In the manifold theory of spiral structure in barred galaxies, the usual assumption is that the spirals rotate with the same pattern speed as the bar. Here, we generalize the manifold theory under the assumption that the spirals rotate with a different pattern speed than the bar. More generally, we consider the case in which one or more modes, represented by the potentials V2, V3, etc., coexist in the galactic disk in addition to the bar’s mode Vbar, but the modes rotate with pattern speeds, Ω2, Ω3, etc., which are incommensurable between themselves and with Ωbar. Through a perturbative treatment (assuming that V2, V3, etc. are small with respect to Vbar), we then show that the unstable Lagrangian points L1 and L2 of the pure bar model (Vbar, Ωbar) are continued in the full model as periodic orbits, in the case of one extra pattern speed, or as epicyclic “Lissajous-like” unstable orbits, in the case of more than one extra pattern speeds. We use GL1 and GL2 to denote the continued orbits around the points L1 and L2. Furthermore, we show that the orbits GL1 and GL2 are simply unstable. As a result, these orbits admit invariant manifolds, which can be regarded as the generalization of the manifolds of the L1 and L2 points in the single pattern speed case. As an example, we computed the generalized orbits GL1, GL2, and their manifolds in a Milky-Way-like model in which bar and spiral pattern speeds were assumed to be different. We find that the manifolds produce a time-varying morphology consisting of segments of spirals or “pseudorings”. These structures are repeated after a period equal to half the relative period of the imposed spirals with respect to the bar. Along one period, the manifold-induced time-varying structures are found to continuously support at least some part of the imposed spirals, except at short intervals around specific times at which the relative phase of the imposed spirals with respect to the bar is equal to ±π/2. The connection of these effects to the phenomenon of recurrent spirals is discussed.
The coalescence of invariant manifolds and the spiral structure of barred galaxies