Bifurcations in Ratra–Peebles quintessence models and their extensions

Abstract

We have used the dynamical system approach in order to investigate the dynamics of cosmological models of the flat Universe with a non-minimally coupled canonical and phantom scalar field and the Ratra-Peebles potential. Applying methods of the bifurcation theory we have found three cases for which the Universe undergoes a generic evolution emerging from either the de Sitter or the static Universe state and finishing at the de Sitter state, without the presence of the initial singularity. This generic class of solutions explains both the inflation and the late-time acceleration of the Universe. In this class inflation is an endogenous effect of dynamics itself.

The non-minimal coupling constant and the primordial de Sitter state

“…The critical point D corresponds to a degenerated critical point with two stable sectors and four saddle sectors of the phase space and its structure can be understood as merger of critical points from the previous diagram. We conclude that not only non-minimal coupling constant can be treated as a bifurcation parameter but also the value of globally constant scalar field potential function [62]. On Fig.…”

“…A cosmological evolution of a universe is represented by trajectories in a space of all states of a model called a phase space. In the recent paper [62] the authors used interesting tools of bifurcation theory in dynamical systems of cosmological origin and found bifurcation values of parameters of the investigated models where qualitative dynamics changes.…”

Stability and bifurcation analysis of interacting f(T) cosmology

“…The generic evolution occurs,when there exists a family of solutions satisfying given initial conditions, while the non-generic evolution takes place if one particular solution exists for given initial conditions. To be specific, when a family of trajectory comes out of a unstable point in the early equilibrium (like unstable node/focus or saddle-node) representing a phase and then finishes in a stable point also representing the same phase is called generic evolution [57]. We mention some generic and non-generic evolution of the universe in tabular form (see Tables 9, 10).…”

Cosmological dynamics and bifurcation analysis of the general non-minimal coupled scalar field models

“…One of the bifurcation theory's novelties is that we can use it to classify the Universe's evolution into two categories: generic and non-generic evolution [21]. While the former occurs for various solutions over a wide range of initial conditions, the latter corresponds to a particular solution for a given initial condition.…”

“…It is worth mentioning that interesting bifurcation scenarios were reported in the Randall-Sundrum braneworld model [33], interacting Veneziano ghost DE [34], Brans-Dicke model [35], nonminimal coupled scalar field model [31,36] etc. Recently, bifurcation scenarios and chaos were discussed in the context of Hořava-Lifshitz gravity [37], non-minimal coupled scalar field with Ratra-Peebles potential [21], interacting f (T ) gravity [38] and bulk viscous cosmology [39]. These recent work show that the study of bifurcation is important in cosmology, giving rise to interesting scenarios.…”