Phase space analysis of sign-shifting interacting dark energy models

The theory of nongravitational interaction between a pressureless dark matter (DM) and dark energy (DE) is a phenomenologically rich cosmological domain which has received magnificent attention in the community. In the present article we have considered some interacting scenarios with some novel features: the interaction functions do not depend on the external parameters of the Universe, rather, they depend on the intrinsic nature of the dark components; the assumption of unidirectional flow of energy between DM and DE has been extended by allowing the possibility of bidirectional energy flow characterized by some sign shifting interaction functions; and the DE equation of state has been considered to be either constant or dynamical in nature. These altogether add new ingredients in this context, and we performed the phase space analysis of each interacting scenario in order to understand their global behavior. According to the existing records in the literature, this combined picture has not been reported elsewhere. From the analyses, we observed that the DE equation of state as well as the coupling parameter(s) of the interaction models can significantly affect the nature of the critical points. It has been found that within these proposed sign shifting interacting scenarios it is possible to obtain stable late time attractors, which may act as global attractors corresponding to an accelerating expansion of the Universe. The overall outcomes of this study clearly highlight that the sign shifting interaction functions are quite appealing in the context of cosmological dynamics, and they deserve further attention.

Figure 1

The phase portrait plot describing model I [Eq. (10)] with $w_d\ge -1$ and $\gamma >0$. In this case we have taken $w_d=-0.95$ and $\gamma =0.4$. We note that one can take any value of $w_d\ge -1$ and any positive value of $\gamma$ in order to get similar graphics. Here, the yellow shaded region represents the accelerated region (i.e., $q<0$), and the pink shaded region corresponds to the decelerated region (i.e., $q>0$).

Figure 2

The phase portrait plot describing model I [Eq. (10)] for $-2<w_d<-1$ and $\gamma >0$. In this case we have taken $w_d=-1.3$ and $\gamma =0.8$. We note that one can take any specific value of $\gamma (>0)$ to draw the plot, however, as long as $\gamma$ decreases, regions I and IV become very small, and they look indistinguishable from one another.

Figure 3

We show the evolution of the CDM density parameter (${\mathrm{\Omega }}_c$), dark energy density parameter (${\mathrm{\Omega }}_d$), and the total equation of state (EOS) parameter ($w_{\mathrm{tot}}$) for model I [Eq. (10)] for $-2<w_d<-1$. We have taken $w_d=-1.3$, $\gamma =0.8$ with the initial conditions $x(N=0)=0.25$, $z(N=0)=0.05$ taken from region II of Fig. 2. For the initial condition on $x(N)$ and $z(N)$ from region I of Fig. 2, again we shall obtain ${\mathrm{\Omega }}_c=0$ and ${\mathrm{\Omega }}_d=1$ at late time. If we choose the initial conditions on $x(N)$ and $y(N)$ from regions III and IV of Fig. 2, we shall reach ${\mathrm{\Omega }}_c={\mathrm{\Omega }}_d=1/2$ in an asymptotic way.

Figure 4

The phase portrait plot describing model I of (10) with $w_d\le -2$ and $\gamma >0$. In this case we have taken $w_d=-2.5$ and $\gamma =0.8$. We note that one can take any value of $w_d\le -2$ and any positive value of $\gamma$ in order to get similar graphics. Here, the yellow shaded region represents the accelerated region (i.e., $q<0$), and the pink shaded region corresponds to the decelerated region (i.e., $q>0$).

Figure 5

Phase portrait plot depicting model II [Eq. (11)] with $\alpha >0$, $\beta >0$ and $w_d\ge -1$. In this case we have taken $w_d=-0.9$, $\alpha =0.5$, and $\beta =0.6$. We note that one can take any value of $w_d\ge -1$ and any positive value of $\alpha$ and $\beta$ in order to get similar graphics. Here, the yellow shaded region represents the accelerated region (i.e., $q<0$), and the pink shaded region corresponds to the decelerated region (i.e., $q>0$).

Figure 6

Phase plot for model II [Eq. (11)] with $\alpha >0$, $\beta >0$ and $-1-\frac{\beta }{\alpha }<w_d<-1$. Here, we have taken $w_d=-1.4$, $\alpha =0.8$, and $\beta =0.9$. Note that one can take any specific value of $\alpha (>0)$ and $\beta (>0)$ to draw the plot, however, as long as $\alpha$ and $\beta$ decrease, regions I and IV become very small, and they look indistinguishable from one another.

Figure 7

We show the evolution of the CDM density parameter (${\mathrm{\Omega }}_c$), dark energy density parameter (${\mathrm{\Omega }}_d$), and the total EOS parameter ($w_{\mathrm{tot}}$) for model II [Eq. (11)]. We have taken the following values of the parameters: $w_d=-1.4$, $\alpha =0.8$, $\beta =0.9$ and the following initial conditions $x(N=0)=0.27$, $z(N=0)=0.08$ from region II of Fig. 6. For the values of initial conditions on $x(N)$ and $y(N)$ from region I, in a similar fashion, we shall obtain ${\mathrm{\Omega }}_c=0$ and ${\mathrm{\Omega }}_d=1$ at late time. Again if we take initial conditions on $x(N)$ and $y(N)$ from regions III and IV of Fig. 6, we shall reach ${\mathrm{\Omega }}_c=\frac{\beta }{\alpha +\beta }$ and ${\mathrm{\Omega }}_d=\frac{\alpha }{\alpha +\beta }$ in an asymptotic fashion.

Figure 8

Phase portrait plot for model II [Eq. (11)] with $\alpha >0$, $\beta >0$ and $w_d\le -1-\frac{\beta }{\alpha }$. In this case we have taken $w_d=-2.4$, $\alpha =0.6$, and $\beta =0.8$. We note that one can take any value of $w_d\le -1-\frac{\beta }{\alpha }$ and any positive value of $\alpha$ and $\beta$ in order to get similar graphics. Here, the yellow shaded region represents the accelerated region (i.e., $q<0$), and the pink shaded region corresponds to the decelerated region (i.e., $q>0$).

Figure 9

Phase plot for model III [Eq. (12)] with $\gamma >0$ and $w_d\ge -1$. In this case we have taken $w_d=-0.9$ and $\gamma =0.5$. We note that one can take any value of $w_d\ge -1$ and any positive value of $\gamma$ in order to get similar graphics. Here, the yellow shaded region represents the accelerated region (i.e., $q<0$), and the pink shaded region corresponds to the decelerated region (i.e., $q>0$).

Figure 10

Phase plot for model III [Eq. (12)] with $\gamma >0$ and $-\frac{3+\sqrt{5}}{2}<w_d<-1$. In this case we have taken $w_d=-1.35$ and $\gamma =0.8$. We note that one can take any specific value of $\gamma (>0)$ to draw the plot, however, as long as $\gamma$ decreases, regions I and IV become very small, and they look indistinguishable from one another.

Figure 11

We display the evolution of the CDM density parameter $({\mathrm{\Omega }}_c)$, dark energy density parameter $({\mathrm{\Omega }}_d)$, and the total equation of state parameter $(w_{\mathrm{tot}})$ for model III [Eq. (12)]. We have chosen the following values of the parameters: $w_d=-1.35$, $\gamma =0.8$ and the following initial conditions: $x(N=0)=0.25$, $z(N=0)=0.07$ from region II of Fig. 10. In addition, if we start any trajectory from region I of Fig. 10, then it converges to the critical point $C_1$. So, for any initial conditions from region I, we shall get ${\mathrm{\Omega }}_c=0$ and ${\mathrm{\Omega }}_d=1$ at late time. Any trajectories starting from regions III and IV of Fig. 10 will converge to the critical point $C_0$. Therefore, if we take initial conditions on $x(N)$ and $z(N)$ from regions III and IV, we shall reach ${\mathrm{\Omega }}_c=\frac{\sqrt{5}-1}{2}$ and ${\mathrm{\Omega }}_d=\frac{3-\sqrt{5}}{2}$ in an asymptotic fashion.

Figure 12

Phase plot for model III [Eq. (12)] with $\gamma >0$ and $w_d\le -\frac{3+\sqrt{5}}{2}$. In this case we have taken $w_d=-2.9$ and $\gamma =0.7$. We note that one can take any value of $w_d$ satisfying $w_d\le -\frac{3+\sqrt{5}}{2}$ and any positive value of $\gamma$ in order to get similar graphics. Here, the yellow shaded region represents the accelerated region (i.e., $q<0$), and the pink shaded region corresponds to the decelerated region (i.e., $q>0$).

Figure 13

Left plot: phase plot for model IV [Eq. (13)] with $w_d>-1$ and $\beta >\alpha >0$. In this case we have used $w_d=-0.92$, $\alpha =0.3$, and $\beta =0.8$. Right plot: phase plot for model IV [Eq. (13)] with $w_d=-1$ and $\beta >\alpha >0$. Particularly, we have taken $w_d=-1$, $\alpha =0.31$, and $\beta =0.9$. We note that the yellow shaded region represents the accelerated region (i.e., $q<0$), and the pink shaded region corresponds to the decelerated region (i.e., $q>0$).

Figure 14

Left plot: phase plot for model IV [Eq. (13)] with $-\frac{\beta }{\alpha }<w_d<-1$ and $\beta >\alpha >0$. For numerical simulation we have used $w_d=-2.2$, $\alpha =1$, and $\beta =6$. Right plot: phase plot for model IV [Eq. (13)] with $w_d\le -\frac{\beta }{\alpha }$ and $\beta >\alpha >0$. For numerical simulation, we have used $w_d=-4.1$, $\alpha =0.2$, and $\beta =0.8$ from the parameter space. In this plot, the yellow shaded region corresponds to the accelerated region (i.e., $q<0$), and the pink shaded region corresponds to the decelerated region (i.e., $q>0$).

Figure 15

We display the evolution of the CDM density parameter $({\mathrm{\Omega }}_c)$, dark energy density parameter $({\mathrm{\Omega }}_d)$, and the total equation of state parameter $(w_{\mathrm{tot}})$ for model IV [Eq. (13)]. We have chosen the following values of the parameters: $w_d=-2.2$, $\alpha =1$, $\beta =6$ and the following initial conditions: $x(N=0)=0.24$, $z(N=0)=0.01$ from region II of the left plot of Fig. 14. In addition, if we start any trajectory from region I of the left plot of Fig. 14, then it converges to the critical point $D_1$. So, for any initial conditions from region I, we shall get ${\mathrm{\Omega }}_c=0$ and ${\mathrm{\Omega }}_d=1$ at late time. Any trajectory starting from regions III and IV of the left plot of Fig. 14 will converge to the critical point $D_0$. Therefore, if we take initial conditions on $x(N)$ and $z(N)$ from regions III and IV, we shall reach ${\mathrm{\Omega }}_c=\frac{\beta -\alpha }{\beta }$ and ${\mathrm{\Omega }}_d=\frac{\alpha }{\beta }$ in an asymptotic fashion.

Figure 16

Left plot: phase plot for model IV [Eq. (13)] with $w_d>-1$ and $\alpha >\beta >0$. In this case we have used $w_d=-0.7$, $\alpha =0.2$, and $\beta =0.1$. Middle plot: phase plot for model IV [Eq. (13)] with $w_d=-1$ and $\alpha >\beta >0$. Particularly, we have chosen $w_d=-1$, $\alpha =0.08$, and $\beta =0.05$. Right plot: phase plot for model IV [Eq. (13)] with $w_d<-1$ and $\alpha >\beta >0$. For numerical simulation we have used $w_d=-3.3$, $\alpha =0.2$, and $\beta =0.1$. Here, the yellow shaded region represents the accelerated region (i.e., $q<0$), and the pink shaded region corresponds to the decelerated region (i.e., $q>0$).

Figure 17

Left plot: phase plot for model V [Eq. (14)] with $w_d\ge -1$ and $\beta <\alpha <0$. In this case we have chosen $w_d=-0.9$, $\alpha =-0.3$, and $\beta =-0.6$. Middle plot: phase plot for model V [Eq. (14)] with $w_d\le \frac{\beta }{\alpha -\beta }$ and $\beta <\alpha <0$. In this case we have chosen $w_d=-2.05$, $\alpha =-0.3$, and $\beta =-0.6$. Right plot: phase plot for model V [Eq. (14)] with $w_d\ge -1$ and $\alpha <\beta <0$. In this case we have chosen $w_d=-0.9$, $\alpha =-0.6$, and $\beta =-0.3$. Here, the yellow shaded region represents the accelerated region (i.e., $q<0$), and the pink shaded region corresponds to the decelerated region (i.e., $q>0$).

Figure 18

Left plot: phase plot for model V [Eq. (14)] with $w_d<-1$ and $\alpha <\beta <0$. In this case we have chosen $w_d=-2$, $\alpha =-2$, and $\beta =-0.1$. Right plot: phase plot for model V [Eq. (14)] with $\frac{\beta }{\alpha -\beta }<w_d<-1$ and $\beta <\alpha <0$. In this case we have taken $w_d=-1.38$, $\alpha =-0.6$, and $\beta =-0.8$. We note that one can take any specific value of $\alpha (<0)$ and $\beta (<0)$ with $\beta <\alpha <0$ to draw the plot, however, as long as $\alpha$ and $\beta$ increase, regions I and IV become very small, and they look indistinguishable from one another.

Figure 19

We display the evolution of the CDM density parameter $({\mathrm{\Omega }}_c)$, dark energy density parameter $({\mathrm{\Omega }}_d)$, and the total equation of state parameter $(w_{\mathrm{tot}})$ for model V [Eq. (14)]. We have chosen the following values of the parameters: $w_d=-1.38$, $\alpha =-0.6$, $\beta =-0.8$ and the following initial conditions: $x(N=0)=0.3$, $z(N=0)=0.1$ from region II of the right plot of Fig. 18. In addition, if we start any trajectory from region I of the right plot of Fig. 18, it converges to the critical point $E_1$. So, for any initial conditions from region I, we shall get ${\mathrm{\Omega }}_c=0$ and ${\mathrm{\Omega }}_d=1$ at late time. Any trajectory starting from regions III and IV of the right plot of Fig. 18 will converge to the critical point $E_0$. Therefore, if we take initial conditions on $x(N)$ and $z(N)$ from regions III and IV, we shall reach ${\mathrm{\Omega }}_c=\frac{\alpha }{\beta }$ and ${\mathrm{\Omega }}_d=\frac{\beta -\alpha }{\beta }$ in an asymptotic fashion.