Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies

In the presence of a strong $m=2$ component in a rotating galaxy, the phase space structure near corotation is shaped to a large extent by the {\it invariant manifolds} of the short period family of unstable periodic orbits terminating at L$_1$ or L$_2$. The main effect of these manifolds is to create robust {\it phase correlations} among a number of chaotic orbits large enough to support a {\it spiral} density wave outside corotation. The phenomenon is described theoretically by soliton-like solutions of a Sine-Gordon equation. Numerical examples are given in an N-Body simulation of a barred spiral galaxy. In these examples, we demonstrate how the projection of unstable manifolds in configuration space reproduces essentially the entire observed bar-spiral pattern.