Dynamics of a rR₁ ringed galaxy. Top: in blue, arrows indicating circulation of matter. In green, intersection of the invariant manifolds with hyperplane S. All of them in the (x, y) plane (image adapted from Athanassoula et al. 2009). Centre: neighbourhood of equilibrium point L₁ in (x, y) plane. Arrows: motion along invariant manifolds (S = stable, U = unstable). Stripped areas: forbidden regions delimited by zero velocity curves (image taken from Athanassoula et al. 2009). Bottom: precessing model with masses GM_b = 0.1, GM_d = 0.9, pattern speed Ω = 0.05, and tilt angle ε = 0. We display the (y, ẏ) projections of the first crossings of the unstable and stable manifolds with the hyperplane S. In cyan: , in green: . See more details in the text.