  The most important theory of the spiral arms of galaxies is the density wave
theory based on the Lin-Shu dispersion relation. However, the density waves
move with the group velocity towards the inner Lindblad resonance and tend to
disappear. Various mechanisms to replenish the spiral waves have been proposed.
Nonlinear effects play an important role near the inner and outer Lindblad
resonances and corotation. The orbits supporting the spiral arms are precessing
ellipses in normal galaxies that extend up to the 4/1 resonance. On the other
hand, in barred galaxies the spiral arms extend along the manifolds of the
unstable periodic orbits at the ends of the bar and they are composed of
chaotic orbits. However these chaotic orbits can be found analytically.
