  We study the role of asymptotic curves in supporting the spiral structure of
a N-body model simulating a barred spiral galaxy. Chaotic orbits with initial
conditions on the unstable asymptotic curves of the main unstable periodic
orbits follow the shape of the periodic orbits for an initial interval of time
and then they are diffused outwards supporting the spiral structure of the
galaxy. Chaotic orbits having small deviations from the unstable periodic
orbits, stay close and along the corresponding unstable asymptotic manifolds,
supporting the spiral structure for more than 10 rotations of the bar. Chaotic
orbits of different Jacobi constants support different parts of the spiral
structure. We also study the diffusion rate of chaotic orbits outwards and find
that chaotic orbits that support the outer parts of the galaxy are diffused
outwards more slowly than the orbits supporting the inner parts of the spiral
structure.
