  We investigate the evolution of families of periodic orbits in a
bisymmetrical potential made up of a two-dimensional harmonic oscillator with
only one quartic perturbing term, in a number of resonant cases. Our main
objective is to compute sufficiently and accurately the position and the period
of the periodic orbits. For the derivation of the above quantities (position
and period) we deploy in each resonance case semi-numerical methods. The
comparison of our semi-numerical results with those obtained by the numerical
integration of the equations of motion indicates that, in every case the
relative error is always less than 1% and therefore, the agreement is more than
sufficient. Thus, we claim that semi-numerical methods are very effective tools
for computing periodic orbits. We also study in detail, the case when the
energy of the orbits is larger than the escape energy. In this case, the
periodic orbits in almost all resonance families become unstable and eventually
escape from the system. Our target is to calculate the escape period and the
escape position of the periodic orbits and also monitor their evolution with
respect to the value of the energy.
