  The theory of the inverse problem is used in order to find a two dimensional
galactic potential generating a mono-parametric family of elliptic periodic
orbits. The potential is made up of a two-dimensional harmonic oscillator with
perturbing terms of third and fourth degree and can be considered to describe
local motion in the central parts of a barred galaxy. Our numerical
calculations indicate that the potential produces also other families of orbits
and there are cases, where distinct chaotic components are also observed. The
numerical experiments suggest that there are regular as well as chaotic orbits
supporting the barred structure. We use a variety of dynamical indicators, in
order to determine the character of motion. Of significant interest, is the
S(g) dynamical spectrum, which is used in order to identify the islandic motion
of resonant orbits and also proves to be a very useful and fast indicator, in
order to distinguish between regular and chaotic motion. Comparison with other
methods for detecting chaos is also discussed. The present results are compared
with outcomes from earlier work.
