Relativistic chaotic scattering: Unveiling scaling laws for trapped trajectories.

In this paper we study different types of phase space structures which appear in the context of relativistic chaotic scattering. By using the relativistic version of the Hénon-Heiles Hamiltonian, we numerically study the topology of different kind of exit basins and compare it with the case of low velocities in which the Newtonian version of the system is valid. Specifically, we numerically study the escapes in the phase space, in the energy plane, and in the ß plane, which richly characterize the dynamics of the system. In all cases, fractal structures are present, and the escaping dynamics is characterized. In every case a scaling law is numerically obtained in which the percentage of the trapped trajectories as a function of the relativistic parameter ß and the energy is obtained. Our work could be useful in the context of charged particles which eventually can be trapped in the magnetosphere, where the analysis of these structures can be relevant.

Influence of the gravitational radius on asymptotic behavior of the relativistic Sitnikov problem.

The Sitnikov problem is a classical problem broadly studied in physics which can represent an illustrative example of chaotic scattering. The relativistic version of this problem can be attacked by using the post-Newtonian formalism. Previous work focused on the role of the gravitational radius λ on the phase space portrait. Here we add two relevant issues on the influence of the gravitational radius in the context of chaotic scattering phenomena. First, we uncover a metamorphosis of the KAM islands for which the escape regions change insofar as λ increases. Second, there are two inflection points in the unpredictability of the final state of the system when λ≃0.02 and λ≃0.028. We analyze these inflection points in a quantitative manner by using the basin entropy. This work can be useful for a better understanding of the Sitnikov problem in the context of relativistic chaotic scattering. In addition, the described techniques can be applied to similar real systems, such as binary stellar systems, among others.

Uncertainty dimension and basin entropy in relativistic chaotic scattering.

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 ß, 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 ß<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 ß≈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.

Global relativistic effects in chaotic scattering.

The phenomenon of chaotic scattering is very relevant in different fields of science and engineering. It has been mainly studied in the context of Newtonian mechanics, where the velocities of the particles are low in comparison with the speed of light. Here, we analyze global properties such as the escape time distribution and the decay law of the Hénon-Heiles system in the context of special relativity. Our results show that the average escape time decreases with increasing values of the relativistic factor ß. As a matter of fact, we have found a crossover point for which the KAM islands in the phase space are destroyed when ß≃0.4. On the other hand, the study of the survival probability of particles in the scattering region shows an algebraic decay for values of ß≤0.4, and this law becomes exponential for ß>0.4. Surprisingly, a scaling law between the exponent of the decay law and the ß factor is uncovered where a quadratic fitting between them is found. The results of our numerical simulations agree faithfully with our qualitative arguments. We expect this work to be useful for a better understanding of both chaotic and relativistic systems.

Weakly noisy chaotic scattering.

The effect of a weak source of noise on the chaotic scattering is relevant to situations of physical interest. We investigate how a weak source of additive uncorrelated Gaussian noise affects both the dynamics and the topology of a paradigmatic chaotic scattering problem as the one taking place in the open nonhyperbolic regime of the Hénon-Heiles Hamiltonian system. We have found long transients for the time escape distributions for critical values of the noise intensity for which the particles escape slower as compared with the noiseless case. An analysis of the survival probability of the scattering function versus the Gaussian noise intensity shows a smooth curve with one local maximum and with one local minimum which are related to those long transients and with the basin structure in phase space. On the other hand, the computation of the exit basins in phase space shows a quadratic curve for which the basin boundaries lose their fractal-like structure as noise turned on.