Panel a: apocentric invariant manifolds in the pure bar model with one pattern speed. The manifolds emanating from the points L₁ and L₂ are plotted in blue and red, respectively. Panel b: apocentric manifolds if we add the spiral potential (intermediate case), assuming that the spirals rotate with the same pattern speed as the bar. The addition of the spiral term enhances the structures described as “lobes”, “bridges”, and “gaps” (see text). Panels c and d: same as in (a) and (b), but with the manifolds computed over a larger length. The black spiral curves correspond to the minima of the imposed spiral potential, given by Eq. (19).