Revealing the evolution, the stability, and the escapes of families of resonant periodic orbits in Hamiltonian systems

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 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 to monitor their evolution with respect to the value of the energy.