in a two-dimensional axisymmetric potential, all orbits are regular, i.e. have vanishing $\lambda_1$. Hence the chaotic regions discussed here are all produced by the influence of the bar alone. Also, the divergence timescale $t_{\lambda}$ in the chaotic regions may be as low as a few orbital times. This property may have important consequences on the early evolution of the distribution function in barred galaxies, as we shall see in the following section.