Escapes in Hamiltonian systems with multiple exit channels: part II

Abstract

We explore the escape dynamics in open Hamiltonian systems with multiple channels of escape continuing the work initiated in Part I. A thorough numerical investigation is conducted distinguishing between trapped (ordered and chaotic) and escaping orbits. The determination of the location of the basins of escape towards the different escape channels and their correlations with the corresponding escape periods of the orbits is undoubtedly an issue of paramount importance. We consider four different cases depending on the perturbation function which controls the number of escape channels on the configuration space. In every case, we computed extensive samples of orbits in both the configuration and the phase space by numerically integrating the equations of motion as well as the variational equations. It was found that in all examined cases regions of non-escaping motion coexist with several basins of escape. The larger escape periods have been measured for orbits with initial conditions in the vicinity of the fractal structure, while the lowest escape rates belong to orbits with initial conditions inside the basins of escape. In addition, we related the model potential with applications in the field of reactive multichannel scattering. We hope that our numerical analysis will be useful for a further understanding of the escape mechanism of orbits in open Hamiltonian systems with two degrees of freedom.

An overview of the escape dynamics in the Hénon–Heiles Hamiltonian system

“…For energy levels beyond the escape energy however, these surfaces open creating exit channels through which the particles can escape to infinity. The literature is replete with studies of such "open" Hamiltonian systems (e.g., [8,25,35,38,45,46,[53][54][55][59][60][61][62]).…”

“…Therefore, it is quite possible to obtain basins of escape, similar to basins of attraction in dissipative systems or even the Newton-Raphson fractal structures. Basins of escape have been studied in several papers (e.g., [1,4,7,8,[59][60][61][62]). The reader can find more details regarding basins of escape in [23].…”

Investigating the planar circular restricted three-body problem with strong gravitational field

“…For energy levels beyond the escape energy however, these surfaces open and exit channels emerge through which the particles can escape to infinity. The literature is replete with studies of such "open" or "leaking" Hamiltonian systems (e.g., [5,16,24,26,31,36,37,45,46,47,51,52,56]).…”

Periodic Orbit Families in the Gravitational Field of Irregular-Shaped Bodies

“…Fukushima (2010) numerically calculated the gravitational acceleration caused by a massive circular ring or disk, which can be used to calculate the equilibrium points and periodic orbits around them. Zotos (2014Zotos ( , 2015a investigated the gravitational potential generated by a 2-D perturbed harmonic oscillator, and discussed the escape orbits and chaotic motion of bodies in the potential.…”