Uncertainty dimension and basin entropy in relativistic chaotic scattering

Abstract

Chaotic scattering is an important topic in nonlinear dynamics and chaos with applications in several fields in physics and engineering. The study of this phenomenon in relativistic systems has received little attention as compared to the Newtonian case. Here we focus our work on the study of some relevant characteristics of the exit basin topology in the relativistic Hénon-Heiles system: the uncertainty dimension, the Wada property, and the basin entropy. Our main findings for the uncertainty dimension show two different behaviors insofar as we change the relativistic parameter β, in which a crossover behavior is uncovered. This crossover point is related with the disappearance of KAM islands in phase space, which happens for velocity values above the ultrarelativistic limit, v>0.1c. This result is supported by numerical simulations and by qualitative analysis, which are in good agreement. On the other hand, the computation of the exit basins in the phase space suggests the existence of Wada basins for a range of β<0.625. We also studied the evolution of the exit basins in a quantitative manner by computing the basin entropy, which shows a maximum value for β≈0.2. This last quantity is related to the uncertainty in the prediction of the final fate of the system. Finally, our work is relevant in galactic dynamics, and it also has important implications in other topics in physics such as as in the Störmer problem, among others.

Chaotic exits from a weakly magnetized Schwarzschild black hole

“…Exit basins (or basins of attraction in the case of dissipative systems) provide a useful way to analyze chaotic conservative dynamical systems. This type of analysis has been applied to various dynamical systems such as the paradigmatic Hénon-Heiles system [16][17][18][19], as well as systems describing black holes [6,8,[20][21][22]]. An exit basin B i , associated with a particular exit E i , is the closure of the set of all initial conditions that escape from the bounded region through exit E i .…”

Transient chaos under coordinate transformations in relativistic systems

“…Some recent works aim at isolating the effects of the variation of the Lorentz factor γ (or, equivalently, β) from the remaining system variables in the context of relativistic chaotic scattering [17,18]. In order to accomplish this, they modify the initial value of β and use it as the only parameter of the dynamical system.…”

“…The Lorentz factor effects on the dynamical properties of the system have already been addressed in relativistic chaotic scattering [17,18]. These works focus on special relativity and study the dynamical regime, the decay law, the basin topology and the fractal dimension of scattering functions.…”

A test for fractal boundaries based on the basin entropy

“…Starting with a similar tiling of phase space, the idea is to compute the probabilities of going to each attractor within a box and exploit Shannon's information entropy. Since their formulation, the basin entropy and the boundary basin entropy have been applied to experiments with cold atoms [8], chaotic scattering [9,10], biological systems [11,12], electronic micro/nanodevices [13], oscillators [14] and astrophysical models [15,16,17], among others.…”