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# A dynamical system analysis of bouncing cosmology with spatial curvature

Soumya Chakraborty<EMAIL_ADDRESS>Department of Mathematics,
Jadavpur University, Kolkata-700 032, West Bengal, India  Sudip Mishra
<EMAIL_ADDRESS>Department of Applied Mathematics, Maulana Abul
Kalam Azad University of Technology, Kolkata-700064 and Haringhata,
Nadia-741249 (main campus), West Bengal, India  Subenoy Chakraborty
<EMAIL_ADDRESS>Department of Mathematics, Jadavpur University,
Kolkata-700 032, West Bengal, India

###### Abstract

The present work deals with a FLRW cosmological model with spatial curvature
and minimally coupled scalar field as the matter content. The curvature term
behaves as a perfect fluid with the equation of state parameter
$\omega_{\mathcal{K}}=-\frac{1}{3}$. Using suitable transformation of
variables, the evolution equations are reduced to an autonomous system for
both power law and exponential form of the scalar potential. The critical
points are analyzed with center manifold theory and stability has been
discussed. Also, critical points at infinity have been studied using the
notion of Poincaré sphere. Finally, the cosmological implications of the
critical points and cosmological bouncing scenarios are discussed. It is found
that the cosmological bounce takes place near the points at infinity when the
non-isolated critical points on the equator of the Poincaré sphere are saddle
or saddle-node in nature.

Scalar field model, bouncing condition, autonomous system, critical points,
stability, center manifold theory, Poincaré sphere.

###### pacs:

98.80.Cq, 98.80.-k, 95.35.+d, 95.36.+x, 05.70.Fh, 05.45.-a, 02.10.Ud, 02.40.Vh

##  I Introduction

The $\Lambda$CDM model is the simplest and very well-known cosmological model
that describes the evolution of the universe from the early inflationary
paradigm to the present era of expansion. Despite being successful, according
to the largest part of the cosmology community, in describing the formation
and evolution of the large scale structure in the universe, $\Lambda$CDM model
has a large family of challenges that have not been resolved. Also, it is not
consistent with many observations e.g. H0-tension [1], missing satellites [2],
cosmological constant problems [3], the initial condition of the universe
before the inflationary era [4, 5, 6, 6, 7] etc.

As the present observations are consistent with the expanding universe so it
is reasonable to choose a model of the universe which keeps expanding from the
very beginning till the present era [8]. However, from theoretical point of
view, the universe may have an era of contraction in the very early epoch
before inflation or the universe may experience a contracting phase at far
future [9, 10, 11]. Friedmann equations tell us that for negative spatial
curvature, the universe is an ever-expanding model, for flat model the
expansion continues until the total energy of the universe is positive, for
positive spatial curvature the expansion will halt at a finite time and there
will be a phase of contraction [12]. Further, for a scalar field cosmology,
there will have a contraction followed by an expanding era for a range of
values of the scalar field. Usually, in the contracting phase, the kinetic
energy of the scalar field increases, and it stands to dominate the evolution.
Then there will be no mechanism to halt this collapse leading to a singularity
[13]. Though theoretically, Einstein field equations for spatial curvature
indicates a possibility for contracting phase in the past or future, however,
there is no observational hint for a possible contracting era.

However, it will be interesting if by some mechanism the contraction can be
halted and again there is expansion - the bouncing model universe. If the
bouncing phase occurs before the plank energy scale, then Einstein’s gravity
may be chosen as an effective field theory. Further, the existence of the
bouncing solution depends on the spatial curvature of the space-time. For flat
or open universe, bouncing phase is characterized by the null energy condition
while for closed model of the universe, the bouncing era occurs when the
curvature term balances the total energy of the universe. Thus there are two
essential ingredients for the bouncing scenario namely the positive spatial
curvature and the scalar potential that gradually becomes flatter as the
scalar field goes away from the minimum. It is to be noted that the transition
from expansion to contraction is possible due to positive spatial curvature.
Due to the suppression of the kinetic energy as the potential becomes
sufficiently flat, the scalar field slow-rolls over a flat part of the
potential and as a result, there is an accelerated contraction of the
universe. Thus for the occurrence of the bouncing scenario, the positive
curvature term should balance the scalar field energy [14, 15, 16, 17, 18].
However, from observational viewpoint, there is no indication of a bouncing
phase of evolution- though $H$ is decreasing but it is positive from
observational data. The greatest advantage of bouncing scenario is avoidance
of possible singularity either in past (Big Bang) or in future (Big crunch).

The present work deals with scalar field cosmology in the background of FLRW
space-time with positive spatial curvature. The potential of the scalar field
is chosen to be flatter as the scalar field goes away from the minimum. One
can induce a transition from expansion to contraction either by the spatial
curvature. The scalar field continues to climb up the potential after passing
through the minimum potential and by converting most of the kinetic energy to
the potential energy. More specifically, for sufficiently flat potential (by
suppressing kinetic energy) the scalar field experiences a slow roll on a flat
plateau of the potential, and as a result, there will be an accelerated
contracting phase of the universe. Further, if the positive spatial curvature
term is equivalent to the scalar field energy during this epoch then a
bouncing scenario will occur. Thus the accelerated contracting phase due to
the flat potential is very much relevant for the bouncing universe [12, 19].

From an observational point of view the bouncing model is not a universally
accepted model because positive curvature can at most be weakly supported by
the latest CMB observation [5, 7] but in addition, when lensing and baryon
acoustic oscillation constraints are incorporated then the flat model is the
best choice. However, if future observations favor positive curvature then one
may interpret it as the remnant of a contracting phase at the very early
universe, firstly, it is to be mentioned that there are few works in the
literature [20, 21] related to the cyclic universe where positive spatial
curvature were considered. Instead of the scalar field, they have used domain
walls for the requirement of the bouncing era.

The entirety of our work is swotted as follows: section II deals with the
basic equations of the present cosmological model. Also, the corresponding
Einstein field equations, matter conservation equations, and the condition for
the bouncing scenarios are also presented in this section. With a suitable
choice of the variables the evolution, equations are converted to an
autonomous system in section III. The non-hyperbolic critical points are
analyzed using center manifold theory and stability analysis has been
discussed in the same section III. We expound compactification and dynamics
around the points at infinity in section IV. We present cosmological
implications in a section V. Finally, the brief discussion and concluding
remarks of the present work are proposed in section VI.

##  II Basic Equations

In the background of Friedmann-Lemaître-Robertson-Walker (FLRW) space-time
with line element

$\displaystyle\,ds^{2}=-\,dt^{2}+a^{2}(t)\left[\frac{\,dr^{2}}{1-\mathcal{K}r^{2}}+r^{2}(\,d\theta^{2}+\sin^{2}\,d\Phi^{2})\right],$
(1)

in co-moving coordinates $(t,r,\theta,\phi)$, where $a(t)$ is the scale factor
and $\mathcal{K}$ is the spatial curvature with $\mathcal{K}=0,1,-1$
corresponding to a flat, closed, or open universe, respectively. We consider
the Einstein-Hilbert action with a real scalar field $\phi$ [12],

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left(\frac{1}{2}R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right),$
(2)

where $R$ is the Ricci scalar and $V(\phi)$ is the scalar potential. Now the
Friedmann equations are given by

$\displaystyle 3\left(H^{2}+\frac{\mathcal{K}}{a^{2}}\right)$
$\displaystyle=\rho_{\phi},$ (3) $\displaystyle
2\left(\dot{H}-\frac{\mathcal{K}}{a^{2}}\right)$
$\displaystyle=-(\rho_{\phi}+p_{\phi}),$ (4)

where the ‘dot’ denotes the derivative with respect to time, and
$H=\frac{\dot{a}}{a}$ is the Hubble parameter. Here $\rho_{\phi}$ and
$p_{\phi}$ are energy density and pressure of the scalar field having
expression

$\displaystyle\rho_{\phi}$ $\displaystyle=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$
(5) $\displaystyle p_{\phi}$
$\displaystyle=\frac{1}{2}\dot{\phi}^{2}-V(\phi).$ (6)

Now the evolution of the scalar field $\phi$ can be written as [12]

$\displaystyle\ddot{\phi}+3H\dot{\phi}+V^{\prime}(\phi)=0,$ (7)

where $V^{\prime}(\phi)$ denotes differentiation of the scalar potential
$V(\phi)$ with respect to $\phi$. Now the above field equations
$(\ref{eq3}-\ref{eq4})$ can be rewritten as

$\displaystyle 3H^{2}$ $\displaystyle=\rho_{\phi}+\rho_{\mathcal{K}},$ (8)
$\displaystyle 2\dot{H}$
$\displaystyle=-(\rho_{\phi}+p_{\phi})-(\rho_{\mathcal{K}}+p_{\mathcal{K})}$
(9)

where

$\displaystyle\rho_{\mathcal{K}}=-\frac{3\mathcal{K}}{a^{2}},\quad\text{and}\quad
p_{\mathcal{K}}=\frac{\mathcal{\mathcal{K}}}{a^{2}}$

are energy density and pressure of the hypothetical curvature fluid having
equation of state parameter $\omega_{\mathcal{K}}=-\frac{1}{3}$. Thus the
density parameter associated with the curvature fluid in the above first
Friedmann equations is given by [12]

$\Omega_{\mathcal{K}}=\frac{\rho_{curv}}{\rho_{critical}}=-\frac{\mathcal{K}}{a^{2}H^{2}}.$

One may note from the first Friedmann equation (i.e., Eq.(3)) that for a given
scalar field energy density if the positive curvature term is significant then
it gradually tries to reduce the expansion of the universe. The expansion can
even be stopped if the curvature term balances the scalar field energy
density. However, from the second Friedmann equation (i.e., Eq.(4)), it is
easy to see that at the very early era when ‘$a$’ is very small then positive
curvature term dominates over the kinetic energy of the scalar field and
consequently $\dot{H}$ is positive (i.e., $H$ is increasing) and the universe
is going through an expanding era. But with the expansion, the scale factor
gradually increases and at one stage the curvature term balances the kinetic
energy term of the scalar field and $\dot{H}$ vanishes at that instant.
Subsequently, due to dominance of the kinetic energy of the scalar field
$\dot{H}$ becomes negative. So $H$ gradually decreases and then there are two
possibilities (i) $H$ reaches a minimum (where $H>0$) and then again $H$
starts increasing indicating a bouncing era at minimum $H$, or (ii) $H$
gradually decreases, passing through zero and then becomes negative,
indicating a contracting phase [12]. This may be considered as an example how
Einstein gravity may also describe a contracting phase for which so far there
are no observational predictions.

On the other hand, during the contracting phase, the scalar field experiences
a negative friction contribution (anti-friction) and if $|H|$ is assumed to be
small compared to the mass or characteristic curvature of the scalar
potential. Then the friction (i.e., second term in equation (7)) will be
negligible compare to the acceleration term and the force term. So, the
evolution equation of the scalar field approximately becomes

$\displaystyle\ddot{\phi}+\frac{\partial V}{\partial\phi}\approx 0.$ (10)

In particular, if $V=\frac{1}{2}m^{2}\phi^{2}$ then from Eq.(10),
$\ddot{\phi}+m^{2}\phi\approx 0$, that is, the scalar field oscillates like a
simple harmonic oscillator. Subsequently, when $|H|$ becomes comparable to (or
larger than) the characteristic mass scale, then the kinetic term gradually
grows rapidly and the universe enters the big crunch. However, to avoid this
ever-contracting evolution, one may note that during the simple harmonic
motion of the scalar field, the kinetic term temporarily vanishes at the two
endpoints. If the potential has a flat nature at the endpoints with
$V(\phi)>0$, then the evolution will be dominated by the potential energy for
a longer time. But as the kinetic term grows much faster so at some epoch it
balances the potential term and also the positive curvature term and the
Hubble parameter vanishes again. This phase is known as the bouncing era.
After this bounce, the universe again starts expanding. The well-known
bouncing solution in cosmology is the de-Sitter bounce solution given by

$a(t)=\sqrt{\frac{3\mathcal{K}}{\Lambda}}\cosh\left(\sqrt{\frac{\Lambda}{3}}t\right),$

where $\mathcal{K}>0$ is the spatial curvature and $\Lambda$ is the positive
cosmological constant [12].

##  III Formation of the autonomous system: critical point and stability
analysis

We try to obtain a qualitative picture of this cosmological model, that’s why
we adopt a dynamical system approach and for that choose the following
dynamical variables [22, 23, 24]

$\displaystyle x:$ $\displaystyle=\frac{\dot{\phi}}{\sqrt{6}H},$ (11)
$\displaystyle y:$ $\displaystyle=\frac{\sqrt{V}}{\sqrt{3}H},$ (12)
$\displaystyle z:$ $\displaystyle=\frac{\sqrt{6}}{\phi},$ (13) $\displaystyle
u:$ $\displaystyle=\frac{1}{\sqrt{3}\dot{a}}.$ (14)

By using these dynamical variables the cosmic evolution equation (in the last
section) can be written in an autonomous system as

$\displaystyle\frac{dx}{dN}$ $\displaystyle=$
$\displaystyle-3x-\sqrt{\frac{3}{2}}\left(\frac{V^{\prime}}{V}\right)y^{2}+3x(x^{2}-\mathcal{K}u^{2}),$
(15) $\displaystyle\frac{dy}{dN}$ $\displaystyle=$ $\displaystyle
3y(x^{2}-\mathcal{K}u^{2})+\sqrt{\frac{3}{2}}\left(\frac{V^{\prime}}{V}\right)xy,$
(16) $\displaystyle\frac{dz}{dN}$ $\displaystyle=$ $\displaystyle-xz^{2},$
(17) $\displaystyle\frac{du}{dN}$ $\displaystyle=$ $\displaystyle
u(2x^{2}-y^{2})$ (18)

together with $N=\ln a$. Using dynamical variables, the first Friedmann
equation (8) can be rewritten as

$\displaystyle\Omega_{\mathcal{K}}+x^{2}+y^{2}=1,$ (19)

where the density parameter due to the curvature term can be expressed as

$\displaystyle\Omega_{\mathcal{K}}=\frac{\rho_{\mathcal{K}}}{3H^{2}}=-3\mathcal{K}u^{2}.$
(20)

It is to be noted that the dynamical variable $u$ is proportional to the
comoving Hubble horizon. The amplitude of the scalar field $\phi$ is inversely
related to the variable $z$. From Eq.(19) we can clearly conclude about the
meaning of the dynamical variables $(\ref{eqn11})$ and $(\ref{eqn12})$:
$x^{2}$ stands for the relative kinetic energy density of the scalar field
while $y^{2}$ stands for its relative potential energy density of $\phi$ and
the total relative energy density parameter due to the scalar field can be
written as [22]

$\displaystyle\Omega_{\phi}=\frac{\rho_{\phi}}{3H^{2}}=x^{2}+y^{2}.$ (21)

The equation of state parameter ($\omega_{\phi}$) can be expressed as

$\displaystyle\omega_{\phi}$ $\displaystyle=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$
(22)

and the equation of state parameter corresponding to the combined matter field
is given by

$\displaystyle\omega_{total}$ $\displaystyle=x^{2}-y^{2}.$ (23)

From (19), (20) and (21), we can conclude that the variables $x$, $y$, and $u$
are not independent, they are dependent by the constraint equation

$\displaystyle x^{2}+y^{2}-3\mathcal{K}u^{2}=1.$ (24)

This equation is nothing but the conservation of energy with proper scaling.
In Proposition (3), we have shown that (15), (16) and (18) are not
independent, in other words Eq.(18) can be obtained by using Eqs.(15), (16)
and the constraint equation (24). Thus we omit equation (18) from the
autonomous system and by substituting (from Eq.(24))

$\mathcal{K}u^{2}=\frac{1}{3}(x^{2}+y^{2}-1)$

into the right hand side of (15) and (16), we get the following modified
autonomous system

$\displaystyle\frac{dx}{dN}$
$\displaystyle=-2x-\sqrt{\frac{3}{2}}\left(\frac{V^{\prime}}{V}\right)y^{2}-xy^{2}+2x^{3},$
(25) $\displaystyle\frac{dy}{dN}$
$\displaystyle=y+\sqrt{\frac{3}{2}}\left(\frac{V^{\prime}}{V}\right)xy+2x^{2}y-y^{3},$
(26) $\displaystyle\frac{dz}{dN}$ $\displaystyle=-xz^{2}.$ (27)

Note that the scalar field model corresponds to stiff matter if potential
energy (P.E.) is negligible compared to kinetic energy (K.E.), while it
corresponds to $\Lambda$CDM if kinetic energy is negligible compared to
potential energy. In between these two extreme cases radiation will occur if
K.E.=2 times the P.E. In that case the dimensionless variables $x$ and $y$ are
not independent and the system becomes a $2$D system. Also as we are
interested in late time cosmology so radiation is not taken into account.

To determine a suitable form of the autonomous system, we consider several
choices of the scalar field potential [25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35]. But only for power-law [33, 34] $V(\phi)=V_{0}\phi^{-\mu}$ and the
exponential dependence [36, 32, 33, 34, 37] as $V(\phi)=V_{1}e^{-\nu\phi}$
where $V_{0},V_{1}>0$ and $\mu,\nu$ are constant parameters, we get the
suitable form of the autonomous system. These two potentials are interesting
because we find that scalar-field potential is “nonlinear” (i.e., the scalar-
field equation of motion is nonlinear); usually it tends to be made from
exponential of the scalar field. We have not succeeded in determining the
general solution of the scalar-field equation of motion (7), but a special
solution (cosmologically relevant) can be found. Somehow particularly, we find
that this solution dominates at large time and a study of phase space shows
that it is an attractive, time-dependent, equilibrium point [37].

###  III.1 Model 1: Power-law potential

At first we shall analyze the above dynamical system for power-law potential.
The above system of first order differential equations
$(\ref{eqn24}-\ref{eqn26})$ reduces to an autonomous system. The justification
for such a power law form is that it satisfies the sufficiently steep
condition namely $\Gamma=\frac{V^{\prime\prime}V}{(V^{\prime})^{2}}\geq 1$. As
a result, the scalar field rolls down such a potential and favors a common
evolutionary path for a wide range of initial conditions [38, 25]. The form of
power law can be expressed as

$V(\phi)=V_{0}\phi^{-\mu}.$

Substituting this into the right hand side of Equation (25) and Equation (26)
and then the system $(\ref{eqn24}-\ref{eqn26})$ modifies to

$\displaystyle\frac{dx}{dN}$
$\displaystyle=-2x+\frac{\mu}{2}y^{2}z-xy^{2}+2x^{3},$ (28)
$\displaystyle\frac{dy}{dN}$ $\displaystyle=y-\frac{\mu}{2}xyz+2x^{2}y-y^{3},$
(29) $\displaystyle\frac{dz}{dN}$ $\displaystyle=-xz^{2}.$ (30)

We have a total of five critical points corresponding to the autonomous system
$(\ref{eqn29}-\ref{eqn31})$ out of which four are isolated critical points and
another point is a line of the critical point. The set of critical points and
the value of cosmological parameters corresponding to the above autonomous
system are shown in Table 1.

Table 1: Table shows the set of critical points, the existence of critical points, and the value of cosmological parameters corresponding to the autonomous system $(\ref{eqn29}-\ref{eqn31})$: |
---
Critical Points
Existence | $x$ | $y$ | $z$ | $\Omega_{\mathcal{K}}$ | $\Omega_{\phi}$ | $\omega_{\phi}$ | $\omega_{total}$ | $q$
|
---
$C_{1}$
For all $\mu\neq 0$ | $0$ | $1$ | $0$ | $0$ | $1$ | $-1$ | $-1$ | $-1$
|
---
$C_{2}$
For all $\mu\neq 0$ | $0$ | $-1$ | $0$ | $0$ | $1$ | $-1$ | $-1$ | $-1$
|
---
$C_{3}$
For all $\mu$ | $1$ | $0$ | $0$ | $0$ | $1$ | $1$ | $1$ | $2$
|
---
$C_{4}$
For all $\mu$ | $-1$ | $0$ | $0$ | $0$ | $1$ | $1$ | $1$ | $2$
|
---
$C_{5}$
For all $\mu$ | $0$ | $0$ | $z_{c}$ | $1$ | $0$ | Not applicable | $0$ | $\frac{1}{2}$

####  III.1.1  Local dynamics around the critical points for power-law
potential

$\bullet$ The critical points $C_{1}$ and $C_{2}$ exist for all $\mu\neq 0$.
Near these points the curvature is negligible. These solutions are completely
dominated by scalar field ($\Omega_{\phi}=1$) which behaves as cosmological
constant ($\omega_{total}=-1$). The cosmic evolution near the points
characterizes the de-Sitter expansion ($q=-1$) of the universe. On the other
hand, we note that $|\Omega_{\mathcal{K}}|\approx 0$. Then from equation (9)
we have $\dot{H}\approx 0$, i.e., the expansion rate $H$ becomes constant.

The eigenvalues of the Jacobian matrix $J(C_{1})\left(J(C_{2})\right)$ are
$\lambda_{1}=-3$, $\lambda_{2}=-2$, and $\lambda_{3}=0$ corresponding to the
eigenvectors $\mathbf{v}^{(1)}=[1,0,0]^{T}$, $\mathbf{v}^{(2)}=[0,1,0]^{T}$,
and $\mathbf{v}^{(3)}=\left[\frac{\mu}{6},0,1\right]^{T}$ respectively. Since
the Jacobian matrix has one zero eigenvalue so we use center manifold theory
to discuss the stability of the critical point. To apply center manifold
theory [39, 35, 31, 40], first, we use the shifting transformation
$x=X,\leavevmode\nobreak\ y=Y\pm 1,\leavevmode\nobreak\ z=Z$ to transform the
critical point $C_{1}(C_{2})$ into the origin. Next, we use a coordinate
transformation $\mathbf{u}=P\mathbf{v}$ where
$\mathbf{u}=[X\leavevmode\nobreak\ Y\leavevmode\nobreak\ Z]^{T}$,
$\mathbf{v}=[u\leavevmode\nobreak\ v\leavevmode\nobreak\ w]^{T}$,
$P=\left[\mathbf{v}^{(1)}\leavevmode\nobreak\
\mathbf{v}^{(2)}\leavevmode\nobreak\ \mathbf{v}^{(3)}\right]$. From the
calculation (see Proposition (4) in the Appendix), the expressions of the
center manifold are given by $u=0$ and
$v=\mp\frac{\mu^{2}}{72}w^{2}+\mathcal{O}(w^{4})$ and the flow on the center
manifold is determined by
$\frac{dw}{dN}=-\frac{\mu}{6}w^{3}+\mathcal{O}(w^{4}).$ Notice that the flow
on the center manifold depends on the sign of $\mu$. If $\mu<0$ then
$\frac{dw}{dN}>0$ for $w>0$ and $\frac{dw}{dN}<0$ for $w<0$. So, for $\mu<0$
the origin is a saddle point, i.e., unstable in nature. Further if $\mu>0$
then $\frac{dw}{dN}<0$ for $w>0$ and $\frac{dw}{dN}>0$ for $w<0$. Thus for
$\mu>0$ the origin is a stable node. The vector field near the origin for both
of the cases are shown in figures (1) and (2). It is to be noted that the new
coordinate system $(u,\leavevmode\nobreak\ v,\leavevmode\nobreak\ w)$ is
topologically equivalent to the old one, hence the origin in the new
coordinate system, i.e., the critical point $C_{1}(C_{2})$ in the old
coordinate system $(x,\leavevmode\nobreak\ y,\leavevmode\nobreak\ z)$ is a
saddle point for $\mu<0$ and a stable node for $\mu>0$.

If we take the projection of the vector field in the $z=0$ plane, that is, on
the $xy$-plane, we can see that the system converges to the critical point
$C_{1}(C_{2})$. Note that the eigenvalues of the Jacobian matrix at the point
$C_{1}(C_{2})$ are real and negative, it follows that the critical points
$C_{1}$ and $C_{2}$ are stable node and asymptotically stable in nature (see
figure (3)).

Near $C_{1}$ and $C_{2}$ the evolution equation (7) takes the form
$\ddot{\phi}+V^{\prime}(\phi)\approx 0$. So the scalar field potential behaves
as a simple harmonic oscillator and for $\mu>0$ the oscillation amplitude
decreases gradually due to the stability of the critical points.

Figure 1:  This figure shows the vector field near the origin corresponding to
the critical point $C_{1}$ for positive and negative values of the parameter
$\mu$. (a) is for $\mu<0$ and (b) is for $\mu>0$. This figure indicates that
for $\mu<0$ the critical point $C_{1}$ behaves as a saddle node and for
$\mu>0$ the critical point $C_{1}$ behaves as a stable node. Figure 2:  This
figure shows the vector field near the origin corresponding to the critical
point $C_{2}$ for positive and negative values of the parameter $\mu$. (a) is
for $\mu<0$ and (b) is for $\mu>0$. This figure indicates that for $\mu<0$ the
critical point $C_{2}$ behaves as a saddle node and for $\mu>0$ the critical
point $C_{2}$ behaves as a stable node.

$\bullet$ The critical points $C_{3}$ and $C_{4}$ exist for
$\mu\in\mathbb{R}$. The critical points correspond to solutions where the
constraint equation (9) is dominated by the kinetic energy of the scalar field
with a stiff equation of state $\omega_{d}=1$. There exists a decelerating
phase of the universe and we also have $|\Omega_{\mathcal{K}}|\approx 0$ near
these critical points. Then from equation (9) we have $\dot{H}$ is negative,
i.e., the expansion rate $H$ decreases.

The eigenvalues of the Jacobian matrix $J(C_{3})\left(J(C_{4})\right)$ are
$4$, $3$, and $0$ corresponding to the eigenvectors $[1,0,0]^{T}$,
$[0,1,0]^{T}$ and $\left[0,0,1\right]^{T}$ respectively. After applying
shifting transformation $x=X+1$, $y=Y$ and $z=Z$ to the autonomous system
$(\ref{eqn29}-\ref{eqn31})$ and then by using the center manifold theory, the
equation of center manifold can be expressed as $X=0,\leavevmode\nobreak\
Y=0.$ This implies that the center manifold completely lying on the $Z$ axis
and the flow on the center manifold is determined by $\frac{dZ}{dN}=\mp
Z^{2}.$ Notice that $\frac{dZ}{dN}<0$ for both of the cases $Z>0$ or $Z<0$ for
the critical point $C_{3}$, and $\frac{dZ}{dN}>0$ for $Z>0$ or $Z<0$ for the
critical point $C_{4}$. From this one concludes that the critical point
$C_{3}(C_{4})$ is a saddle node, that is, along the $Z$ direction the orbits
approach critical point $C_{3}$ for positive(negative) $Z$ and move away from
it for negative(positive) $Z$. The flow on the center manifold is shown in
figure (4) which implies that the critical points $C_{3}$ and $C_{4}$ behave
as a saddle-node.

Now if we take the projection of the vector field in the $z=0$ plane, that is,
on the $xy$-plane, the system diverges from the critical point $C_{3}(C_{4})$.
Note that the eigenvalues are real and positive which implies that the
critical point $C_{3}(C_{4})$ is unstable in nature due to the unstable node
type instability (see figure (3)).

If the kinetic energy grows much faster than the radiation, matter, or
curvature to prevent the kinetic energy from dominating the universe, the
universe continues to contract until its size becomes zero. So the Big Crunch
may happen if the flow of the vector field goes towards $C_{3}$ or $C_{4}$
asymptotically. As the critical points are saddle in nature, to avoid the Big
Crunch we need to fine-tune the initial condition.

Figure 3:  Profile of the global analysis in finite phase space for several
values of $\mu$. This figure shows the projection of the vector field on the
$xy$ plane corresponding to the autonomous system $(\ref{eqn29}-\ref{eqn31})$.
The horizontal axis represents variable ‘x’ and the vertical axis represents
variable ‘y’. This phase plot indicates that in $xy$ plane the critical points
$C_{1}$ and $C_{2}$ behave as a stable node, $C_{3}$ and $C_{4}$ behave as an
unstable node, and the critical point $C_{5}$ behaves as a saddle node for all
$\mu$. Figure 4:  This figure shows flow direction on the center manifold
corresponding to the critical point $C_{3}$ and $C_{4}$. (a) is for the
critical point $C_{3}$ and (b) is for the critical point $C_{4}$. The
direction of the flow on the center manifold indicates that both of the
critical points are saddle-node in nature.

$\bullet$ The set of critical points $C_{5}$ exist for all $\mu\in\mathbb{R}$.
This set of critical points is normally hyperbolic where the stability is
confirmed by the signature of the remaining non-vanishing eigenvalues. The
constraint equation (9) is dominated by spatial curvature near this set of
points. In this case, the kinetic term is subdominant compared to the
curvature contribution in eq. (9). Then, for $\mathcal{K}>0$, $\dot{H}$ is
positive, and the expansion rate $H$ increases. On the other hand, for
$\mathcal{K}<0$, $\dot{H}$ is negative, and the expansion rate $H$ decreases.
Note that the line of critical point $C_{5}$ arises on the $z$ axis so to
determine the vector field near $C_{5}$ we consider only the first two
equations of the autonomous system $(\ref{eqn29}-\ref{eqn31})$ and taking
$z=const.$ Due to the presence of one positive and one negative eigenvalue
corresponding to the first two equations of the autonomous system, by Hartman-
Grobman theorem, we conclude that the line of critical points $C_{5}$ behave
as a saddle node (figure (5)) for all $\mu$. So depending on the initial
condition there exists a late time (along stable eigen direction) decelerating
dust-dominated universe.

The eigenvalues of the Jacobian matrix at all critical points corresponding to
the autonomous system $(\ref{eqn29}-\ref{eqn31})$ and the nature of critical
points are shown in Table 2.

Table 2: Table shows the eigenvalues $(\lambda_{1},\leavevmode\nobreak\ \lambda_{2},\leavevmode\nobreak\ \lambda_{3})$ of the Jacobian matrix corresponding to the autonomous system $(\ref{eqn29}-\ref{eqn31})$ at each critical points and the nature of all critical points: |
---
Critical Points
$\leavevmode\nobreak\ \leavevmode\nobreak\ \lambda_{1}$ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \lambda_{2}$ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \lambda_{3}$ | Nature of Critical points
|
---
$C_{1}$
$-3$ | $-2$ | $0$ | Saddle node for $\mu<0$ and stable node for $\mu>0$.
|
---
$C_{2}$
$-3$ | $-2$ | $0$ | Saddle node for $\mu<0$ and stable node for $\mu>0$.
|
---
$C_{3}$
$4$ | $3$ | $0$ | Saddle node for all $\mu$ (unstable node in $xy$-plane).
|
---
$C_{4}$
$4$ | $3$ | $0$ | Saddle node for all $\mu$ (unstable node in $xy$-plane).
|
---
$C_{5}$
$-2$ | $1$ | $0$ | Saddle node for all $\mu$.
Figure 5:  This figure shows the projection of the vector field near $C_{5}$
on $z=const.$ plane. This phase plot indicates that the line of critical
points $C_{5}$ behaves as a saddle node for all $\mu$.

###  III.2 Model 2: Exponential potential

As before this choice of the potential simplifies the system of equations
$(\ref{eqn24}-\ref{eqn26})$ into an autonomous system. The choice of
exponential potential corresponds to an extreme example of quintessence [38,
25]. Exponential potential will remain subdominant if it was so initially.
Further, nucleosynthesis constraints the energy density of the quintessence
field to be smaller than the associated background energy density at the early
era. The form of exponential dependence can be expressed as

$V(\phi)=V_{0}e^{-\nu\phi}.$

Substituting this into the right hand side of Eq.(25) and Eq.(26) and then the
system $(\ref{eqn24}-\ref{eqn26})$ modifies to

$\displaystyle\frac{dx}{dN}$ $\displaystyle=-2x+\sqrt{\frac{3}{2}}\nu
y^{2}-xy^{2}+2x^{3},$ (31) $\displaystyle\frac{dy}{dN}$
$\displaystyle=y-\sqrt{\frac{3}{2}}\nu xy+2x^{2}y-y^{3},$ (32)
$\displaystyle\frac{dz}{dN}$ $\displaystyle=-xz^{2}.$ (33)

We can see that the first two equations of the autonomous system
$(\ref{equ30}-\ref{equ32})$ are independent of the variable $z$. So when we
are going to determine the critical points, we can see that most of the
physical meaningful critical points are lying on the $xy$-plane. Thus the last
equation of the autonomous system, that is, equation (33) is not interesting
to provide the stability criteria of the critical points corresponding to the
autonomous system $(\ref{equ30}-\ref{equ32})$. Further, note that the variable
$z$ can not impact physically on this model as all the expressions of the
cosmological parameters are independent of $z$. That’s why it is more
interesting if we consider $z=0$ and analyze the stability of the critical
points in $xy$ plane. We have a total of seven physically meaningful isolated
critical points corresponding to the autonomous system
$(\ref{equ30}-\ref{equ31})$. The set of critical points corresponding to this
autonomous system and the value of cosmological parameters are shown in Table
3.

####  III.2.1 Local dynamics around the critical points for exponential
potential

Table 3: Table shows the set of critical points, the existence of critical points, and the value of cosmological parameters corresponding to the autonomous system $(\ref{equ30}-\ref{equ31})$: |
---
Critical Points
Existence | $x$ | $y$ | $\Omega_{\mathcal{K}}$ | $\Omega_{\phi}$ | $\omega_{\phi}$ | $\omega_{total}$ | $q$
|
---
$P_{0}$
For all $\nu$ | $0$ | $0$ | $1$ | $0$ | Not applicable | $0$ | $\frac{1}{2}$
|
---
$P_{1}$
For all $\nu$ | $1$ | $0$ | $0$ | $1$ | $1$ | $1$ | $2$
|
---
$P_{2}$
For all $\nu$ | $-1$ | $0$ | $0$ | $1$ | $1$ | $1$ | $2$
|
---
$P_{3}$
For $0\leq\nu^{2}\leq 6$ | $\frac{\nu}{\sqrt{6}}$ | $\sqrt{1-\frac{\nu^{2}}{6}}$ | $0$ | $1$ | $\frac{\nu^{2}}{3}-1$ | $\frac{\nu^{2}}{3}-1$ | $\frac{1}{2}(\nu^{2}-2)$
|
---
$P_{4}$
For $0\leq\nu^{2}\leq 6$ | $\frac{\nu}{\sqrt{6}}$ | $-\sqrt{1-\frac{\nu^{2}}{6}}$ | $0$ | $1$ | $\frac{\nu^{2}}{3}-1$ | $\frac{\nu^{2}}{3}-1$ | $\frac{1}{2}(\nu^{2}-2)$
|
---
$P_{5}$
For all $\nu\neq 0$ | $\frac{1}{\nu}\sqrt{\frac{2}{3}}$ | $\frac{2}{\sqrt{3}\nu}$ | $1-\frac{2}{\nu^{2}}$ | $\frac{2}{\nu^{2}}$ | $-\frac{1}{3}$ | $-\frac{2}{3\nu^{2}}$ | $\frac{1}{2}\left(1-\frac{2}{\nu^{2}}\right)$
|
---
$P_{6}$
For all $\nu\neq 0$ | $\frac{1}{\nu}\sqrt{\frac{2}{3}}$ | $-\frac{2}{\sqrt{3}\nu}$ | $1-\frac{2}{\nu^{2}}$ | $\frac{2}{\nu^{2}}$ | $-\frac{1}{3}$ | $-\frac{2}{3\nu^{2}}$ | $\frac{1}{2}\left(1-\frac{2}{\nu^{2}}\right)$

$\bullet$ The critical point $P_{0}$ exists for all $\nu$. For this solution,
the kinetic term is subdominant compared to the curvature contribution in eq.
(9). Then, for $\mathcal{K}>0$, $\dot{H}$ is positive, and the expansion rate
$H$ increases. On the other hand, for $\mathcal{K}<0$, $\dot{H}$ is negative,
and the expansion rate $H$ decreases.

The eigenvalues of the Jacobian matrix $J(P_{0})$ are $\lambda_{1}=-2$, and
$\lambda_{2}=1$. Since $J(P_{0})$ has no zero eigenvalue, the fixed point
$P_{0}$ is hyperbolic. Further, note that $\lambda_{1}<0$ and $\lambda_{2}>0$,
thus by using the Hartman-Grobman theorem, we conclude that the critical point
$P_{0}$ behaves as a saddle node (see figure (6)). Depending on the initial
condition there exists a late time (along stable eigen direction) decelerating
dust-dominated universe.

$\bullet$ Similar to the case of $C_{3}$ and $C_{4}$, the critical point
$P_{1}$ and $P_{2}$ exist for all $\nu$ and correspond to the solutions where
the constraint equation (9) is dominated by the kinetic energy of the scalar
field with a stiff equation of state $\omega_{d}=1$. There exists a
decelerating phase of the universe and we also have
$|\Omega_{\mathcal{K}}|\approx 0$ near these critical points. Then from
equation (9) we have $\dot{H}$ is negative, i.e. the expansion rate $H$
decreases.

The eigenvalues of the Jacobian matrix $J(P_{1})$ are $\lambda_{1}=4$, and
$\lambda_{2}=3-\sqrt{\frac{3}{2}}\nu$. Notice that the eigenvalue
$\lambda_{1}$ is always positive which implies that the critical point $P_{1}$
is always unstable in nature but the type of instability is determined by the
nature of the eigenvalue $\lambda_{2}$. The critical point $P_{1}$ is
hyperbolic for $\nu\neq\sqrt{6}$ and nonhyperbolic for $\nu=\sqrt{6}$. For
$\nu<\sqrt{6}$, the eigenvalue $\lambda_{2}$ is positive which implies that
the critical point $P_{1}$ is an unstable node. For $\nu>\sqrt{6}$, we have
$\lambda_{2}<0$ and consequently the critical point $P_{1}$ is a saddle node.
To determine the stability of $P_{1}$ for $\nu=\sqrt{6}$, we use center
manifold theory which we have already applied to discuss the stability of the
critical point $C_{1}$. To apply this theory, first, we use a shifting
transformation $(x=X+1,y=Y)$ to transform the fixed point $P_{1}$ into the
origin. Then using the similar arguments that we have used to determine the
stability analysis of $C_{1}$, we obtained that the expression of the center
manifold is $X=-\frac{1}{2}Y^{2}+\mathcal{O}(Y^{3})$ and the flow on the
center manifold is determined by the equation
$\frac{dY}{dN}=-\frac{3}{2}Y^{3}+\mathcal{O}(Y^{4})$. Note that
$\frac{dY}{dN}<0$ while $Y>0$ and $\frac{dY}{dN}>0$ for $Y<0$. This implies
that the flow on the center manifold is stable but the origin is a saddle
node. As the transformed system $(X,Y)$ is topologically equivalent to the old
one, the critical point $P_{1}$ also exhibits saddle-node type instability for
$\nu=\sqrt{6}$. Hence, the critical point $P_{1}$ is a saddle-node for
$\nu\geq\sqrt{6}$ and unstable node for $\nu<\sqrt{6}$ (see figure (6)).

The eigenvalues of the Jacobian matrix $J(P_{2})$ are $\lambda_{1}=4$, and
$\lambda_{2}=3+\sqrt{\frac{3}{2}}\nu$. Notice that the eigenvalue
$\lambda_{1}$ is always positive which implies that the critical point $P_{2}$
is always unstable in nature but the type of instability is determined by the
nature of the eigenvalue $\lambda_{2}$. The critical point $P_{2}$ is
hyperbolic for $\nu\neq-\sqrt{6}$ and nonhyperbolic for $\nu=-\sqrt{6}$. For
$\nu>-\sqrt{6}$, the eigenvalue $\lambda_{2}$ is positive which implies that
the critical point $P_{2}$ is an unstable node. For $\nu<-\sqrt{6}$, we have
$\lambda_{2}<0$ and consequently the critical point $P_{2}$ is a saddle node.
To determine the stability of $P_{2}$ for $\nu=\sqrt{6}$, we use center
manifold theory and to apply this theory, first, we use a shifting
transformation $(x=X-1,y=Y)$ to transform the fixed point $P_{2}$ into the
origin. Then the expression of the center manifold is given by
$X=\frac{1}{2}Y^{2}+\mathcal{O}(Y^{3})$ and the flow on the center manifold is
determined by the equation
$\frac{dY}{dN}=-\frac{3}{2}Y^{3}+\mathcal{O}(Y^{4})$. Note that
$\frac{dY}{dN}<0$ while $Y>0$ and $\frac{dY}{dN}>0$ for $Y<0$. This implies
that the flow on the center manifold is stable but the origin is a saddle
node. Hence, the critical point $P_{2}$ is a saddle-node for
$\nu\leq-\sqrt{6}$ and unstable node for $\nu>-\sqrt{6}$ (see figure (6)).

So the Big Crunch can be avoided for $-\sqrt{6}<\nu<\sqrt{6}$ due to the
instability of the critical points. One may note that depending on the initial
condition there exists a late time (along stable eigen direction) kinetic
dominated decelerating universe.

$\bullet$ The fixed point $P_{3}$ and $P_{4}$ exist for $\nu^{2}\leq 6$.
Varying $\nu$ we get a non-isolated set of critical points which are
completely dominated by kinetic energy ($\Omega_{\phi}=1$ and $\dot{\phi}\neq
0$). The DE can represent either quintessence or any other exotic type fluid
depending on the parameter $\nu$. Specially, the depletion
$\nu^{2}<2\leavevmode\nobreak\ (\nu\neq 0)$ implies that the scalar field
behaves as quintessence like fluid and there exists an accelerated universe
(since for this case, $-1<\omega_{total}<-\frac{1}{3},q<0$) near these
critical points whereas the scalar field DE behaves as dust for
$\nu^{2}=3\leavevmode\nobreak\ (\omega_{\phi}=0)$ and in this case the
solutions insinuate dust-dominated decelerating phase
($\omega_{total}=0,\leavevmode\nobreak\ q=\frac{1}{2},\leavevmode\nobreak\
\Omega_{\mathcal{K}}=0,\leavevmode\nobreak\ \Omega_{\phi}=1$) of the cosmic
evolution.

The eigenvalues of the Jacobian matrix $J(P_{3})$ are
$\lambda_{1}=-2+\nu^{2}$, and $\lambda_{2}=\frac{1}{2}(\nu^{2}-6)$. The
critical point $P_{3}$ is hyperbolic for $\nu\neq\pm\sqrt{2},\pm\sqrt{6}$
because $\lambda_{1}=0$ for $\nu=\pm\sqrt{2}$ and $\lambda_{2}=0$ for
$\nu=\pm\sqrt{6}$. Consequently, the critical point $P_{3}$ is nonhyperbolic
if $\nu=\pm\sqrt{2}$ or $\nu=\pm\sqrt{6}$.

Hyperbolic case $\left(\nu\neq\pm\sqrt{2},\pm\sqrt{6}\right)$: As the region
of existence of the critical point is $-\sqrt{6}\leq\nu\leq\sqrt{6}$, to
determine the stability of this critical point in this case, we consider three
intervals (i) $-\sqrt{6}<\nu<-\sqrt{2}$, (ii) $-\sqrt{2}<\nu<\sqrt{2}$, and
(iii) $\sqrt{2}<\nu<\sqrt{6}$. For $\nu\in(-\sqrt{6},-\sqrt{2})$, we have
$\lambda_{1}>0$ and $\lambda_{2}<0$, it follows that the critical point
$P_{3}$ is a saddle node. For $\nu\in(-\sqrt{2},\sqrt{2})$, we can see that
$\lambda_{1},\lambda_{2}<0$ and this implies that $P_{3}$ is a stable node and
asymptotically stable in nature. Lastly, for $\nu\in(\sqrt{2},\sqrt{6})$, we
have $\lambda_{1}>0$ and $\lambda_{2}<0$, thus in this interval of $\nu$ the
critical point $P_{3}$ behaves as a saddle node (see figure (6)).

Nonhyperbolic case $\left(\nu=\sqrt{2}\leavevmode\nobreak\
\text{or}\leavevmode\nobreak\ \nu=-\sqrt{2}\leavevmode\nobreak\
\text{or}\leavevmode\nobreak\ \nu=\sqrt{6}\leavevmode\nobreak\
\text{or}\leavevmode\nobreak\ \nu=-\sqrt{6}\right)$:

We use the center manifold theory to determine the stability of $P_{3}$ for
this case. To apply this theory, first, we use a shifting transformation
$\left(x=X+\frac{\nu}{\sqrt{6}},y=Y+\sqrt{1-\frac{\nu^{2}}{6}}\right)$ to
transform the critical point $P_{3}$ into the origin. Next, we use a
coordinate transformation $\mathbf{u}=P\mathbf{v}$ where
$\mathbf{u}=[X\leavevmode\nobreak\ Y]^{T}$, $\mathbf{v}=[U\leavevmode\nobreak\
V]^{T}$, $P=[\mathbf{v_{1}}\leavevmode\nobreak\ \mathbf{v_{2}}]$ and
$\mathbf{v_{1}}$, and $\mathbf{v_{2}}$ are the eigenvectors corresponding to
the eigenvalues $\lambda_{1}$, and $\lambda_{2}$ respectively. For
$\nu=\sqrt{2}$, the equations of center manifold is given by
$U=\sqrt{\frac{3}{2}}V^{2}+\mathcal{O}(V^{3})$, and the flow on the center
manifold is determined by $\frac{dV}{dN}=4\sqrt{6}V^{2}+\mathcal{O}(V^{3})$.
Note that $\frac{dV}{dN}>0$ for $V>0$ or $V<0$ which implies that the origin
is a saddle node (see figure (6)). For $\nu=-\sqrt{2}$, the expression of the
center manifold and the flow on the center manifold are the same as of
$\nu=\sqrt{2}$ case and consequently, the origin exhibits a similar type of
stability in this case also. For $\nu=\sqrt{6}$ the critical point $P_{3}$
converts into $P_{1}$ and for $\nu=-\sqrt{6}$, $P_{3}$ converts into $P_{2}$
and the stability analysis of these critical points already discussed.

Figure 6:  Profile of the global analysis in finite phase space for several
values of $\nu$. The horizontal axis represents variable $`$x’ and the
vertical axis represents variable $`$y’. (a) corresponds to $\nu=-\sqrt{6}$ :
$P_{0}$ and $P_{2}$ are saddle node, $P_{1}$ is an unstable node, and $P_{5}$
and $P_{6}$ are spiral sink, (b) corresponds to $\nu=-\sqrt{2}$ : $P_{0}$,
$P_{3}$ and $P_{4}$ are saddle node, $P_{1}$ and $P_{2}$ are unstable node,
(c) corresponds to $\nu=0$ : $P_{0}$ is a saddle node, $P_{1}$ and $P_{2}$ are
unstable node, and $P_{3}$ and $P_{4}$ are stable node, (d) corresponds to
$\nu=\frac{1}{2}$ : where $P_{0}$ is a saddle node, $P_{1}$ and $P_{2}$ are
unstable node, $P_{3}$ and $P_{4}$ are stable node, and $P_{5}$ and $P_{6}$
are saddle node, (e) corresponds to $\nu=1$ : $P_{0}$ is a saddle node,
$P_{1}$ and $P_{2}$ are unstable node, $P_{3}$ and $P_{4}$ are stable node,
and $P_{5}$ and $P_{6}$ are saddle node, (f) corresponds to $\nu=\sqrt{2}$ :
$P_{0}$, $P_{3}$ and $P_{4}$ are saddle node, $P_{1}$ and $P_{2}$ are unstable
node, (g) corresponds to $\nu=2$ : $P_{0}$, $P_{3}$ and $P_{4}$ are saddle
node, $P_{1}$ and $P_{2}$ are unstable node, and $P_{5}$ and $P_{6}$ are
spiral sink, (h) corresponds to $\nu=\sqrt{6}$ : $P_{0}$ and $P_{1}$ are
saddle node, $P_{2}$ is an unstable node, and $P_{5}$ and $P_{6}$ are spiral
sink, and (i) corresponds to $\nu=3$ : $P_{0}$ and $P_{1}$ are saddle node,
$P_{2}$ is an unstable node, and $P_{5}$ and $P_{6}$ are spiral sink.

The eigenvalues of the Jacobian matrix $J(P_{4})$ are
$\lambda_{1}=-2+\nu^{2}$, and $\lambda_{2}=\frac{1}{2}(\nu^{2}-6)$. The
critical point $P_{4}$ is hyperbolic for $\nu\neq\pm\sqrt{2},\pm\sqrt{6}$
because $\lambda_{1}=0$ for $\nu=\pm\sqrt{2}$ and $\lambda_{2}=0$ for
$\nu=\pm\sqrt{6}$. Consequently, the critical point $P_{4}$ is nonhyperbolic
if $\nu=\pm\sqrt{2}$ or $\nu=\pm\sqrt{6}$.

Hyperbolic case $\left(\nu\neq\pm\sqrt{2},\pm\sqrt{6}\right)$: As the region
of existence of the critical point is $-\sqrt{6}\leq\nu\leq\sqrt{6}$, to
determine the stability of this critical point in this case, we consider three
intervals (i) $-\sqrt{6}<\nu<-\sqrt{2}$, (ii) $-\sqrt{2}<\nu<\sqrt{2}$, and
(iii) $\sqrt{2}<\nu<\sqrt{6}$. For $\nu\in(-\sqrt{6},-\sqrt{2})$, we have
$\lambda_{1}>0$ and $\lambda_{2}<0$ which implies that the critical point
$P_{4}$ is a saddle node. Now while $\nu\in(-\sqrt{2},\sqrt{2})$, we can see
that $\lambda_{1},\lambda_{2}<0$ and this implies that $P_{4}$ is a stable
node and asymptotically stable in nature. Lastly, for
$\nu\in(\sqrt{2},\sqrt{6})$, we have $\lambda_{1}>0$ and $\lambda_{2}<0$ which
means that the critical point $P_{4}$ is a saddle node (see figure (6)).

Nonhyperbolic case $\left(\nu=\sqrt{2}\leavevmode\nobreak\
\text{or}\leavevmode\nobreak\ \nu=-\sqrt{2}\leavevmode\nobreak\
\text{or}\leavevmode\nobreak\ \nu=\sqrt{6}\leavevmode\nobreak\
\text{or}\leavevmode\nobreak\ \nu=-\sqrt{6}\right)$:

We use the center manifold theory to determine the stability of $P_{4}$ for
this case. To apply this theory, first, we use a shifting transformation
$\left(x=X+\frac{\nu}{\sqrt{6}},y=Y-\sqrt{1-\frac{\nu^{2}}{6}}\right)$ to
transform the critical point $P_{4}$ into the origin. Next, we use a
coordinate transformation $\mathbf{u}=P\mathbf{v}$ where
$\mathbf{u}=[X\leavevmode\nobreak\ Y]^{T}$, $\mathbf{v}=[U\leavevmode\nobreak\
V]^{T}$, $P=[\mathbf{v_{1}}\leavevmode\nobreak\ \mathbf{v_{2}}]$ and
$\mathbf{v_{1}}$, and $\mathbf{v_{2}}$ are the eigenvectors corresponding to
the eigenvalues $\lambda_{1}$, and $\lambda_{2}$ respectively. For
$\nu=\sqrt{2}$, the equations of center manifold is given by
$U=-\sqrt{\frac{3}{2}}V^{2}+\mathcal{O}(V^{3})$, and the flow on the center
manifold is determined by $\frac{dV}{dN}=-4\sqrt{6}V^{2}+\mathcal{O}(V^{3})$.
Note that $\frac{dV}{dN}<0$ for $V>0$ or $V<0$ which concludes that the origin
is a saddle node (see figure (6)). For $\nu=-\sqrt{2}$, the expression of the
center manifold and the flow on the center manifold are the same as of
$\nu=\sqrt{2}$ case and consequently, the origin exhibits a similar type of
stability in this case also. For $\nu=\sqrt{6}$ the critical point $P_{4}$
converts into $P_{1}$ and for $\nu=-\sqrt{6}$, $P_{4}$ converts into $P_{2}$
and the stability analysis of these critical points already discussed.

Therefore, one can conclude that the points $P_{3}$ and $P_{4}$ correspond to
kinetic dominated solutions that behave as quintessence like fluid
($\nu^{2}<2$) and there exists an accelerated universe at late times, although
cosmological coincidence problem can not be alleviated by these points. On the
other hand, late time decelerated solution ($2\leq\nu^{2}\leq 6$) depends on
the initial condition of the autonomous system of model 2.

$\bullet$ Scaling solutions represented by the critical points $P_{5}$ and
$P_{6}$ exist for all $\nu\neq 0$. Varying $\nu$ we get a non-isolated set of
critical points where the nature (hyperbolic or normally-hyperbolic) of points
depends on the values of $\nu$. Near these critical points, scalar field DE
behaves as a quintessence boundary ($\omega_{\phi}=-\frac{1}{3}$). As
$0\leq\Omega_{\phi}\leq 1$ when $\mathcal{K}<0$ (from eq.(24)), so for
cosmological viability the parameter $\nu$ must satisfy $\nu^{2}\geq 2$. In
this case, there exists always a decelerating universe. On the other hand, for
$\mathcal{K}>0$ there exists an accelerating universe when $\nu^{2}<2$. The
cosmic coincidence problem can be alleviated by these points depending on the
values of $\nu$.

The eigenvalues of the Jacobian matrices $J(P_{5})$ and $J(P_{6})$ are same
and those are
$\lambda_{1}=\frac{-\nu^{2}+\sqrt{(8-3\nu^{2})\nu^{2}}}{\nu^{2}}$, and
$\lambda_{2}=\frac{-\nu^{2}-\sqrt{(8-3\nu^{2})\nu^{2}}}{\nu^{2}}$. Thus the
critical points $P_{5}$ and $P_{6}$ are hyperbolic for $\nu\neq\pm\sqrt{2}$
and nonhyperbolic for $\nu=\pm\sqrt{2}$. For $\nu^{2}<2$, we can observe that
$\lambda_{1}>0$ and $\lambda_{2}<0$ which concludes that $P_{5}$ and $P_{6}$
saddle point and unstable in nature. For $2<\nu^{2}<\frac{8}{3}$, we can see
that $\lambda_{1},\lambda_{2}<0$ which implies that the critical points
$P_{5}$ and $P_{6}$ are asymptotically stable in nature due to the stable node
type stability. For $\nu^{2}>\frac{8}{3}$, the eigenvalues are complex
conjugate to each other with negative real part which implies that the
critical points $P_{5}$ and $P_{6}$ are spiral sink and asymptotically stable
in nature (see figure (6)). For $\nu=\pm\sqrt{2}$, the coordinate of
$P_{5}(P_{6})$ is the same as of $P_{3}(P_{4})$ for $\nu=\pm\sqrt{2}$ and the
stability analysis corresponding to these two cases already discussed. From
the discussion, we conclude that both of the critical points $P_{5}$ and
$P_{6}$ behave as a saddle node for $\nu=\pm\sqrt{2}$. The eigenvalues of the
Jacobian matrix at all critical points corresponding to the autonomous system
$(\ref{equ30}-\ref{equ31})$ and the nature of critical points are shown in
Table 4.

Therefore, the kinetic dominated solution at $\nu^{2}=2$ is saddle-node in
nature. For sufficiently flat potential $\nu^{2}\leq 2$ one may get an
accelerating universe at late time depending on the initial condition of the
model 2. Depending on some parameter restrictions, the decelerating universe
is the late-time attractor described by these points.

If we consider our analysis in the three-dimensional coordinate system, that
is, in $(x,y,z)$ coordinate system, then we can observe that for all possible
cases, the critical points are nonhyperbolic in nature. To determine the
stability of the critical points by applying the center manifold theory, we
conclude that for all critical points, the center manifolds are lying on the
$z$-axis and the flows on the center manifold are unstable in nature.

Table 4: Table shows the eigenvalues $(\lambda_{1},\leavevmode\nobreak\ \lambda_{2})$ of the Jacobian matrix corresponding to the autonomous system $(\ref{equ30}-\ref{equ31})$ at each critical points and the nature of all critical points: |
---
Critical points
$\lambda_{1}$ | $\lambda_{2}$ | Nature of Critical points
|
---
$P_{0}$
$-2$ | $1$ | Saddle node for all $\nu$.
|
---
$P_{1}$
$4$ | $3-\sqrt{\frac{3}{2}}\nu$ | Saddle node for $\nu\geq\sqrt{6}$ and unstable node for $\nu<\sqrt{6}$.
|
---
$P_{2}$
$4$ | $3+\sqrt{\frac{3}{2}}\nu$ | Saddle node for $\nu\leq-\sqrt{6}$ and unstable node for $\nu>-\sqrt{6}$.
|
---
$P_{3}$
$-2+\nu^{2}$ | $\frac{1}{2}(\nu^{2}-6)$ | Stable node for $\nu^{2}<2$ and saddle-node for $2\leq\nu^{2}\leq 6$.
|
---
$P_{4}$
$-2+\nu^{2}$ | $\frac{1}{2}(\nu^{2}-6)$ | Stable node for $\nu^{2}<2$ and saddle-node for $2\leq\nu^{2}\leq 6$.
|
---
$P_{5}$
$\frac{-\nu^{2}+\sqrt{(8-3\nu^{2})\nu^{2}}}{\nu^{2}}$ | $\frac{-\nu^{2}-\sqrt{(8-3\nu^{2})\nu^{2}}}{\nu^{2}}$ |  |
---
Saddle nodet for $\nu^{2}\leq 2$, stable node for $2<\nu^{2}\leq\frac{8}{3}$,
and spiral sink while $\nu^{2}>\frac{8}{3}$.
|
---
$P_{6}$
$\frac{-\nu^{2}+\sqrt{(8-3\nu^{2})\nu^{2}}}{\nu^{2}}$ | $\frac{-\nu^{2}-\sqrt{(8-3\nu^{2})\nu^{2}}}{\nu^{2}}$ |  |
---
Saddle node for $\nu^{2}\leq 2$, stable node for $2<\nu^{2}\leq\frac{8}{3}$,
and spiral sink while $\nu^{2}>\frac{8}{3}$.

##  IV Compactification and the dynamics around the critical points at
infinity

The idea to compactify the space $\mathbb{R}^{n}$ by the addition of points at
infinity and to map them into finite points is frequently used in the setting
of two-dimensional differential equations [41, 42, 43]. An early study of
analyzing the global behavior of a planar dynamical system via
compactification was carried out by Bendixson using the stereographic
projection of the sphere (often called Bendixson sphere) onto the plane.
However, this stereographic projection has the drawback of obliterating the
“different directions at infinity” [44]. French mathematician Henri Poincaré
overcame this difficulty by projecting $\mathbb{R}^{2}$ onto the Poincaré
hemisphere through its center.

In this compactification scheme, we draw straight lines starting from the
center of the unit sphere $S^{2}=\\{(X,Y,Z)\in\mathbb{R}^{3}\mid
X^{2}+Y^{2}+Z^{2}=1\\}$ to $xy$-plane which is tangent to $S^{2}$ at either
the north or south pole. It is to be noted that each straight line meets the
unit sphere once and the $xy$ plane once. Let a straight line intersects the
sphere at $(X_{1},Y_{1},Z_{1})$ and the $xy$-plane at $(x_{1},y_{1})$, then
our projective transformation is realized by the transformation

$\displaystyle x_{1}=\frac{X_{1}}{Z_{1}},\leavevmode\nobreak\
y_{1}=\frac{Y_{1}}{Z_{1}}.$ (34)

So an arbitrary point $(x,y)$ on the $xy$-plane can be expressed uniquely by a
point $(X,Y,Z)$ on the upper/lower half of the sphere as (34) [44]. This
scheme has the advantage that the critical points of different directions at
infinity are spread out along the equator of the sphere.

Given an autonomous system of differential equations on $\mathbb{R}^{2}$

$\displaystyle x^{\prime}=P(x,y),$ (35) $\displaystyle y^{\prime}=Q(x,y)$ (36)

where $P$ and $Q$ are polynomial functions of $x$ and $y$. We can write (35)
and (36) in the form of a single differential equation

$\frac{dy}{dx}=\frac{Q(x,y)}{P(x,y)}$

which yields

$\displaystyle Q(x,y)dx-P(x,y)dy=0.$ (37)

Now for any point $(x,y)$, using (34), we obtain

$\displaystyle dx=\frac{ZdX-XdZ}{Z^{2}},\leavevmode\nobreak\ dy=\frac{ZdY-
YdZ}{Z^{2}}.$ (38)

Hence, the differential equation (37) can be represented as

$\displaystyle Q(ZdX-XdZ)-P(ZdY-YdZ)=0$ (39)

where

$P=P(x,y)=P\left(X/Z,Y/Z\right)$

and

$Q=Q(x,y)=Q\left(X/Z,Y/Z\right).$

Let $m$ be the maximum degree of the terms in $P$ and $Q$. We multiply both
side of the equation (39) by $Z^{m}$ to eliminate $Z$ from the denominator and
obtain

$\displaystyle ZQ^{*}dX-ZP^{*}dY+(YP^{*}-XQ^{*})dZ=0$ (40)

where

$P^{*}(X,Y,Z)=Z^{m}P(X/Z,Y/Z)$

and

$Q^{*}(X,Y,Z)=Z^{m}Q(X/Z,Y/Z)$

are polynomials in $(X,Y,Z)$.

The equator of $S^{2}$ can be expressed by $\\{(X,Y,0)|X^{2}+Y^{2}=1\\}$. So,
the critical points of (40) on the equator of $S^{2}$ where $Z=0$ are given by
the equation

$\displaystyle XQ^{*}-YP^{*}=0.$ (41)

On the equator of the Poincaré sphere we can derive

$\displaystyle XQ^{*}-YP^{*}$ $\displaystyle=$ $\displaystyle
XQ_{m}(X,Y)-YP_{m}(X,Y)$ $\displaystyle=$
$\displaystyle\mathcal{S}\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{(say)},$

where $P_{m}$ and $Q_{m}$ are homogeneous $m^{th}$ degree polynomials in $x$
and $y$ of the system $(\ref{p2})-(\ref{p3})$. So the critical points at
infinity are the set $\\{(X,Y,0)\mid X^{2}+Y^{2}=1\leavevmode\nobreak\
\mbox{and}\leavevmode\nobreak\ \mathcal{S}=0\\}$. Here we do not consider the
Eq.(30) of the autonomous system $(\ref{eqn29}-\ref{eqn31})$ and Eq.(33) of
the autonomous system $(\ref{equ30}-\ref{equ32})$. It is to be noted that we
get vanishing center manifold equation and vanishing flow near the point at
infinity for both of the models for the two equations (30) and (33). So it is
convenient to study the characteristics of the vector field on $xy$ plane for
the autonomous systems $(\ref{eqn29}-\ref{eqn31})$ and
$(\ref{equ30}-\ref{equ32})$. Then both of the above autonomous systems can be
converted to the following two-dimensional system

$\displaystyle\frac{dx}{dN}$ $\displaystyle=-2x-xy^{2}+2x^{3},$ (43)
$\displaystyle\frac{dy}{dN}$ $\displaystyle=y+2x^{2}y-y^{3}.$ (44)

For the autonomous system $(\ref{eqn74}-\ref{eqn75})$, we obtain the equation
(41) as

$\displaystyle XQ_{3}(X,Y)-YP_{3}(X,Y)=X(2X^{2}Y-Y^{3})-Y(-XY^{2}+2X^{3})=0.$
(45)

Hence, the critical points on the Poincaré sphere are $(X_{c},Y_{c},0)$ with
$X_{c}^{2}+Y_{c}^{2}=1$.

It is to be noted that the critical points are non-isolated in nature and the
flow in a neighborhood of any critical point $(X_{c},Y_{c},0)$ on the equator
of the Poincaré sphere $S^{2}$, except the points $(0,\pm 1,0)$, is
topologically equivalent to the flow defined by the following system

$\displaystyle\begin{split}\pm\frac{dv}{dN}&=vw^{m}P\left(\frac{1}{w},\frac{v}{w}\right)-w^{m}Q\left(\frac{1}{w},\frac{v}{w}\right)\\\
\pm\frac{dw}{dN}&=w^{m+1}P\left(\frac{1}{w},\frac{v}{w}\right)\end{split}$
(46)

where

$\displaystyle P\left(\frac{1}{w},\frac{v}{w}\right)$ $\displaystyle=$
$\displaystyle-\frac{2}{w}-\frac{v^{2}}{w^{3}}+\frac{2}{w^{3}},$
$\displaystyle Q\left(\frac{1}{w},\frac{v}{w}\right)$ $\displaystyle=$
$\displaystyle\frac{v}{w}+2\frac{v}{w^{3}}-\frac{v^{3}}{w^{3}}.$

or equivalently,

$\displaystyle\frac{dv}{dN}$ $\displaystyle=3vw^{2},$ (47)
$\displaystyle\frac{dw}{dN}$ $\displaystyle=-2w+2w^{3}+v^{2}w.$ (48)

by considering negative signs. Note that $(v_{c},0)$ is non-isolated critical
points for the above system (47). The eigenvalues of the Jacobian matrix at
$(v_{c},0)$ are $0$ and $-2+v_{c}^{2}$. For $|v_{c}|\leq 1$, we get so
$-2+v_{c}^{2}<0$ which follows that the vector field near the critical point
$(v_{c},0)$ is stable along $w$ axis. On the other hand, for $w\neq 0$ and
$v>0$, we have form equation (47) that $\frac{dv}{dN}>0$ and for $v<0$, we
have $\frac{dv}{dN}<0$. One may note that the flow comes towards the line of
critical points rapidly compare to it goes away from the critical line. So we
can conclude that except the points $(0,\pm 1,0)$, the other non-isolated
points on the equator of the Poincaré sphere are saddle node (see figure
(7(a))) in nature and cosmological bounce may happen near the points at
infinity (see section V.3).

Similarly, the flow in a neighborhood of any critical point $(X_{c},Y_{c},0)$
on the equator of the Poincaré sphere $S^{2}$, except the points $(\pm
1,0,0)$, is topologically equivalent to the flow defined by the following
system

$\displaystyle\begin{split}\pm\frac{du}{dN}&=uw^{m}Q\left(\frac{u}{w},\frac{1}{w}\right)-w^{m}P\left(\frac{u}{w},\frac{1}{w}\right)\\\
\pm\frac{dw}{dN}&=w^{m+1}Q\left(\frac{u}{w},\frac{1}{w}\right)\end{split}$
(49)

where

$\displaystyle P\left(\frac{u}{w},\frac{1}{w}\right)$ $\displaystyle=$
$\displaystyle-2\frac{u}{w}-\frac{u}{w^{3}}+2\frac{u^{3}}{w^{3}},$
$\displaystyle Q\left(\frac{u}{w},\frac{1}{w}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{w}+2\frac{u^{2}}{w^{3}}-\frac{1}{w^{3}}.$

or equivalently,

$\displaystyle\frac{du}{dN}$ $\displaystyle=3uw^{2},$ (50)
$\displaystyle\frac{dw}{dN}$ $\displaystyle=-w+2u^{2}w+w^{3}.$ (51)

by considering positive signs. Notice that $(u_{c},0)$ is non-isolated
critical points for the above system (50). The eigenvalues of the Jacobian
matrix at $(u_{c},0)$ are $0$ and $-1+2u_{c}^{2}$. Since $|u_{c}|\leq 1$ so
$-1\leq-1+2u_{c}^{2}\leq 1$ which follows that the vector field near the
critical point $(u_{c},0)$ is repelling along $w$ axis if
$\frac{1}{\sqrt{2}}<u_{c}\leq 1$ or $-1\leq u_{c}<-\frac{1}{\sqrt{2}}$ and
attracting along $w$ axis if $-\frac{1}{\sqrt{2}}<u_{c}<\frac{1}{\sqrt{2}}$.
On the other hand, for $w\neq 0$ and $u>0$, we have form equation (50) that
$\frac{du}{dN}>0$ and for $u<0$, we have $\frac{du}{dN}<0$. One may note that
the flow comes towards the line of critical points rapidly compare to it goes
away from the critical line (see figure (7(b))). So the vector field near the
$\left(\pm\frac{1}{\sqrt{2}},0\right)$ are also saddle in nature. Cosmological
bounce may take place near the points at infinity when the non-isolated
critical points on the equator of the Poincaré sphere are saddle in nature
(see section V.3).

(a)

(b)

Figure 7: Vector fields near non-isolated critical points except $(0,\pm 1,0)$
are shown in figure (a), and vector fields near non-isolated critical points
except $(\pm 1,0,0)$ are shown in figure (b) on the equator of the Poincaré
sphere.

##  V Cosmological Implication

###  V.1 Cosmological Implication for Model $1$

For the critical points $C_{1}$ and $C_{2}$, as along the eigendirection of
$x$, $x$ decreases to $0$ so $\dot{\phi}\rightarrow 0$. This implies
$\phi=const.$ and it indicates that the scalar field behaves as a cosmological
constant. Further, due to the presence of negative eigenvalue corresponding to
$y$ coordinate, along the eigendirection of $y$, $y\rightarrow\pm 1$ which
follows that $V(\phi)\rightarrow 3H^{2}=Constant$. So we have the Hubble
parameter to be constant. Thus the scale factor grows exponentially. Hence,
the critical points described the quasi-de Sitter expansion in the early
inflationary era.

Now from the stability analysis of $C_{1}$ and $C_{2}$, we have that
$z\rightarrow 0$ for $\mu>0$, $z\rightarrow\infty$ for $\mu<0$ along the
eigen-direction of $z$ which implies that $\phi\rightarrow\infty$ for $\mu>0$
and $\phi\rightarrow 0$ for $\mu<0$. Thus for $\mu<0$ the scalar field behaves
as a cosmological constant and for $\mu>0$ the observation is not
cosmologically significant.

For the critical points $C_{3}$ and $C_{4}$, as along the eigen-direction
$x\rightarrow\infty$ so $\dot{\phi}\rightarrow\infty$ and this is not
interesting from a cosmological point of view. Further due to the presence of
positive eigenvalue corresponding to $y$ coordinate along the eigendirection
of $y$, $y\rightarrow\infty$ which follows that $V(\phi)\rightarrow\infty$ and
this is not interesting from a cosmological point of view. Now from the
stability analysis of $C_{3}$ and $C_{4}$, we have $z\rightarrow\infty$ along
the eigendirection of $z$ which implies that $\phi\rightarrow 0$ and so the
scalar field behaves as the cosmological constant.

###  V.2 Cosmological Implication for Model $2$

For the critical point $P_{0}$ : along the eigen-direction of $x$ as
$t\rightarrow\infty$, $x\rightarrow 0$ which follows that
$\dot{\phi}\rightarrow 0$. This implies $\phi=const.$ and it indicates that
the scalar field behaves as a cosmological constant. Further, along the eigen-
direction of $y$ as $t\rightarrow\infty$, $y\rightarrow\infty$ which follows
that $V(\phi)\rightarrow\infty$. For the critical point $P_{1}$ : along the
eigen-direction of $x$ as $t\rightarrow\infty$, $x\rightarrow\infty$ which
follows that $\dot{\phi}\rightarrow\infty$. Further along the eigen-direction
of $y$ as $t\rightarrow\infty$, $y\rightarrow 0$ for $\nu>\sqrt{6}$ and
$y\rightarrow\infty$ for $\nu<\sqrt{6}$. It follows that $V\rightarrow 0$ for
$\nu>\sqrt{6}$ and $V\rightarrow\infty$ for $\nu<\sqrt{6}$. For the critical
point $P_{2}$ : along the eigen-direction of $x$ as $t\rightarrow\infty$,
$x\rightarrow\infty$ which follows that $\dot{\phi}\rightarrow\infty$. Further
along the eigen-direction of $y$ as $t\rightarrow\infty$, $y\rightarrow 0$ for
$\nu>-\sqrt{6}$ and $y\rightarrow\infty$ for $\nu<-\sqrt{6}$. It follows that
$V\rightarrow 0$ for $\nu>-\sqrt{6}$ and $V\rightarrow\infty$ for
$\nu<-\sqrt{6}$. These results are not interesting from a cosmological point
of view.

For the critical points $P_{3}$ and $P_{4}$ : along the eigen-direction of $x$
as $t\rightarrow\infty$, $x\rightarrow\frac{\nu}{\sqrt{6}}$ if
$|\nu|<\sqrt{2}$ and $x\rightarrow\infty$ if $\nu>\sqrt{2}$ or
$\nu<-\sqrt{2}$. It follows that as $t\rightarrow\infty$,
$\dot{\phi}\rightarrow\nu
H\implies\phi\rightarrow\nu\lim\limits_{t\rightarrow\infty}\ln
a(t)\implies\phi\rightarrow\infty$ for $|\nu|<\sqrt{2}$ and also
$\dot{\phi}\rightarrow\infty$ if $\nu>\sqrt{2}$ or $\nu<-\sqrt{2}$. As the
critical points $P_{3}$ and $P_{4}$ both exist for $0\leq\nu^{2}\leq 6$, so as
$t\rightarrow\infty$, $y\rightarrow\pm\sqrt{1-\frac{\nu^{2}}{6}}$ which
follows that $V\rightarrow 3H^{2}\left(1-\frac{\nu^{2}}{6}\right)$, that is,
for $\nu=0$ it leads to an accelerated era of evolution. Further, for the
critical points $P_{5}$ and $P_{6}$ : along the eigen-directions of $x$ and
$y$ as $t\rightarrow\infty$, $x\rightarrow\frac{1}{\nu}\sqrt{\frac{2}{3}}$ and
$y\rightarrow\frac{2}{\sqrt{3}\nu}$ respectively while
$|\nu|>\frac{2\sqrt{2}}{\sqrt{3}}$ which follows that as $t\rightarrow\infty$,
$\dot{\phi}\rightarrow\frac{2}{\nu}$ and $V\rightarrow\frac{4H^{2}}{\nu^{2}}$
while $|\nu|>\frac{2\sqrt{2}}{\sqrt{3}}$. Further for
$\nu\in\left(-\sqrt{2},0\right)\cup\left(0,\sqrt{2}\right)$ as
$t\rightarrow\infty$, $x\rightarrow\infty$ and
$y\rightarrow\frac{2}{\sqrt{3}\nu}$ which follows that as
$t\rightarrow\infty$, $\dot{\phi}\rightarrow\infty$ and
$V\rightarrow\frac{4H^{2}}{\nu^{2}}$ while
$\nu\in\left(-\sqrt{2},0\right)\cup\left(0,\sqrt{2}\right)$. It leads to an
accelerated phase of the universe for $\nu=\pm\frac{2}{\sqrt{3}}$.

(a)

(b)

Figure 8: The figures show the time evolution of cosmological parameters of
our cosmological model. In panel (a), Power potential: Late time solutions of
cosmological parameters. For $\mu=1$, the kinetic energy dominated late time
accelerated solutions are attracted towards the phantom boundary. In panel
(b), Exponential potential: Late time solutions of cosmological parameters.
For $\nu^{2}<2$, the kinetic energy dominated late time accelerated solutions
are attracted towards the quintessence era which agrees with the present
observed accelerated universe $(-0.56<q<-0.49)$.

###  V.3 Cosmological Bouncing Scenarios

Generally speaking, the cosmological bounce takes place for $\dot{a}\approx 0$
and $\ddot{a}>0$ or $\ddot{a}<0$. On the other hand, one can derive that
$\ddot{a}>0$ or $\ddot{a}<0$ iff $\rho_{\phi}+3p_{\phi}<0$ or
$\rho_{\phi}+3p_{\phi}>0$ respectively (see Proposition (1) in Appendix). For
$\ddot{a}>0$ the universe experiences an accelerating phase and for
$\ddot{a}<0$ the universe experiences decelerating phase of evolution. Now for
$\dot{a}\approx 0$ we have $\dot{{H}}\approx\frac{\ddot{a}}{a}$. So, in terms
of Hubble parameter $H$, we can say that the bounce takes place when $H\approx
0$ and $\dot{H}>\leavevmode\nobreak\ \text{or}\leavevmode\nobreak\ <0$. We
also note that $\ddot{a}>\leavevmode\nobreak\ \text{or}\leavevmode\nobreak\
<0$ iff $y^{2}>2x^{2}$ or $y^{2}<2x^{2}$ respectively (see Proposition (2) in
the Appendix and Table (5)).

$\bullet$ For model 1, as mentioned previously, near the critical points
$C_{1}-C_{4}$ the solutions are completely dominated by scalar field and
$\dot{{H}}\approx 0$ due to the condition $|\Omega_{\mathcal{K}}|\approx 0$.
So bounce does not take place if the initial conditions are given near these
points. Nevertheless, near $C_{3}$ and $C_{4}$ (unstable in nature), the
spatial curvature is negligible, and $\dot{{H}}$ is negative. As expected, we
also note that $y^{2}<2x^{2}$ near these critical points and $\ddot{a}<0$ (see
Table (5)) for all values of $\mu$. So, if the Hubble parameter takes positive
sign (expanding universe) near these critical points, the expansion rate
begins to decrease (as $\ddot{a}<0$) and the trajectory goes towards the
critical points at infinity (Minkowski limit with $H\rightarrow 0$) in the
compactified phase space (see section (IV)). The trajectory soon commences to
go away from those critical points (at infinity) which are saddle/saddle node
in nature and $H$ becomes soon negative, i.e., the universe begins to
contract.

On the other hand, near the critical point $C_{5}$, $\dot{H}$ is positive for
positive spatial curvature. So, if the Hubble parameter takes negative sign
(contracting universe) near these critical points, the expansion rate begins
to increase and the trajectory goes towards the critical points at infinity
(Minkowski limit with $H\rightarrow 0$) in the compactified phase space (see
section (IV)). The trajectory soon commences to go away from those critical
points (at infinity) which are saddle/saddle nodes in nature and contrary to
the previous instance, $H$ becomes soon positive, i.e., the universe begins to
expand. The cosmological bounce takes place for $\mu>0$ when the Hubble
parameter $H$ changes its sign and the trajectory asymptotically goes towards
$C_{1}$ or $C_{2}$ (see table 5).

So on the proviso that the bounce takes place, there may appear homoclinic and
heteroclinic orbits due to the existence of saddle points. So, there are
several prospects to acquire various late-time solutions which depend on the
initial condition; for instance, we acquire scalar field dominated time de-
Sitter solution to the evolution of the universe if the trajectory
asymptotically goes towards $C_{1}$ or $C_{2}$ (see figure (3)) on the $XY$
plane, late time decelerating dust-dominated universe if the trajectory
asymptotically goes towards $C_{5}$ along stable eigen direction.

$\bullet$ For model 2, we have discussed earlier that near $P_{0}$ (saddle for
all $\nu$) the kinetic term is subdominant compared to the curvature
contribution. If initially the universe is in the contracting phase then near
$P_{0}$ the expansion rate $H$ increases for positive spatial curvature.
Depending on the initial condition if the flow goes away from $P_{0}$ towards
infinity (as $H\rightarrow 0$) where the Hubble parameter changes its sign
(when critical points at infinity are saddle/saddle nodes in nature) then the
universe begins to expand. The expansion rate decreases soon when the spatial
curvature contribution becomes negligible.

On the other hand, near $P_{1}$ and $P_{2}$ (saddle-node / unstable node), the
solutions are dominated by the kinetic energy of the scalar field and
$\dot{{H}}$ is negative. As expected, it is also to be noted that
$y^{2}<2x^{2}$ near these critical points and $\ddot{a}<0$ (see Table (5)) for
all values of $\nu$. Similar to the case of $C_{3}$ and $C_{4}$ in Model 1,
the cosmological bounce takes place when the Hubble parameter changes its sign
at Minkowski limit with $H\rightarrow 0$, i.e., at critical points (at
infinity) which are saddle/saddle-node in nature.

The non-isolated critical points $P_{3}$ and $P_{4}$ are completely dominated
by kinetic energy and saddle-node in nature for $2\leq\nu^{2}\leq 6$ and
$\dot{{H}}$ is negative near these points. Unsurprisingly, $y^{2}<2x^{2}$ near
these critical points and $\ddot{a}<0$ (see Table (5)) for $2<\nu^{2}\leq 6$.
So depending on the initial condition, the cosmological bounce may take place
if the flow starts near these critical points. The non-isolated critical
points $P_{5}$ and $P_{6}$ are saddle in nature for $\nu^{2}\leq 2$. If
$\nu\rightarrow 0$, then the critical points are dominated by spatial
curvature and the sign of $\dot{{H}}$ depends on the sign of $\mathcal{K}$. If
$\nu^{2}\approx 2$, then the critical points are completely dominated by
kinetic energy and $\dot{{H}}<0$. Similar to the previous case, the initial
condition plays an important role in initiating the cosmological bounce.

Similar to model 1 when it comes to the case of cosmological bounce, there may
appear homoclinic and heteroclinic orbits due to the existence of saddle
points. So, there are several prospects to acquire various late-time solutions
which depend on the initial condition; for instance, we come by late time
decelerating dust-dominated universe if trajectory asymptotically approaches
towards $P_{0}$, late-time kinetic energy dominated solutions with a stiff
equation of state if trajectory asymptotically approaches towards $P_{1}$ or
$P_{2}$, late time kinetic energy dominated accelerated solution as trajectory
asymptotically approaches towards $P_{3}$ or $P_{4}$, late time scaling
solutions where DE behaves as quintessence boundary if trajectory
asymptotically approaches towards $P_{5}$ or $P_{6}$.

Table 5: Table shows the sign of ‘$\ddot{a}$’ near the critical points of
Model 1 and 2 in terms of dynamical variables (for details, see Proposition 2
in Appendix)

.  Model Critical points $y^{2}>2x^{2}\leavevmode\nobreak\
i.e.\leavevmode\nobreak\ \ddot{a}>0$ $y^{2}<2x^{2}\leavevmode\nobreak\
i.e.\leavevmode\nobreak\ \ddot{a}<0$ $C_{1}$ Satisfied for all $\mu\neq 0$ Not
applicable $C_{2}$ Satisfied for all $\mu\neq 0$ Not applicable Model $1$
$C_{3}$ Not applicable Satisfied for all $\mu$ $C_{4}$ Not applicable
Satisfied for all $\mu$ $C_{5}$ Not applicable Not applicable $P_{0}$ Not
applicable Not applicable $P_{1}$ Not applicable Satisfied for all $\nu$
$P_{2}$ Not applicable Satisfied for all $\nu$ Model $2$ $P_{3}$ Satisfied for
$0\leq\nu^{2}<{2}$ Satisfied for $2<\nu^{2}\leq{6}$ $P_{4}$ Satisfied for
$0\leq\nu^{2}<{2}$ Satisfied for $2<\nu^{2}\leq{6}$ $P_{5},P_{6}$ Not
applicable Not applicable

##  VI Brief discussion and concluding remarks

We have studied the cosmological model with spatial curvature in the
background of FLRW metric where the scalar field is minimally coupled and
plays a role of dark matter content. Two models have been discussed here. In
the first model, potential $V(\phi)$ of scalar field is taken as power-law
function of scalar field $\phi$. Whereas in the second interaction model, the
potential term is taken as the exponential function of scalar field $\phi$.
Since the cosmological evolution equations are non-linear and complicated in
nature, we have performed dynamical system analysis to accomplish the
qualitative behavior of the cosmological model. We have explained all possible
scenarios of the bouncing universe and the corresponding late-time solutions.

We have obtained five non-hyperbolic type critical points from model 1 and
from model 2 we have obtained seven critical points which are characterized
(hyperbolic and non-hyperbolic) by the parameter $\nu$. To study the nature of
hyperbolic critical points we apply linear stability theory (Hartman-Grobman
theorem). On the other hand, to analyze non-hyperbolic points center manifold
theory is employed to obtain the exact dynamical nature of the points on the
sets. Since at the bouncing scenarios, some variables become singular, we have
performed the dynamical analysis around the critical points at infinity in
section IV. At infinity, the critical points are normally hyperbolic and
saddle/saddle-node in nature (see figure (7)). Model 1 can not alleviate the
cosmic coincidence problem whereas model 2 can take the edge off the cosmic
coincidence problem depending on the parameter $\nu$.

In Model $1$, there are five non-hyperbolic equilibrium points. The first two
equilibrium points $C_{1}$ and $C_{2}$ describe the flat model of the universe
with a cosmological constant and they describe the phantom barrier of
accelerated expansion. The point $C_{1}$ represents an expanding model while
$C_{2}$ corresponds to a contracting model of the universe. The other three
equilibrium points $C_{3}$, $C_{4}$, $C_{5}$ represent a decelerating phase of
expansion. From a cosmological point of view only equilibrium point $C_{5}$ is
interesting, it describes a dust era of evolution. Further, the critical
points $C_{1}$ and $C_{2}$ are purely DE dominant $(\Omega_{\phi}=1)$ and they
are analogous to the observationally favored $\Lambda$CDM model. The critical
points $C_{3}$ and $C_{4}$ are not of much interest. Here the result matter
behaves as stiff fluid which is effective at the very early era when quantum
effects are important. The last critical point $C_{5}$ has no effect of DE
$(\Omega_{\phi}=0,\leavevmode\nobreak\ \Omega_{\mathcal{K}}=1)$ and the
resulting fluid behaves as dust.

In Model $2$, there are seven equilibrium points all of which are non-
hyperbolic in nature. The dust era of decelerated expansion is characterized
by the equilibrium point $P_{0}$. The equilibrium points $P_{1}$ and $P_{2}$
are not interesting from a cosmological point of view. The other four
equilibrium points $P_{3}$, $P_{4}$, $P_{5}$, and $P_{6}$ represent scaling
cosmological solutions. The equilibrium points $P_{3}$ and $P_{4}$ describe a
flat FLRW model with an accelerating phase of expansion for $\nu^{2}<2$ and
the scalar field describes a dark energy model. The equilibrium points $P_{5}$
and $P_{6}$ also describe the decelerating era of expansion for $\nu^{2}>2$
and here curvature contribution of the matter is non-zero and the scalar field
behaves as perfect fluid at the quintessence barrier. Further, if $\nu>2$ then
curvature matter contribution will dominate over the scalar field while if
$\sqrt{2}<\nu<2$ then the scalar field is the dominant matter component.
Moreover, from cosmological view point the critical point $P_{0}$ describes a
dust era of evolution having no effect of DE $(\Omega_{\phi}=0)$. Due to
unacceptable cosmological parameters the critical points $P_{1}$ and $P_{2}$
may be discarded. The critical points $P_{3}$ and $P_{4}$ are completely
dominated by DE $(\Omega_{\phi}=1)$. For $0\leq\nu^{2}\leq 2$ the present
model describes the late time accelerated phase (as predicted by observational
data) and $\nu=0$ describes the observationally supported $\Lambda$CDM model
(as expected $V(\phi)$ becomes $V_{0}$, a constant), while for $2<\nu^{2}\leq
3$ the model goes back to the decelerated era of evolution with $\nu^{2}=3$
represents the dust epoch. Lastly the critical points $P_{5}$ and $P_{6}$ have
similar behavior as $P_{3}$ and $P_{4}$ and they describe the $\Lambda$CDM
model with the choice $\nu^{2}=2/3$. Thus it is possible to have an analogy of
the present models with the well-known $\Lambda$CDM cosmology. Moreover, if
$\nu^{2}<2/3$ then the critical points $P_{5}$ and $P_{6}$ may go beyond
$\Lambda$CDM and they may be related to other tensions in cosmology. Though
both the potential models are cosmologically viable yet model $2$ has an edge
over model $1$. In model $2$, there are scaling solutions (corresponding to
critical points $P_{3}-P_{6}$) indicating the possibility of decelerating era
of expansion to the present accelerated expansion phase. However, such type of
expansion is not possible for power law of potential.

Thus, in summary, the present cosmological model based on the spatial
curvature with power law and exponential potential may describe different
evolutionary phases of the universe for which bounce takes place. To study the
bouncing universe we need to analyze the behavior of the vector field at the
Minkowski limit with $H\rightarrow 0$, i.e., at critical points (at infinity)
which are saddle/saddle-node in nature. Depending on the initial condition,
either the transition comes to pass from expansion to contraction and vice
versa, from contraction to expansion of the universe. Although, so far, there
is neither any observational evidence that substantiates the bouncing nature
of the Universe nor any evidence of contracting phase of evolution.

The present model describes a change in the universe’s expansion rate (i.e.,
from expansion to contraction) so it may be possible to have an explanation or
it may alleviate the standard $H_{0}$-tension. The uncoupled scalar field
models do not really increase the $H_{0}$ value [7, 45]. Though a coupled
scalar field model can reduce the $H_{0}$ tension, a proper choice of the
coupling function between the scalar field and dark matter [46], it is
possible to increase the $H_{0}$ value compared to the $\Lambda$CDM based
Plank’s estimation [7]. As a result, the obtained value of $H_{0}$ could be
closer to the SH0ES value [47]. One may note that the inclusion of curvature
in the cosmological models does not offer any solution to the Hubble constant
tension, rather the tension on $H_{0}$ increases. Therefore, even if the
consideration of the curvature of the Universe gives a complete picture of the
underlying cosmological model, however, the increasing tension in $H_{0}$
needs special attention with the growing sensitivity in the upcoming
astronomical probes. In addition, two other additional tensions related to
$H_{0}$-tension in cosmology, namely the $\sigma_{8}$ tension and the galaxy
rotation curves problem may have some issues which we shall study in a future
work.

###### Acknowledgements.

The authors thank the anonymous referees whose comments and suggestions
improved the quality of the paper. Soumya Chakraborty thanks CSIR, Govt. of
India for providing Senior Research Fellowship (CSIR Award No:
09/096(1009)/2020-EMR-I) for the Ph.D. work.

Authors’ comment: This article describes entirely theoretical research. So
data sharing is not applicable to this article as no datasets were generated
or analyzed during the current study.

Conflict of Interest: The authors declare that they have no conflict of
interest.

##  VII Appendix

###### Proposition 1

If $\rho_{\phi}$ is the energy density and $p_{\phi}$ is the pressure of the
scalar field (given by equations (5) and (6)) then the second order derivative
of the scale factor $a(t)$ with respect to the cosmic time $t$, that is,
$\ddot{a}>0$ or $\ddot{a}<0$ iff $\rho_{\phi}+3p_{\phi}<0$ or
$\rho_{\phi}+3p_{\phi}>0$ respectively.

Proof: The expression of Hubble parameter $H(t)$ in terms of scale factor
$a(t)$ is defined by

$H=\frac{\dot{a}}{a}.$

Taking differentiation on both sides of the above with respect to $t$ yields

$\dot{H}=\frac{\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}.$

Substituting $H=\frac{\dot{a}}{a}$ in left hand side of Eq.(8) yields

$\displaystyle 3\frac{\dot{a}^{2}}{a^{2}}$
$\displaystyle=\frac{1}{2}\dot{\phi}^{2}+V(\phi)-3\frac{\mathcal{K}}{a^{2}},$
$\displaystyle\frac{\dot{a}^{2}}{a^{2}}$
$\displaystyle=\frac{1}{3}\left(\frac{1}{2}\dot{\phi}^{2}+V(\phi)-3\frac{\mathcal{K}}{a^{2}}\right).$
(52)

Plugging the values of $\dot{H}$ in left hand side of Eq.(9) and using (52)
yields

$\displaystyle 2\left(\frac{\ddot{a}}{a}-\frac{\dot{a}^{2}}{a^{2}}\right)$
$\displaystyle=-\dot{\phi}^{2}+2\frac{\mathcal{K}}{a^{2}},$ $\displaystyle
2\frac{\ddot{a}}{a}$
$\displaystyle=2\frac{\dot{a}^{2}}{a^{2}}-\dot{\phi}^{2}+2\frac{\mathcal{K}}{a^{2}},$
$\displaystyle 2\frac{\ddot{a}}{a}$
$\displaystyle=\frac{2}{3}\left(\frac{1}{2}\dot{\phi}^{2}+V(\phi)-3\frac{\mathcal{K}}{a^{2}}\right)-\dot{\phi}^{2}+2\frac{\mathcal{K}}{a^{2}},$
$\displaystyle 2\frac{\ddot{a}}{a}$
$\displaystyle=\frac{1}{3}\dot{\phi}^{2}+\frac{2}{3}V(\phi)-\dot{\phi}^{2},$
$\displaystyle\frac{\ddot{a}}{a}$
$\displaystyle=-\frac{1}{3}(\dot{\phi}^{2}-V(\phi)).$ (53)

From equations (5) and (6), we have

$\displaystyle\rho_{\phi}$ $\displaystyle=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$
$\displaystyle p_{\phi}$ $\displaystyle=\frac{1}{2}\dot{\phi}^{2}-V(\phi).$

Note that

$\rho_{\phi}+3p_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi)+\frac{3}{2}\dot{\phi}^{2}-3V(\phi)=2(\dot{\phi}^{2}-V(\phi))$

which implies

$\dot{\phi}^{2}-V(\phi)=\frac{1}{2}(\rho_{\phi}+3p_{\phi}).$

Using the above result on the right-hand side of (53) yields

$\frac{\ddot{a}}{a}=-\frac{1}{6}(\rho_{\phi}+3p_{\phi}).$ (54)

Now one may note from the equation (54) that $\ddot{a}>0$ iff

$\displaystyle-\frac{1}{6}(\rho_{\phi}+3p_{\phi})$ $\displaystyle>0,$
$\displaystyle\rho_{\phi}+3p_{\phi}$ $\displaystyle<0.$

and $\ddot{a}<0$, iff

$\displaystyle\rho_{\phi}+3p_{\phi}$ $\displaystyle>0.$

This completes the proof.

###### Proposition 2

The second order derivative of the scale factor $a(t)$ with respect to the
cosmic time $t$, that is, $\ddot{a}>0$ or $\ddot{a}<0$ iff $y^{2}>2x^{2}$ or
$y^{2}<2x^{2}$ respectively where $x$ and $y$ are the dynamical variables
given by the equations (11) and (12).

Proof: In the previous proposition, we have shown that at the point of
bouncing condition

$\rho_{\phi}+3p_{\phi}<0\leavevmode\nobreak\ \text{or}\leavevmode\nobreak\
\rho_{\phi}+3p_{\phi}>0.$

From the equations (5) and (6), we have

$\displaystyle\rho_{\phi}$ $\displaystyle=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$
$\displaystyle p_{\phi}$ $\displaystyle=\frac{1}{2}\dot{\phi}^{2}-V(\phi).$

Plugging the expressions of $\rho_{\phi}$ and $p_{\phi}$ into the first
bouncing condition (i.e., $\rho_{\phi}+3p_{\phi}<0$) yields

$\displaystyle\frac{1}{2}\dot{\phi}^{2}+V(\phi)+3\left(\frac{1}{2}\dot{\phi}^{2}-V(\phi)\right)$
$\displaystyle<0,$ $\displaystyle 2\dot{\phi}^{2}-2V(\phi)$ $\displaystyle<0,$
$\displaystyle\dot{\phi}^{2}$ $\displaystyle<V(\phi).$ (55)

In terms of dynamical variables, from equations (11) and (12), we have

$\displaystyle\dot{\phi}$ $\displaystyle=\sqrt{6}Hx,$ $\displaystyle V(\phi)$
$\displaystyle=3H^{2}y^{2}.$

Substituting these into (55) yields

$\displaystyle(\sqrt{6}Hx)^{2}$ $\displaystyle<3H^{2}y^{2},$ $\displaystyle
6H^{2}x^{2}$ $\displaystyle<3H^{2}y^{2},$ $\displaystyle y^{2}$
$\displaystyle>2x^{2}.$

Similarly, by substituting the values of $p_{\phi}$ and $\rho_{\phi}$ from the
equations (5) and (6) into the second bouncing condition (i.e.,
$\rho_{\phi}+3p_{\phi}>0$), we can show that $y^{2}<2x^{2}$. This completes
the proof.

###### Proposition 3

Equation (18) can be obtained by using equations (15), (16) and the constraint
equation (24).

Proof: By taking differentiation both sides of (24) with respect to $N$ yields

$\displaystyle 2x\frac{dx}{dN}+2y\frac{dy}{dN}-6\mathcal{K}u\frac{du}{dN}=0,$
$\displaystyle\frac{du}{dN}=\frac{1}{3\mathcal{K}u}\left\\{x\frac{dx}{dN}+y\frac{dy}{dN}\right\\}.$

Now by substituting the equations (15) and (16) in the right hand side of
above yields

$\displaystyle\frac{du}{dN}$
$\displaystyle=\frac{1}{3\mathcal{K}u}\left\\{-3x^{2}-\sqrt{\frac{3}{2}}\left(\frac{V^{\prime}}{V}\right)xy^{2}+3x^{4}-3\mathcal{K}u^{2}x^{2}+3y^{2}x^{2}-3\mathcal{K}u^{2}y^{2}+\sqrt{\frac{3}{2}}\left(\frac{V^{\prime}}{V}\right)xy^{2}\right\\}$
$\displaystyle=\frac{1}{3\mathcal{K}u}\\{-3\mathcal{K}u^{2}(x^{2}+y^{2})+3x^{2}(y^{2}+x^{2}-1)\\}.$

From the equation (24), we have

$x^{2}+y^{2}-1=3\mathcal{K}u^{2}.$

Plugging this into the second term of the right-hand side of the last
expression yields

$\displaystyle\frac{du}{dN}$
$\displaystyle=\frac{1}{3\mathcal{K}u}\\{9\mathcal{K}u^{2}x^{2}-3\mathcal{K}u^{2}(x^{2}+y^{2})\\}$
$\displaystyle=\frac{1}{3\mathcal{K}u}\\{3\mathcal{K}u^{2}(3x^{2}-x^{2}-y^{2})\\}$
$\displaystyle=u(2x^{2}-y^{2}).$

This completes the proof.

###### Proposition 4

The expressions of the center manifold corresponding to the critical point
$C_{1}$ are given by $u=0$ and
$v=-\frac{\mu^{2}}{72}w^{2}+\mathcal{O}(w^{4})$, and the flow on the center
manifold is determined by
$\frac{dw}{dN}=-\frac{\mu}{6}w^{3}+\mathcal{O}(w^{4})$.

Proof: The Jacobian matrix at the critical point $C_{1}$ can be put as

$J(C_{1})=\begin{bmatrix}-3&0&\frac{\mu}{2}\\\ 0&-2&0\\\ 0&0&0\end{bmatrix}.$
(56)

The eigenvalues of the above matrix are $-3$, $-2$ and $0$ with $[1,0,0]^{T}$,
$[0,1,0]^{T}$ and $\left[\frac{\mu}{6},0,1\right]^{T}$ are the corresponding
eigenvectors respectively. To apply center manifold theory, first, we
transform the coordinates into a new system $x=X,\leavevmode\nobreak\
y=Y+1,\leavevmode\nobreak\ z=Z$, such that the critical point $C_{1}$ moves to
the origin. By using the eigenvectors of the Jacobian matrix $J(C_{1})$, we
introduce another set of new coordinates $(u,\leavevmode\nobreak\
v,\leavevmode\nobreak\ w)$ in terms of $(X,\leavevmode\nobreak\
Y,\leavevmode\nobreak\ Z)$ as

$\begin{bmatrix}u\\\ v\\\ w\end{bmatrix}=\begin{bmatrix}1&0&-\frac{\mu}{6}\\\
0&1&0\\\ 0&0&1\end{bmatrix}\begin{bmatrix}X\\\ Y\\\ Z\end{bmatrix}$ (57)

and in these new coordinate system, the equations $(\ref{eqn29}-\ref{eqn31})$
are transformed into

$\begin{bmatrix}\frac{du}{dN}\\\ \frac{dv}{dN}\\\
\frac{dw}{dN}\end{bmatrix}=\begin{bmatrix}-3&\leavevmode\nobreak\
\leavevmode\nobreak\ 0&0\\\ \leavevmode\nobreak\ \leavevmode\nobreak\
0&-2&0\\\ \leavevmode\nobreak\ \leavevmode\nobreak\ 0&\leavevmode\nobreak\
\leavevmode\nobreak\ 0&0\end{bmatrix}\begin{bmatrix}u\\\ v\\\
w\end{bmatrix}+\begin{bmatrix}non\\\ linear\\\ terms\end{bmatrix}.$ (58)

By center manifold theory there exists a continuously differentiable function
$h:\mathbb{R}$$\rightarrow$$\mathbb{R}^{2}$ such that

$\displaystyle h(w)=\begin{bmatrix}u\\\ v\\\
\end{bmatrix}=\begin{bmatrix}a_{1}w^{2}+a_{2}w^{3}+\mathcal{O}(w^{4})\\\
b_{1}w^{2}+b_{2}w^{3}+\mathcal{O}(w^{4})\end{bmatrix}.$ (59)

Differentiating both sides with respect to $N$, we get

$\displaystyle\frac{du}{dN}$ $\displaystyle=$
$\displaystyle(2a_{1}w+3a_{2}w^{2})\frac{dw}{dN}+\mathcal{O}(w^{3})$ (60)
$\displaystyle\frac{dv}{dN}$ $\displaystyle=$
$\displaystyle(2b_{1}w+3b_{2}w^{2})\frac{dw}{dN}+\mathcal{O}(w^{3})$ (61)

where $a_{i}$, $b_{i}$ $\in\mathbb{R}$. We are only concerned about the non-
zero coefficients of the lowest power terms in CMT as we analyze the stability
in an arbitrarily small neighborhood of the origin. Comparing coefficients
corresponding to power of $w$ both sides of (60) and (61), we get $a_{i}=0$
and $b_{1}=\frac{\mu^{2}}{72}$, $b_{2}=0$. So, the center manifold can be
written as

$\displaystyle u$ $\displaystyle=$ $\displaystyle 0,$ (62) $\displaystyle v$
$\displaystyle=$ $\displaystyle-\frac{\mu^{2}}{72}w^{2}+\mathcal{O}(w^{4})$
(63)

and the flow on the center manifold is determined by

$\displaystyle\frac{dw}{dN}$ $\displaystyle=$
$\displaystyle-\frac{\mu}{6}w^{3}+\mathcal{O}(w^{4}).$ (64)

## References

  * [1] Fumiya Okamatsu, Toyokazu Sekiguchi, and Tomo Takahashi.  ${H}_{0}$ tension without cmb data: Beyond the $\mathrm{\Lambda}\mathrm{CDM}$.  Phys. Rev. D, 104:023523, Jul 2021.
  * [2] Antonino Del Popolo and Morgan Le Delliou.  Small scale problems of the $\lambda$cdm model: A short review.  Galaxies, 5(1), 2017.
  * [3] H. A. P. Macedo, L.S. Brito, J.F. Jesus, and M.E.S. Alves.  Cosmological constraints on $\lambda(t)cdm$ models.  The European Physical Journal C, 83, 16 December 2023.
  * [4] Mariusz P. Dabrowski.  Oscillating Friedman cosmology.  Annals Phys., 248:199–219, 1996.
  * [5] Peter W. Graham, Bart Horn, Shamit Kachru, Surjeet Rajendran, and Gonzalo Torroba.  A Simple Harmonic Universe.  JHEP, 02:029, 2014.
  * [6] Peter W. Graham, Bart Horn, Surjeet Rajendran, and Gonzalo Torroba.  Exploring eternal stability with the simple harmonic universe.  JHEP, 08:163, 2014.
  * [7] N. Aghanim et al.  Planck 2018 results. VI. Cosmological parameters.  Astron. Astrophys., 641:A6, 2020.  [Erratum: Astron.Astrophys. 652, C4 (2021)].
  * [8] Alexei A. Starobinsky.  Future and origin of our universe: Modern view.  Grav. Cosmol., 6:157–163, 2000.
  * [9] A. A. Starobinskii.  On a nonsingular isotropic cosmological model.  Soviet Astronomy Letters, 4:82–84, February 1978.
  * [10] Pierre-Philippe Dechant, Anthony N. Lasenby, and Michael P. Hobson.  Anisotropic, nonsingular early universe model leading to a realistic cosmology.  Phys. Rev. D, 79:043524, Feb 2009.
  * [11] Genly León Torres.  Qualitative analysis and characterization of two cosmologies including scalar fields.  PhD thesis, Marta Abreu Central U., Cuba, 2010.
  * [12] Hiroki Matsui, Fuminobu Takahashi, and Takahiro Terada.  Non-singular bouncing cosmology with positive spatial curvature and flat scalar potential.  Physics Letters B, 795:152–159, 2019.
  * [13] Ujjaini Alam, Varun Sahni, and A. A. Starobinsky.  The Case for dynamical dark energy revisited.  JCAP, 06:008, 2004.
  * [14] D. Battefeld and Patrick Peter.  A Critical Review of Classical Bouncing Cosmologies.  Phys. Rept., 571:1–66, 2015.
  * [15] Robert Brandenberger and Patrick Peter.  Bouncing Cosmologies: Progress and Problems.  Found. Phys., 47(6):797–850, 2017.
  * [16] Anna Ijjas and Paul J. Steinhardt.  Bouncing Cosmology made simple.  Class. Quant. Grav., 35(13):135004, 2018.
  * [17] J. K. Singh, Kazuharu Bamba, Ritika Nagpal, and S. K. J. Pacif.  Bouncing cosmology in $f(R,T)$ gravity.  Phys. Rev. D, 97(12):123536, 2018.
  * [18] A. S. Agrawal, Francisco Tello-Ortiz, B. Mishra, and S. K. Tripathy.  Bouncing Cosmology in Extended Gravity and Its Reconstruction as Dark Energy Model.  Fortsch. Phys., 70(1):2100065, 2022.
  * [19] Pritha Bari and Kaushik Bhattacharya.  Evolution of scalar and vector cosmological perturbations through a bounce in metric f(r) gravity in flat FLRW spacetime.  Journal of Cosmology and Astroparticle Physics, 2019(11):019–019, nov 2019.
  * [20] Adam Moss, Douglas Scott, and James P. Zibin.  No evidence for anomalously low variance circles on the sky.  Journal of Cosmology and Astroparticle Physics, 2011(04):033, apr 2011.
  * [21] Dylan L. Jow and Douglas Scott.  Re-evaluating evidence for hawking points in the cmb.  Journal of Cosmology and Astroparticle Physics, 2020(03):021, mar 2020.
  * [22] Sebastian Bahamonde, Christian G. Böhmer, Sante Carloni, Edmund J. Copeland, Wei Fang, and Nicola Tamanini.  Dynamical systems applied to cosmology: dark energy and modified gravity.  Phys. Rept., 775-777:1–122, 2018.
  * [23] Jibitesh Dutta and Subenoy Chakraborty.  Generalized second law of thermodynamics for interacting dark energy in the DGP braneworld.  Int. J. Theor. Phys., 50:2383–2390, 2011.
  * [24] A. Savaş Arapoğlu, Ezgi Yalçınkaya, and A. Emrah Yükselci.  Dynamical System Analysis of a Five-Dimensional Cosmological Model.  Astrophys. Space Sci., 363(10):215, 2018.
  * [25] Bharat Ratra and P. J. E. Peebles.  Cosmological consequences of a rolling homogeneous scalar field.  Phys. Rev. D, 37:3406–3427, Jun 1988.
  * [26] Scott Dodelson, Manoj Kaplinghat, and Ewan Stewart.  Solving the Coincidence Problem : Tracking Oscillating Energy.  Phys. Rev. Lett., 85:5276–5279, 2000.
  * [27] L. Arturo Urena-Lopez and Tonatiuh Matos.  A New cosmological tracker solution for quintessence.  Phys. Rev. D, 62:081302, 2000.
  * [28] Philippe Brax and Jerome Martin.  Quintessence and supergravity.  Phys. Lett. B, 468:40–45, 1999.
  * [29] Paul J. Steinhardt, Li-Min Wang, and Ivaylo Zlatev.  Cosmological tracking solutions.  Phys. Rev. D, 59:123504, 1999.
  * [30] Joshua A. Frieman, Christopher T. Hill, Albert Stebbins, and Ioav Waga.  Cosmology with ultralight pseudo Nambu-Goldstone bosons.  Phys. Rev. Lett., 75:2077–2080, 1995.
  * [31] Sudip Mishra and Subenoy Chakraborty.  Stability and bifurcation analysis of interacting f(T) cosmology.  Eur. Phys. J., C79(4):328, 2019.
  * [32] Savas Arapoglu and A. Emrah Yükselci.  Dynamical system analysis of quintessence models with exponential potential - revisited.  Modern Physics Letters A, 34, 11 2017.
  * [33] Nandan Roy and Narayan Banerjee.  Quintessence scalar field: A dynamical systems study.  The European Physical Journal Plus, 129, 02 2014.
  * [34] S. Surendra Singh and Chingtham Sonia.  Dynamical system perspective of cosmological models minimally coupled with scalar field.  Adv. High Energy Phys., 2020:1805350, 2020.
  * [35] Soumya Chakraborty, Sudip Mishra, and Subenoy Chakraborty.  A dynamical system analysis of cosmic evolution with coupled phantom dark energy with dark matter.  International Journal of Modern Physics D, 10 2021.
  * [36] Goutam Mandal, Soumya Chakraborty, Sudip Mishra, and Sujay Kr. Biswas.  A study of interacting scalar field model from the perspective of the dynamical systems theory.  Phys. Dark Univ., 40:101210, 2023.
  * [37] Sudip Mishra.  Dynamics around nonhyperbolic equilibrium and bifurcation analysis of various cosmological models.  PhD thesis, Jadavpur University, Department of Mathematics, India, 2021\.
  * [38] Varun Sahni.  Dark matter and dark energy.  Lect. Notes Phys., 653:141–180, 2004.
  * [39] Lawrence Perko.  Differential Equations and Dynamical Systems.  Springer-Verlag, Berlin, Heidelberg, 1991.
  * [40] Trupti Patil, Sukanta Panda, Manabendra Sharma, and Ruchika.  Dynamics of interacting scalar field model in the realm of chiral cosmology.  Eur. Phys. J. C, 83(2):131, 2023.
  * [41] Mariam Bouhmadi-López, João Marto, João Morais, and César M. Silva.  Cosmic infinity: A dynamical system approach.  JCAP, 03:042, 2017.
  * [42] Harry Gingold.  Approximation of unbounded functions via compactification.  Journal of Approximation Theory, 131(2):284–305, 2004.
  * [43] Genly Leon and Felipe Orlando Franz Silva.  Generalized scalar field cosmologies: a global dynamical systems formulation.  Class. Quant. Grav., 38(1):015004, 2021.
  * [44] Uri Elias and Harry Gingold.  Critical points at infinity and blow up of solutions of autonomous polynomial differential systems via compactification.  Journal of Mathematical Analysis and Applications, 318(1):305–322, 2006.
  * [45] Weiqiang Yang, M. Shahalam, Barun Pal, Supriya Pan, and Anzhong Wang.  Constraints on quintessence scalar field models using cosmological observations.  Phys. Rev. D, 100(2):023522, 2019.
  * [46] Adrià Gómez-Valent, Ziyang Zheng, Luca Amendola, Christof Wetterich, and Valeria Pettorino.  Coupled and uncoupled early dark energy, massive neutrinos, and the cosmological tensions.  Phys. Rev. D, 106(10):103522, 2022.
  * [47] Adam G. Riess et al.  A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km s-1 Mpc-1 Uncertainty from the Hubble Space Telescope and the SH0ES Team.  Astrophys. J. Lett., 934(1):L7, 2022.