Chaotic Dynamics In The Curved Spacetime

In this thesis we investigate the chaotic dynamics in the curved space-time in the phase space (?) and using the Poincare section method. This work is the junction between the nonlinear dynamics and general relativity.At first we give several examples of chaotic system to prove the limitation of the classical Newtonian dynamical theory when we investigate the complicated dynamical system. As an overview, we introduce the general theory of the nonlinear dynamics in Chapter I.In chapter II we investigate the evolution of the test particle in several gravitational fields. By considering the Newtonian limit, the gravitational field potential is a nonintegrable Henon-Heiles structure. The potential has a reflection symmetry about an equatorial plane and the motion, which restricts to the equatorial plane, is integral. To investigate a nonplanar stellar motion deviating from the planar one, we approximately consider the motion as a bi-dimensional harmonic oscillator perturbed by higher order terms, so we obtain Henon-Heiles potential, which is nonintegrable. With performing the evolution of the motion of the test particle in the phase space, we can obviously see that the motion randomly oscillates without any period and that the motion of the test particle sensitively depends on the initial conditions and parameters. Byperforming Poincare sections for different values of the parameters and initial conditions, we find the chaotic motion of the test particle and further conform that the chaotic motion of the test particle in the gravitational field becomes obvious when the electoral (magnetic) dipole increases.In chapter III we investigate the dynamical evolution of different early universes models. In the past years Oliveira et al found that preinflationary Friedmann-Robertson-Walk and Bianchi IX universes present a very complicated dynamics with the existence of critical points of saddle-center-type and saddle-type in the phase space. The topology of the phase space about the saddle centers is characterized by homoclinical cylinders emanating from unstable periodical motions and transversal crossing of the cylinders results in a chaotic dynamics. When we investigate the dynamical evolution of the Yang-Mills (YM) field in the Bianchi I cosmology background, we find that the long-time behavior of the evolution is highly sensitive to the choice of the initial conditions. By using the Poincare section method, we further illustrate that the dynamical evolution of the YM field in Bianchi I cosmology background has typically chaotic property. We also find that the evolution of coupled scalar fields system during inflation may change from a regular motion into the chaotic motion when the energy density and the coupling constant of the system increase.