Dynamics and chaos in the unified scalar field cosmology. II. System in a finite box

We revisit the global dynamics of unified dark matter cosmological models and analyze it in a new dynamical system setting. In particular, by defining a suitable set of variables we obtain a bounded variable space, a feature that allows a better control of the critical elements of the system. First, we give a comprehensive cosmological interpretation of the critical points. Then, we turn our focus on particular representative trajectories with physically motivated initial conditions studied in the first paper of the series, and we discuss how the scale factor relates to the equation of state parameter. We review and complement these results in the light of the new variable approach by discussing the issue of whether the system is chaotic or not.

Figure 1

Invariant sets $X_H=1$ (left panel) and $X_H=-1$ (right panel). The points identify sources (blue), saddles (black) and sinks (green).

Figure 2

Invariant sets $X_{\varphi }=1$ (left panel) and $X_{\varphi }=-1$ (right panel). The points identify sources (blue), saddles (black) and sinks (green).

Figure 3

Invariant sets $X_{\partial U}=\sqrt{3/2}$ (left panel) and $X_{\partial U}=-\sqrt{3/2}$ (right panel). The points identify sources (blue), saddles (black) and sinks (green).

Figure 4

Top panel: Five representative trajectories in the variable space; the initial conditions corresponding to the numbers are given in the text; the dots indicate the attractors. Bottom panel: The evolution of the effective EoS parameter as a function of the cosmic time for the same initial conditions.

Figure 5

The scale factor $a$ and the EoS parameter $w$ as functions of the cosmic time $t$ for a trajectory starting from $(-1/\sqrt{2},0.8,-\sqrt{3/2}+0.03)$. $a_b$ shows the evolution of the scale factor in the matter-dominated era, if we assume constant $w$.

Figure 6

A portion of the plane $X_{\partial U}=0$. The numbers 1 and 2 identify the initial conditions for the respective trajectories of Fig. 4. The colors correspond to the basins of attraction of the three sinks for the points in the plane.

Figure 7

The basins of attraction of the three sinks for the points in the planes $X_{\partial U}=0$ (left panel), $X_H=0$ (central panel) and $X_{\varphi }=0$ (right panel). The colors correspond to the basins of attraction of the three sinks indicated in Fig. 6.