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 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.

Published in Nonlinear Dynamics (NODY) journal

References in corpus (5)

• Orbit classification in the meridional plane of a disk galaxy model with a spherical nucleus
• Trapped and escaping orbits in an axially symmetric galactic-type potential
• Investigating the nature of motion in 3D perturbed elliptic oscillators displaying exact periodic orbits
• Application of new dynamical spectra of orbits in Hamiltonian systems
• Are semi-numerical methods an effective tool for locating periodic orbits in 3D potentials?

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• Revealing the influence of dark matter on the nature of motion and the families of orbits in axisymmetric galaxy models
• Determining the nature of orbits in disk galaxies with non spherical nuclei
• Unveiling the Influence of Dark Matter in Axially Symmetric Galaxies
• Escape dynamics in a Hamiltonian system with four exit channels
• Escape dynamics and fractal basin boundaries in Seyfert galaxies
• Orbit classification of low and high angular momentum stars
• Determining the type of orbits in the central regions of barred galaxies
• Revealing the network of periodic orbits in galaxy models with a prolate or an oblate dark matter halo component