  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.
