9 Resonances and stochasticity

A worthful exercise is to determine what happens to the periodic orbits which are initially in (outer) 2/1 resonance before the growth of the bar. This is illustrated in Fig. 15, which highlights for two different space positions at $t=120t_{\rm b}$ the points in the u-v plane corresponding to trajectories which were on such orbits at t=0. The diagrams are built by integrating backwards the trajectories passing through a Cartesian u-v grid and marking all the points on this grid originating from initial orbits with $(\omega_{\phi}+2\omega_{\rm R}-\Omega_{\rm P})/\Omega_{\rm P}<10^{-2}$. The darkness of the points reflects the angle $\alpha$ between the major axis of the initial resonant orbit and the major axis of the then vanishing bar potential, with darker points standing for smaller angles, i.e. resonant orbits with apocentre closer to the bar major axis.

Figure 15: For two different space positions and after 120 bar rotations, traces in the u-v plane of the trajectories which were in OLR with the unperturbed rotating potential at t=0 (bar strength F=0.10). The points are colour-coded according to the angle $\alpha$ between the line joining the apocentres of the initial resonant orbit and the major axis of the vanishing bar potential: black if $\vert\alpha \vert<45^{\circ }$, light grey if $\vert\alpha \vert>70^{\circ }$, and dark grey for intermediate angles.

At $\varphi=90^{\circ}$ and $R/R_{\hbox{\tiny OLR}}=0.9$, there is a wide region of regular orbits around most of the 2/1 resonance curve (see Fig. 7). The trajectories with small $\alpha$'s clearly fall in the inner part of this region, while those with larger $\alpha$'s appear rather on its edge and are spread within the neighbouring chaotic regions. In addition to the regular versus chaotic phase space decoupling, the fact that the OLR curve is associated with a valley in the planar velocity distribution (see Fig. 12) is also because the phase space density in the chaotic regions bounding the regular orbit arc around this curve is forced to be a function of H only, hence lowering the density on the low-v side and increasing it on the other side relative to the initial densities. At $\varphi =25^{\circ }$ and $R/R_{\hbox{\tiny OLR}}=1.1$, the part of the 2/1resonance curve within the u>0 chaotic region has no nearby small-$\alpha$points and is embedded in a broad cloud of high-$\alpha$ points.

Hence, the initial 2/1 resonant orbits more nearly aligned with the bar major axis end into regular regions, while only those more nearly perpendicular to this axis are scattered into chaotic regions. This is consistent with the stability properties of the low-eccentricity x₁(1) and x₁^*(2) orbits in the full barred potential, which are both 2/1 periodic orbits.