Rings, spirals and manifolds
  Two-armed, grand design spirals and inner and outer rings in barred galaxies can be due to orbits guided by the manifolds emanating from the vicinity of the L1 and L2 Lagrangian points, located at the ends of the bar. We first summarise the necessary theoretical background and in particular we describe the dynamics around the unstable equilibrium points in barred galaxy models, and the corresponding homoclinic and heteroclinic orbits. We then discuss two specific morphologies and the circulation of material within the corresponding manifolds. We also discuss the case where mass concentrations at the end of the bar can stabilise the L1 and L2 and the relevance of this work to the gas concentrations in spirals and rings. 
E. Athanassoula, M. Romero-Gomez, J.J. Masdemont