Phase Space Structure in the vicinity of vertical Lyapunov orbits around L1,2 in a barred galaxy model

The picture shows the phase space structure and the orbital diffusion from the vicinity of the vertical Lyapunov periodic orbits (vLPOs) around the unstable Lagrangian points L1,2 in a 3D barred galaxy model. In particular, we see the (x,y) projection of the (x,y,p_x,p_y) 4D space of section (we consider the Poincare section z=0 with pz>0), for the consequents and
invariant manifolds associated with the diffusion of 3D orbits that form the neighborhood of vLPOs around L1 and L2 for a value of energy E=-0.165. The initial conditions of the two vLPOs are indicated with black squares, while those of the vertical POs around L4,5 with black stars. Furthermore, the central cross corresponds to the initial conditions of the POs of the z-axis
family (coincides with the location of L3), black triangles mark the location of the two sao POs (stable anomalous orbits - initially stable bifurcations of the z-axis family) and the X symbols those of the two uao ones (unstable anomalous orbits - initially unstable bifurcations of the z-axis family). The 190 consequents of the double-crescent orbit (that is formed in the
neighborhood of the vLPOs of L1,2 - see Katsanikas and Patsis 2022) are given with heavy red dots. The 2D projection of the unstable invariant manifolds of the simple unstable vLPOs around L1 and L2 are drawn with green and cyan colors respectively. We see that the projections of the tori around the stable L4,5 are drawn with blue, while those around the sao POs with magenta color. We observe that the points forming the double-crescent are sticky on the unstable invariant manifolds of the simple unstable POs around the Lagrangian points L1,2. The manifolds are guided from the invariant tori that are found around the stable vertical periodic orbits associated with the Lagrangian points L4,5. This landscape offers an illustrative representation of the main structures existing in the phase space we study.

References:
1) M.Katsanikas and P.Patsis, The Phase Space Structure in the vicinity of vertical Lyapunov orbits around L1,2 in a barred galaxy model, MNRAS, 516, 5232-5243 (2022).