  The aim of this work is to review and also explore even further the escape
properties of orbits in a dynamical system of a two-dimensional perturbed
harmonic oscillator, which is a characteristic example of open Hamiltonian
systems. In particular, we conduct a thorough numerical investigation
distinguishing between trapped (ordered and chaotic) and escaping orbits,
considering only unbounded motion for several energy levels. It is of
particular interest, 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 examination into three different cases depending on the
function of the perturbation term which determines the number of escape
channels on the physical space. In every case, we computed extensive samples of
orbits in both the physical and the phase space by integrating numerically the
equations of motion as well as the variational equations. In an attempt to
determine the regular or chaotic nature of trapped motion, we applied the SALI
method as a chaos detector. It was found, that in all studied cases regions of
trapped orbits coexist with several basins of escape. It was also observed,
that for energy levels very close to the escape value the escape times of
orbits are large, while for values of energy much higher than the escape energy
the vast majority of orbits escape very quickly or even immediately to
infinity. The larger escape periods have been measured for orbits with initial
conditions in the boundaries of the escape basins and also in the vicinity of
the fractal structure. Most of the current outcomes have been compared with
previous related work. We hope that our results will be useful for a further
understanding of the escape mechanism of orbits in open Hamiltonian systems
with two degrees of freedom.
