  In this work, we try to shed some light to the nature of orbits in a
three-dimensional potential of a perturbed harmonic oscillator with eight
possible channels of escape, which was chosen as an interesting example of open
three-dimensional Hamiltonian systems. In particular, we conduct a thorough
numerical investigation distinguishing between regular and chaotic orbits as
well as between trapped and escaping orbits, considering unbounded motion for
several values of the energy. In an attempt to discriminate safely and with
certainty between ordered and chaotic motion, we use the Smaller ALingment
Index (SALI) detector, computed by integrating numerically the basic equations
of motion as well as the variational equations. Of particular interest, is to
locate the basins of escape towards the different escape channels and connect
them with the corresponding escape periods of the orbits. We split our study
into three different cases depending on the initial value of the $z$ coordinate
which was used for launching the test particles. We found, that when the orbits
are started very close to the primary $(x,y)$ plane the respective grids
exhibit a high degree of fractalization, while on the other hand for orbits
with relatively high values of $z_0$ several well-formed basins of escape
emerge thus, reducing significantly the fractalization of the grids. It was
also observed, that for values of energy very close to the escape energy the
escape times of orbits are large, while for energy levels much higher than the
escape energy the vast majority of orbits escape extremely fast or even
immediately to infinity. We hope our outcomes to be useful for a further
understanding of the escape process in open 3D Hamiltonian systems.
