Stability and bifurcation analysis of interacting f(T) cosmology
Abstract
The present work deals with dynamical system analysis of Interacting f(T) cosmology. Einstein field equations are second order nonlinear differential equations. So it is very difficult to solve them analytically. We can draw the vector field and analyze the stability of the universe in different phase by dynamical system analysis. By suitable transformation of variables the Einstein field equations are converted to an autonomous system. The critical points are determined and the stability of the equilibrium points are examined by center manifold theory. Possible bifurcation scenarios have also been explained.
1 Introduction
Several cosmological observations of the Supernovae type Ia [1, 2, 3], cosmic microwave background radiation [4, 5], large scale structure [6, 7, 8], baryon acoustic oscillations [9, 10] and weak lensing [11, 12, 13, 14, 15, 16, 17, 18] suggest the fact that our universe is now experiencing an accelerated expansion. Many theories are formulated to explain this latetime acceleration. However, these theories can be divided mainly in two categories fulfilling the criteria of a homogeneous and isotropic universe. First kind of theory assumes a mysterious matter component with negative pressure, dubbed as dark energy (DE) [11, 12, 13, 14, 15, 16, 17, 18] which accounts for about 70% of total energy. This model of exotic fluid seems to have an equation of state \(\omega \) close to \(1\) which gives birth of “antigravity” [19, 20] and makes it phenomenologically analogous to a cosmological \(\Lambda \)term or to a vacuum energy. The observations have shown that about 5% of universe’s energy content is baryonic matter. However, observations of rotation galaxy curves, galaxy clustering and galaxy Xray emission have shown that 26% of the matter in the universe is dark matter. Cosmologists are inclined to suspect dark energy and dark matter are thought to be responsible for the current acceleration of the universe and the dynamics of the galaxy respectively.
It is not achievable to derive explicit form of the interaction between dark energy (DE) and dark matter (DM) from first principle. A specific coupling has to be considered relating to the phenomena. Moreover, the inevitable interaction between the dark components is natural to consider in the framework of the field theory and to diminish the possibility of coincidence problem an appropriate interaction between DE and DM has to be taken care of. On the other hand, it was revealed that the late integrated Sachs–Wolfe effect has the unique ability to provide insight into the coupling between dark sectors. Further, the dark energy decays into matter at a rate proportional to Hubble length. So, it is customary to consider a phenomenological form of interaction between dark matter and dark energy since they are dominant components of the cosmic configuration now a days. In fact, the rate of exchange of energy density in the dark sector is given by the interaction term.
However, standard cosmological framework of general relativity (GR) does not match with the overwhelming abundance of observational evidences for cosmic speed up [21]. To solve this paradox, one could follow in search for solution of the accelerating cosmic expansion enigma is to develop several modified theories of which f(R)gravity theory gets much attention [22, 23, 24, 25, 26, 27, 28]. The first inferred f(R) models proposed rely on negative power of R, which become dominant at small curvature, in the gravitational action to generate the latetime deSitter stage, but suffered from severe instabilities in the presence of matter. In the f(R)gravity theory the geometric Lagrangian density is replaced by a suitably chosen algebraic function f(R), where R is the variable.
2 Basic equations
 (i)
Q=Q(H\(\rho _m\)),
 (ii)
Q=Q(H\(\rho _{\phi }\)),
 (iii)
Q=Q(H\(\rho _m\),H\(\rho _{\phi }\)).
Critical points (CPs) and corresponding cosmological parameters
CPs  x  y  \(\Omega _m\)  \(\omega _{\phi }\)  \(\Omega _{\phi }\)  \(\omega _{Tot}\)  q 

\(P_1\)  1  0  0  1  1  1  2 
\(P_2\)  \(\)1  0  0  1  1  1  2 
\(P_3\)  \(\frac{\lambda }{\sqrt{6}}\)  \(\sqrt{1\frac{\lambda ^2}{6}}\)  0  \(\frac{\lambda ^2}{3}1\)  1  \(\frac{\lambda ^2}{3}1\)  \( \frac{\lambda ^2}{2}1\) 
\(P_4\)  \(\frac{\lambda }{\sqrt{6}}\)  \(\sqrt{1\frac{\lambda ^2}{6}}\)  0  \(\frac{\lambda ^2}{3}1\)  1  \(\frac{\lambda ^2}{3}1\)  \(\frac{\lambda ^2}{2}1\) 
\(P_5\)  \(\frac{\alpha 3}{\sqrt{6}\lambda }\)  \(\sqrt{\frac{\alpha }{3}+\frac{(\alpha 3)^2}{6\lambda ^2}}\)  \(\frac{3\alpha }{3}(1\frac{3\alpha }{\lambda ^2})\)  \(\frac{\alpha }{\frac{(\alpha 3)^2}{\lambda ^2}+\alpha }\)  \(\frac{(\alpha 3)^2}{3\lambda ^2}+ \frac{\alpha }{3}\)  \(\frac{\alpha }{3}\)  \(\frac{1\alpha }{2}\) 
\(P_6\)  \(\frac{\alpha 3}{\sqrt{6}\lambda }\)  \(\sqrt{\frac{\alpha }{3}+\frac{(\alpha 3)^2}{6\lambda ^2}}\)  \(\frac{3\alpha }{3}(1\frac{3\alpha }{\lambda ^2})\)  \(\frac{\alpha }{\frac{(\alpha 3)^2}{\lambda ^2}+\alpha }\)  \(\frac{(\alpha 3)^2}{3\lambda ^2}+ \frac{\alpha }{3}\)  \(\frac{\alpha }{3}\)  \(\frac{1\alpha }{2}\) 
\(P_7\)  \(\sqrt{\frac{\alpha }{3}}\)  0  \(1+\frac{\alpha }{3}\)  1  \(\frac{\alpha }{3}\)  \(\frac{\alpha }{3}\)  \(\frac{\alpha }{3}\) 
\(P_8\)  \(\sqrt{\frac{\alpha }{3}}\)  0  \(1+\frac{\alpha }{3}\)  1  \(\frac{\alpha }{3}\)  \(\frac{\alpha }{3}\)  \(\frac{\alpha }{3}\) 
Eigenvalues of the Jacobian matrix at CPs and corresponding nonhyperbolic constrains
CPs  \(\lambda _1\)  \(\lambda _2\)  \(\lambda _3\)  nonhyperbolic constrains 

\(P_1\)  \(3+\alpha \)  \(3+\sqrt{\frac{3}{2}}\lambda \)  \(3+\alpha \)  \(\alpha =3\), \(\lambda =\sqrt{6}\) 
\(P_2\)  \(3+\alpha \)  \(3\sqrt{\frac{3}{2}}\lambda \)  \(3+\alpha \)  \(\alpha =3\), \(\lambda =\sqrt{6}\) 
\(P_3\)  \(\lambda ^2+\alpha 3\)  \(\frac{\lambda ^2}{2}3\)  \(\lambda ^2+\alpha 3\)  \(\lambda ^2+\alpha =3\), \(\lambda =\pm \sqrt{6}\) 
\(P_4\)  \(\lambda ^2+\alpha 3\)  \(\frac{\lambda ^2}{2}3\)  \(\lambda ^2+\alpha 3\)  \(\lambda ^2+\alpha =3\), \(\lambda =\pm \sqrt{6}\) 
\(P_5\)  \(\frac{(A_1+B_2)+\sqrt{(A_1+B_2)^24(A_1B_2B_1A_2)}}{2}\)  \(\frac{(A_1+B_2)\sqrt{(A_1+B_2)^24(A_1B_2B_1A_2)}}{2}\)  0  Any \(\alpha \) and \(\lambda \) 
\(P_6\)  \(\frac{(A_1+B_2)+\sqrt{(A_1+B_2)^24(A_1B_2B_1A_2)}}{2}\)  \(\frac{(A_1+B_2)\sqrt{(A_1+B_2)^24(A_1B_2B_1A_2)}}{2}\)  0  Any \(\alpha \) and \(\lambda \) 
3 Stability analysis
In this section, We investigate the stability of nonhyperbolic critical points for some choices of \(\alpha \) and \(\lambda \). Most of the cases stability of nonhyperbolic critical points can be determined by center manifold (CM) theory. For this we first perform coordinate transformation so that the critical point moves to the origin.
3.1 Critical point \(P_1\)
\(\bullet \) The critical point is nonhyperbolic for \(\alpha =3\). As Hartman–Grobman Theorem can not be used to analyze the critical point, we shall use center manifold theory (CMT).
By CMT there exits a continuously differentiable function \(h:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) such that \(Y=h(X,{\bar{\Omega }}_m)=aX^2+bX{\bar{\Omega }}_m +c{\bar{\Omega }}_m^2 + \text {higher power terms}\), where \(a,b,c\in {\mathbb {R}}\) [56]. We only concern about the nonzero coefficients of the lowest power terms in CMT as we analyze arbitrary small neighborhood of the origin.

The critical point is nonhyperbolic for \(\lambda =\sqrt{6}\). So by CMT there exists a continuously differentiable function \(h:{\mathbb {R}}\rightarrow {\mathbb {R}}^2\) such that
3.2 Critical point \(P_2\)

The critical point is nonhyperbolic for \(\alpha =3\). Similar to the point \(P_1\), we get the center manifold Y = 0. This two dimensional center manifold coincide with the two dimensional center subspace (X\({\bar{\Omega }}_m\))plane near the origin. The flow on the CM is determined by the following equations

The critical point is nonhyperbolic for \(\lambda =\sqrt{6}\). By similar procedure as stated for \(P_1\), the center manifold at the origin is
Next we consider \(\lambda \ne 0\) for the existence of the critical points \(P_3\) to \(P_6\) and to make the corresponding autonomous system continuously differentiable. For this, we consider \(\sqrt{6}<\lambda <\sqrt{6}\) same as \(\lambda \in (\sqrt{6},0) ~\cup ~(0,\sqrt{6})\).
3.3 Critical point \(P_3\)
If we project the vector field on the plane which is parallel to XYplane, the origin is a saddlenode and unstable in nature for \(\alpha + \lambda ^2 =3\) and \(\sqrt{6}<\lambda <\sqrt{6}\) (Figs. 5, 6).
3.4 Critical point \(P_4\)
If we project the vector field on the plane which is parallel to XYplane, the origin is a saddlenode and unstable in nature for \(\alpha + \lambda ^2 =3\) and \(\sqrt{6}<\lambda <\sqrt{6}\) (Figs. 7, 8).
3.5 Critical point \(P_5\)
Next we analyze the system for arbitrary \(\alpha \). In this case, the eigenvalues of Jacobian matrix become very complicated to characterize. For this reason we state some ranges of \(\alpha \) and corresponding character of the vector field.
\(D>0\): ranges of \(\alpha \) and \(\lambda \) for distinct real eigenvalues
\(\alpha ~(\approx )\)  \(\lambda ~(\approx )\) 

\(\)15.0858  \( \sqrt{6}<\lambda <0\) 
\(0<\lambda <1.69251\)  
\(1.69251<\lambda <\sqrt{6}\) 
\(\alpha \)  \(\lambda \) 

\(\)2  \(\pm 2\) 
\(\)2  \(\pm 1\) 
\(\)1  \(\pm 1\) 
0  ± 1 
\(D<0\): ranges of \(\alpha \) and \(\lambda \) for complex conjugate eigenvalues
\(\alpha ~(\approx )\)  \(\lambda ~(\approx )\) 

0  \(\sqrt{6}<\lambda < 1.36381\) 
2.9123  \(0.462046<\lambda < 0.0962196\) 
\(0.0962196<\lambda < 0.04546\) 
\(\alpha \)  \(\lambda \) 

\(\pm 1\)  \(\)2 
\(D=0\): values of \(\alpha \) and \(\lambda \) for \(\lambda _1\)=\(\lambda _2\)
\(\alpha ~(\approx )\)  \(\lambda ~(\approx )\) 

0  \(\)1.36381 
2.9123  \(\)0.0962196 
The Tables 3, 4 and 5 present the values of \(\alpha \) and \(\lambda \) for different character of eigenvalues. But all these critical points are not cosmologically viable as there is no phase transition of the evolution of the universe at these critical points. So we skip the discussion of these critical points in detail.
3.6 Critical point \(P_6\)
4 Bifurcation analysis and cosmological upshot
In this section, we find the change of stability of all critical points of the autonomous system (16–18) by varying the parameters \(\alpha \) and \(\lambda \). We prepare bifurcation diagram corresponding to the sudden changes of stability of the critical points for small perturbation of the parameters and identify the phase transition of the universe. Finally we categorize the evolution of the universe corresponding to the evolutionary path without singularity.
Stability of \(P_1\) and \(P_2\) according to range of parameters
\(\alpha \), \(\lambda \)  Nature  Stability 

\(\alpha <3\), \(\lambda >\sqrt{6}\)  Hyperbolic  \(P_1\): saddlenode 
\(P_2\): stablenode  
\(\alpha <3\), \(\sqrt{6}<\lambda <\sqrt{6}\)  Hyperbolic  \(P_1\): saddlenode 
\(P_2\): saddlenode  
\(\alpha <3\), \(\lambda <\sqrt{6}\)  Hyperbolic  \(P_1\): stablenode 
\(P_2\): saddlenode  
\(\alpha >3\), \(\lambda >\sqrt{6}\)  Hyperbolic  \(P_1\): unstablenode 
\(P_2\): saddle  
\(\alpha >3\), \(\sqrt{6}<\lambda <\sqrt{6}\)  Hyperbolic  \(P_1\): unstablenode 
\(P_2\): unstablenode  
\(\alpha >3\), \(\lambda <\sqrt{6}\)  Hyperbolic  \(P_1\): saddle 
\(P_2\): unstablenode  
\(\alpha >3\), \(\lambda =\sqrt{6}\)  \(P_1\): nonhyperbolic  \(P_1\): saddle 
\(P_2\): hyperbolic  \(P_2\): unstablenode  
\(\alpha <3\), \(\lambda =\sqrt{6}\)  \(P_1\): nonhyperbolic  \(P_1\): stablenode 
\(P_2\): hyperbolic  \(P_2\): saddlenode  
\(\alpha >3\), \(\lambda =\sqrt{6}\)  \(P_1\): hyperbolic  \(P_1\): unstablenode 
\(P_2\): nonhyperbolic  \(P_2\): saddle  
\(\alpha <3\), \(\lambda =\sqrt{6}\)  \(P_1\): hyperbolic  \(P_1\): saddle 
\(P_2\): nonhyperbolic  \(P_2\): stablenode  
\(\alpha =3\)  Nonhyperbolic  \(P_1\): saddlenode 
\(P_2\): saddlenode  
\(\alpha =3\), \(\lambda =\sqrt{6}\)  \(P_1\): nonhyperbolic  \(P_1\): undetermined 
\(P_2\): hyperbolic  \(P_2\): saddlenode  
\(\alpha =3\), \(\lambda =\sqrt{6}\)  \(P_1\): hyperbolic  \(P_1\): saddlenode 
\(P_2\): nonhyperbolic  \(P_2\): undetermined 
Stability of \(P_3\) and \(P_4\) according to range of parameters
\(\alpha \), \(\lambda \)  Nature  Stability 

\(\lambda ^2+\alpha =3\), \(\sqrt{6}<\lambda <0\)  Nonhyperbolic  \(P_3\): saddlenode 
\(P_4\): saddlenode  
\(\lambda ^2+\alpha =3\), \(0<\lambda <\sqrt{6}\)  Nonhyperbolic  \(P_3\): saddlenode 
\(P_4\): saddlenode  
\(\lambda ^2+\alpha <3\), \(\sqrt{6}<\lambda <\sqrt{6}\)  Hyperbolic  \(P_3\): stablenode 
\(P_4\): stablenode  
\(\lambda ^2+\alpha >3\), \(\lambda >\sqrt{6}\)  Hyperbolic  \(P_3\): saddle 
\(P_4\): saddle  
\(\lambda =\sqrt{6}\)  Same as \(P_1\)  Same as \(P_1\) 
\(\lambda =\sqrt{6}\)  Same as \(P_2\)  Same as \(P_2\) 
Stability of \(P_5\) and \(P_6\) according to range of parameters
\(\alpha \), \(\lambda \)  Nature  Stability 

\(\alpha =3\), \(\lambda =\pm \sqrt{6}\)  Nonhyperbolic  \(P_5\): saddlenode 
\(P_6\): saddlenode  
\(\alpha =3\), \(\sqrt{6}<\lambda <\sqrt{6}\)  Nonhyperbolic  \(P_5\): saddle/saddlenode 
\(P_6\): saddle/saddlenode 
The local bifurcation diagram for the critical point \(P_1\) is shown in Fig. 9. The transition is visible in this diagram for the line \(\alpha =3\) where for \(\lambda >\sqrt{6}\), \(P_1\) undergoes stablenode to saddle via nonhyperbolic saddlenodes (blue line) and for \(\lambda <\sqrt{6}\), it passes through nonhyperbolic saddlenodes (blue line) from stablenode to saddle. The another transition is visible in this diagram for the line \(\lambda =\sqrt{6}\) where for \(\alpha <3\), \(P_1\) undergoes a change from saddlenode to stablenode via nonhyperbolic stable nodes (black line) and a change from unstable to saddle via nonhyperbolic saddle (red line) for \(\alpha >3\). Thus \(\alpha =3\) and \(\lambda =\sqrt{6}\) are lines of bifurcation values and \(P_1\) is a bifurcation point.
Figure 10 represents the local bifurcation diagram for \(\alpha =3\) line and \(\lambda =\sqrt{6}\) line. The phase transition is visible for the line \(\alpha =3\) where for \(\lambda >\sqrt{6}\), \(P_2\) undergoes stablenode to saddle via nonhyperbolic saddlenodes (blue line) and for \(\lambda <\sqrt{6}\), it passes through nonhyperbolic saddlenodes (blue line) from saddlenode to unstablenode. The another transition is visible in this diagram for the line \(\lambda =\sqrt{6}\) where for \(\alpha <3\), \(P_2\) undergoes a change from saddle to unstablenode via nonhyperbolic stable nodes (black line) and a change from unstable to saddle via nonhyperbolic saddle (red line) for \(\alpha >3\). Cosmological evolution at \(P_2\) is same as of \(P_1\).
Another local bifurcation is shown in Fig. 11 for the critical points \(P_3\) and \(P_4\) which exist for \(\sqrt{6}<\lambda <\sqrt{6}\). The transition is visible in this diagram for the curve \(\alpha +\lambda ^2=3\). \(P_3\) and \(P_4\) undergo a change from stablenode (\(\alpha +\lambda ^2<0\)) to saddle (\(\alpha +\lambda ^2>0\)) via nonhyperbolic curve \(\alpha +\lambda ^2=3\). On this curve \(P_3\) and \(P_4\) are saddlenode in nature. Thus \(\alpha +\lambda =3\) is a curve of bifurcation values. In particular, (\(\alpha ,\lambda \))=(3, 0) is a bifurcation value.
Different phasegeneric evolution decelerating to accelerating evolution
\(\alpha \) and \(\lambda \)  Starting point  Early phase  End point  Late phase 

\(1>\alpha >3, \sqrt{2}<\lambda <\sqrt{2}\)  Unstable node: \(P_1\) or \(P_2\)  Free scalar field and decelerating expansion  Stable node: \(P_3\) or \(P_4\)  Exponential de Sitter or accelerating power law expansion in phantom divide line 
Furthermore, combining Figs. 9, 10 and 11 with Table 8 and fixing \(\alpha =3\) we can notice that bifurcation takes place for \(\lambda =\pm \sqrt{6}\). This phenomenon is shown in Fig. 12. In this diagram \(P_1\) and \(P_2\) overlap by their Ycoordinates but their stability is opposite for \(\lambda <\sqrt{6}\) and \(\lambda >\sqrt{6}\): \(P_1\) is stablenode (green line) and \(P_2\) is unstablenode (red dotted line) for \(\lambda <\sqrt{6}\) and in turn, the stability is reverse for \(\lambda >\sqrt{6}\).
When \(\lambda \) crosses \(\sqrt{6}\) from left, \(P_1\) becomes unstable and two stable and two unstable fixed points appear in the picture. It is so to speak a combination of the subcritical and supercritical pichfork bifurcation: the fixed points \(P_1\), \(P_3\) and \(P_4\) behave like supercritical pichfork bifurcation, while the fixed points \(P_1\), \(P_5\) and \(P_6\) behave like subcritical pichfork bifurcation. Similar behavior of dynamics can be found for \(\lambda =\sqrt{6}\) also.
At \(P_5\) and \(P_6\) we have \(\omega _{total}=\frac{\alpha }{3}\) and \(q=\frac{1\alpha }{2}\), so for \(\alpha =3\) the critical point represents cosmological constant era with exponential deSitter expansion. For \(3<\alpha <3\) the critical point represents phantom phase of the universe and for \(\alpha >3\) the universe would be in the case of BigRip singularity. The universe experiences accelerating expansion for \(\alpha >1\) and decelerating expansion for \(\alpha <1\).
Nongeneric evolution at bifurcation values
Bifurcation values  Starting point  Early phase  End point  Late phase 

\(\alpha =3, \lambda \approx 0\)  Saddlenode/saddle: \(P_3\) or \(P_4\)  Cosmological constant and exponential de Sitter expansion  Homoclinic orbit  Cosmological constant and exponential de Sitter expansion 
\(\alpha \approx 3, \lambda =\pm \sqrt{6}\)  Unstablenode/saddlenode/saddle: \(P_1\)  Free scalar field decelerating expansion with stiff matter era  Homoclinic orbit/saddle :\(P_2\)/saddle: \(P_5\) or \(P_6\)  Free scalar field decelerating expansion with stiff matter era 
\(\alpha \approx 3, \lambda =\pm \sqrt{6}\)  Unstablenode/saddlenode/saddle: \(P_2\)  Free scalar field decelerating expansion with stiff matter era  Homoclinic orbit/saddle: \(P_1\)/saddle: \(P_5\) or \(P_6\)  Free scalar field decelerating expansion with stiff matter era 
Figure 13 presents the local phase transition of cosmological evolution for \(\lambda =0\) and \(\alpha =3\). The only transition of the cosmological solution is visible from nonphantom model to phantom one via vanishing kinetic energy for \(\alpha =3\). So the cosmological model is structurally unstable in the case of nongeneric evolution of the universe. On the other hand, Fig. 14 presents the universe passes through a free scalar field stiff fluid era for \(\lambda =\pm \sqrt{6}\) and \(\alpha =3\) to transit nonvanishing scalar field to nonvanishing one.
5 Discussion
An unstable fixed point may be considered as an initial position of the universe and a small perturbation scamper the start of the trajectory of the universe. The initial position of the universe drives a unique trajectory which starts from an arbitrary close neighborhood of an unstable critical point (CP) and stops to an arbitrary close neighborhood of a stable critical point or goes to infinity. As our phase space is finite region of infinite paraboloid, so we skip the infinity case. The qualitative global behavior of the trajectory of the universe is described by inspecting the local behavior of critical points.
The critical points \(P_1\) to \(P_4\) are shown to be nonhyperbolic equilibrium points for certain choices of the parameters \(\alpha \) and \(\lambda \). The critical point \(P_1\) is nonhyperbolic in nature for the choices \(\alpha =3\) or \(\lambda =\sqrt{6}\) or both. However, \(\alpha =3\) and \(\lambda =\sqrt{6}\) is not chosen simultaneously as the vector field is undetermined. From Figs. 1a, b, we see that for \(\alpha =3\) and \(\lambda \ne \sqrt{6}\) the critical point \(P_1\) is a saddlenode and unstable in nature while for \(\lambda =\sqrt{6}\), \(P_1\) is unstable saddle for \(\alpha >3\) and stable node for \(\alpha <3\). Cosmologically, the critical point is not interesting as it represents a scalar field dominated decelerating era. Note that the critical point \(P_2\) has identical behavior as \(P_1\) from the cosmological point of view. The nonhyperbolic critical point \(P_3\) is analyzed for the parametric restriction \(\alpha +\lambda ^2=3\) and \(\lambda ^2 \ne 6\) so that we have unstable saddlenode as shown in Figs. 5 and 6. From cosmological point of view, the equilibrium points \(P_3\) and \(P_4\) are equivalent and both of them are dark energy dominated. The model represents accelerated era of expansion for \(\lambda ^2<2\) while it will be in decelerating phase for \(\lambda ^2>2\). The equilibrium points \(P_5\) and \(P_6\) are nonhyperbolic in nature without any restriction on the parameters \(\alpha \) and \(\lambda \). Both the critical points represent cosmological scaling solution. The effective single fluid will be of quintessence nature if \(1<\alpha <3\) while phantom nature is characterized by the restriction \(\alpha >3\).
The application of bifurcation theory allows us to distinguish some classes of nongeneric and generic evolutionary scenarios for either cosmological constant or free scalar field. There are nonsingular evolutionary paths from decelerating phase to accelerating phase of the universe. There are two types of initial state from which the universe starts evolution. In the first scenario, the universe is emergent from decelerating expansion with free scalar field and in second one, the universe is emergent from the exponential de Sitter expansion with cosmological constant era. From the bifurcation analysis, we have obtained a lines of bifurcation values varying \(\alpha \) and \(\lambda \). We also notice phase transition of the universe at the bifurcation value \(\alpha =3\) or \(\lambda =\pm \sqrt{6}\).
Notes
Acknowledgements
The author S. Mishra is grateful to CSIR, Govt. of India for giving Senior Research Fellowship (CSIR Award No: 09/096 (0890)/2017EMRI) for the Ph.D work.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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