The Peculiarities of the Cosmological Models Based on Non-Linear Classical and Phantom Fields with Minimal Interaction. I. The Cosmological Model Based on Scalar Singlet

Abstract: A detailed comparative qualitative analysis and numerical simulation of evolution of the cosmological models based on classical and phantom scalar fields with self-action was performed. The phase portraits of the dynamic systems of classical and phantom fields and their projections to the Poincare sphere were constructed. It was shown that the phase trajectories of the corresponding dynamic systems can be split by bifurcation trajectories into 2,4 or 6 different dynamic streams corresponding to different pairwise symmetric histories of the Universe depending on the parameters of the scalar field's model. The phase space of such systems becomes multiply connected, the ranges of negative total effective energy unavailable for motion, getting appear there. In the case when attracting centers are situated inside these ranges, the phase trajectories of the classical scalar field in the infinite future tend to limit cycles, winding onto the boundaries of these ranges. The phase trajectories of the scalar field, in turn, get away from the boundaries of ranges with null effective energy and in the infinite future are wound onto one of the symmetrical focuses (centers). Thus, the situations when the Universe, in case of classical scalar field, begins its history with the inflation and ends it up in the Euclidean future, or, in the case of phantom scalar field, in opposite, has the Euclidian start and proceeds to inflation mode after anomalous burst of the acceleration, both become possible. The potentials of scalar fields on the surface of null curvature are distinct from zero and thereby define a certain vacuum state.