Trapped and escaping orbits in an axially symmetric galactic-type
  potential
  In the present article, we investigate the behavior of orbits in a time
independent axially symmetric galactic type potential. This dynamical model can
be considered to describe the motion in the central parts of a galaxy, for
values of energies larger than the energy of escape. We use the classical
method of the surface of section, in order to visualize and interpret the
structure of the phase space of the dynamical system. Moreover, the Lyapunov
Characteristic Exponent (LCE), is used in order to make an estimation of the
degree of the chaoticity of the orbits in our galactic model. Our numerical
calculations suggest that in this galactic type potential, there are two kinds
of orbits: (i) escaping orbits and (ii) trapped orbits which do not escape at
all. Furthermore, a large number of orbits of the dynamical system, display
chaotic motion. Among the chaotic orbits, there are orbits that escape fast and
also orbits that remain trapped for vast time intervals. When the value of the
test particle's energy exceeds slightly the energy of escape, the amount of the
trapped regular orbits increases, as the the value of the angular momentum
increases. Therefore, the extent of the chaotic regions observed in the phase
plane decreases as the value of the energy increases. Moreover, we calculate
the average value of the escape period of the chaotic orbits and we try to
correlate it with the value of the energy and also with the maximum value of
the z component of the orbits. In addition, we find that the value of the LCE
corresponding to each chaotic region, for different values of the energy,
increases exponentially as the value of the energy increases. Some theoretical
arguments in order to support the numerically obtained outcomes are presented.
