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

Abstract

We investigate the evolution of families of periodic orbits in a bisymmetrical potential made up of a twodimensional 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.

New families of periodic orbits for a galactic potential

“…This potential has been studied by many authors, see for instance Deprit and Elipe [4], Caranicolas [3], Elipe and Deprit [5], Elipe [6], Arribas et al [2], Zotos [10,11,12,13], Zotos and Caranicolas [14], Zotos and Carpintero [15], ...…”

Determining the type of orbits in the central regions of barred galaxies

“…3(b)). Furthermore, the diagram shown in Figure 3(c) is another type of "characteristic" diagram (Sellwood & Wilkinson 1993;Binney & Tremaine 2008;Zotos 2013), where the value of the Jacobi integral E J is plotted against the coordinate where the minor axis of the bar crosses the y-axis. As can be seen in Figures 1 and 2, all the higher resonant orbits encountered in our potential (i.e., the 2:3, 3:4, 3:5, 4:7 and 5:9 resonant orbits) have complicated shapes thus crossing the y-axis multiple times and at several positions.…”

Revealing the Network of Periodic Orbits in Galaxy Models with a Prolate or an Oblate Dark Matter Halo Component

“…A periodic orbit is stable if only the stability index (S.I.) (Zotos 2013) is between -2 and +2. This diagram helps us monitor the evolution of S.I.…”