  This paper discusses phase space structures encountered in the neighborhood
of periodic orbits with high order multiplicity in a 3D autonomous Hamiltonian
system with a potential of galactic type. We consider 4D spaces of section and
we use the method of color and rotation [Patsis and Zachilas 1994] in order to
visualize them. As examples we use the case of two orbits, one 2-periodic and
one 7-periodic. We investigate the structure of multiple tori around them in
the 4D surface of section and in addition we study the orbital behavior in the
neighborhood of the corresponding simple unstable periodic orbits. By
considering initially a few consequents in the neighborhood of the orbits in
both cases we find a structure in the space of section, which is in direct
correspondence with what is observed in a resonance zone of a 2D autonomous
Hamiltonian system. However, in our 3D case we have instead of stability
islands rotational tori, while the chaotic zone connecting the points of the
unstable periodic orbit is replaced by filaments extending in 4D following a
smooth color variation. For more intersections, the consequents of the orbit
which started in the neighborhood of the unstable periodic orbit, diffuse in
phase space and form a cloud that occupies a large volume surrounding the
region containing the rotational tori. In this cloud the colors of the points
are mixed. The same structures have been observed in the neighborhood of all
m-periodic orbits we have examined in the system. This indicates a generic
behavior.
