Chaotic Motion Of Particles Coupling To Einstein Tensor In The Schwarzschild-Melvin Black Hole Spacetime

Chaos is a kind of non-periodic motion occurred in the non-linear dynamical systems, which is very sensitive to the initial conditions. It presents the intrinsic randomicity of deterministic systems. Since General relativity is a highly nonlinear theory played an important role in the modern physics, it is very important to study the chaotic motion in this theory. In this thesis, we will investigate the chaotic motion of the particle coupling to Einstein tensor in the Schwarzschild-Melvin spacetime by the fast Lyapunov indicator, power spectrum, the Poincare section, bifurcation and the basins of attraction.In the first chapter, we presented a brief introduction of chaos and the theo-retical model of scalar field coupling to the Einstein tensor.In the second chapter, we investigated the chaotic motion of the particle cou-pling to Einstein tensor in the Schwarzschild-Melvin black hole spacetime. Mak-ing use of the methods of the fast Lyapunov indicator, the power spectrum and Poincare section, we confirmed that chaos emerges in the motion of the coupled particle in Schwarzschild-Melvin spacetime. By analyzing the Poincare section, we found that with the increase of magnetic induction B, the chaotic region of this system increases firstly and then decreases. When the coupling parameter is negative, the chaotic region decreases with the strength of the coupling between Einstein tensor and particle. When the coupling parameter is positive, the chaotic region first decreases and then increases with the strength of the coupling. From the bifurcation diagram, we found that with the increase of magnetic induction B and coupling parameter, the motion states of the coupled particle could be trans- formed among periodic, quasi-periodic and chaotic motions. Through plotting the basins of attraction for the coupled particles with different initial conditions, we found that the dynamical system of the particle coupling to Einstein tensor possesses the self-similar structures with fractal geometry in Schwarzschild-Melvin black hole spacetime.Finally, we presented a summary and make some prospects of the chaotic phenomena in black hole spacetimes.