Relativistic Chaotic Scattering

Abstract

The phenomenon of chaotic scattering is very relevant in different fields in science and engineering, and it has been mainly studied in the context of Newtonian mechanics. In this chapter, we study the chaotic scattering considering the special relativity, not just for high velocities but also small ones. We indeed research on global properties of any chaotic scattering system as the escape time distribution and the decay law. Moreover, we study some relevant characteristics of the system exit basin topology as the uncertainty dimension, the Wada property and the basin entropy. As a propotypical chaotic scattering model, we use the relativistic Hénon-Heiles Hamiltonian. Our results show that the average escape time decreases with increasing values of the relativistic factor β. We have found a cross-over point for which the KAM islands in the phase space are destroyed when β ≃ 0.4. The study of the survival probability of the particles in the scattering region shows an algebraic decay for values of β ≤ 0.4, and this law becomes exponential for β > 0.4. A scaling law between the exponent of the decay law and the β factor is uncovered. With regards to the exit basin topology, our main findings for the uncertainty dimension show two different behaviors insofar we change the relativistic parameter β. These are related with the disappearance of KAM islands in phase space. Moreover, the computation of the exit basins in the phase space suggests the existence of Wada basins for β > 0.625. We have also studied the evolution of the exit basins by computing the basin entropy. It shows a maximum value for β ≈ 0.2. Our work might be relevant to galactic dynamics and it also has important applications as, for example, in the Störmer problem.