Analysis of the escape in systems with four exit channels

Abstract

In this paper, we have performed a numerical investigation of the escape of a particle from two different dynamical systems with the same number of exit channels. We have chosen specific values of the parameters of the systems so that the openings of the potential well in both systems are approximately of the same size. We have found that, in the galactic system, the distribution of the times of escape follows a sequential pattern that has never been detected before. Moreover, we have proved that this pattern is directly related to the geometry of the stable manifolds to the Lyapunov orbits located at the openings of the potential.Finally, we have shown that the different nature of the two systems affects the way the escape occurs, due to the difference in the geometry of the manifolds to the Lyapunov orbits in both systems.

Effect of the mass ratio on the escape in the 4-body ring problem

“…If a particle has not left the system for an integration time of 100 units of time, we will consider that it remains trapped in it. In this way, we analyze the dependence of the distribution of short times of escape on β and C and, also, we show that the sequential pattern in the distribution of short times of escape that Navarro et al [22] found in the galactic problem can also be found in the 4-body problem when the value of the Jacobi constant is close to its critical value and, then, the architecture of the projections of the stable manifolds to the Lyapunov orbits located at the openings of the potential well on the surface of section arises clearly.…”

“…for ν 1, 2, 3, 4, and being x * ν and y * ν the coordinates of the peripheral primaries. These coordinates can be easily expressed as a function of the angle between the central body and two successive peripheral primaries [21,22]. The Jacobi integral of motion C of this system is given by…”

“…The analysis of the geometric structures that govern the escape of a particle from an open Hamiltonian system has focused the efforts of many research groups over the last decades [1-8, 10, 11, 13-22, 24-26]. One of the systems that has aroused much of that interest is a particular case of the N-body problem: the N-body ring problem [18][19][20][21][22]. This attention is due to the fact that this system models a wide range of celestial systems, like planetary rings, asteroid belts and some type of formations of stars and planets, to cite some examples.…”

“…Also, Navarro and Martínez-Belda [20] carry out a numerical exploration of the N-body ring problem in order to analyze the distribution of times of escapes, for N 5, 6, 7, 8 primaries, and considering a fixed value of the mass ratio β 2. Moreover, they fulfill a detailed calculation of the basins of escapes for these four values of N. Navarro et al [22] try to find out common properties in the escape from open Hamiltonian systems with the same number of openings in the potential well. In their work, they compare a galactic potential with the 4-body ring problem, pointing to a characteristic way in which the fast escape occurs in this type of systems.…”

Dependence of the probability of escape on the Jacobi constant in the N-body ring problem without central body

“…The problem of escaping particles from open Hamiltonian systems is one of the most analyzed topics in nonlinear dynamics [12][13][14][15][16][17][18][19][20][21][22][23][24][25]. In this kind of systems, there exists a finite energy of escape, E e , such that if the energy of the particle is smaller than E e , the equipotential surfaces are closed and the escape from the system is impossible.…”