Abstract
The escape of a particle from a dynamical system depends on the intersection between the ingoing and outgoing asymptotic trajectories to certain periodic orbits placed at the openings of the curves of zero velocity of the system. Although many efforts have been devoted to the analysis of the escape from potentials presenting multiple openings, there are still few studies on potentials with only one opening. In this article, we clarify the way in which the energy affects the escape in this type of systems, showing that, contrary to what one could expect, there are several bifurcations for certain values of the energy.
Highlights
The escape of a particle from a dynamical system depends on the intersection between the ingoing and outgoing asymptotic trajectories to certain periodic orbits placed at the openings of the curves of zero velocity of the system
N avarro[13] investigates the geometry of the curves that delimite the escape domains by determining the intersection of the ingoing and outgoing asymptotic trajectories to the Lyapunov orbit with an apropiate surface of section, for a fixed value of the energy. We describe how these limiting curves evolve as the energy of the system varies, showing that the intersection between the ingoing and outgoing asymptotic trajectories to the Lyapunov orbit takes place in a way that relies on the energy of the system
For values of the energy in the interval Iν = (Hν,[1], Hν,2), the tip of each tongue belonging to Ws,4(φ) is contained in the area enclosed by Wu,1(φ) and, the area delimited by these two sets (the tip of each tongue andWu,1(φ) ) corresponds to initial conditions of orbits coming from the infinity and escaping from the galaxy after intersecting the surface of section at four different points
Summary
We analyze the motion of a particle in the (r, z) meridian plane near the central part of an axially symmetric galaxy modeled by a galactic type potential of the form. There exists a value of the energy, known as energy of escape and denoted by Hc , such that if the energy of the test particle exceeds Hc , the curves of zero velocity open at one place and particles may escape from the system. For those values of H, there is an unstable periodic orbit at the exit of the potential. As the value of the energy of the system grows, the size of the opening becomes bigger
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