# Variational symmetries and superintegrability in multifield cosmology

Alex Giacomini<EMAIL_ADDRESS>Instituto de Ciencias Físicas y
Matemáticas, Universidad Austral de Chile, Valdivia, Chile  Esteban González
<EMAIL_ADDRESS>Universidad de Santiago de Chile (USACH), Facultad
de Ciencia, Departamento de Física, Chile  Genly Leon<EMAIL_ADDRESS>Departamento de Matemáticas, Universidad Católica del Norte, Avda. Angamos
0610, Casilla 1280 Antofagasta, Chile  Andronikos Paliathanasis
<EMAIL_ADDRESS>Instituto de Ciencias Físicas y Matemáticas, Universidad
Austral de Chile, Valdivia, Chile  Institute of Systems Science, Durban
University of Technology, Durban 4000, South Africa

###### Abstract

We consider a spatially flat Friedmann–Lemaître–Robertson–Walker background
space with an ideal gas and a multifield Lagrangian consisting of two
minimally coupled scalar fields which evolve in a field space of constant
curvature. For this cosmological model we classify the potential function for
the scalar fields such that variational point symmetries exist. The
corresponding conservation laws are calculated. Finally, analytic solutions
are presented for specific functional forms of the scalar field potential in
which the cosmological field equations are characterized as a Liouville
integrable system by point symmetries. The free parameters of the cosmological
model are constrained in order to describe analytic solutions for an
inflationary epoch. Finally, stability properties of exact closed-form
solutions are investigated. These solutions are scaling solutions with
important physical properties for the cosmological model.

Scalar field; Multifield Cosmology; Chiral Cosmology; Variational symmetries;
Conservation laws; Integrability

###### pacs:

98.80.-k, 95.35.+d, 95.36.+x

##  I Introduction

Scalar fields play an important role in the theoretical description of two
accelerated expansion phases of the observed universe Teg1 ; Teg2 ; Teg3 ;
Teg4 ; planck2015 ; su1 ; su2 . The early acceleration phase of the universe
known as inflation is attributed to domination for a finite time of the
inflaton Aref1 ; guth . The inflaton is a single scalar field model that
drives the dynamics; providing a matter source in the universe which can
display antigravitating behavior. Because of the latter antigravitating
property, scalar fields have been used in the literature as dark energy models
de1 ; de2 ; de3 ; de4 ; de5 ; de6  to overpass the problems of
$\Lambda$-cosmology per1 . Moreover, because of the additional degrees of
freedom provided by scalar fields, the scalar field models can be used as
unified models for the description of the dark matter and dark energy un1 ;
un2 ; un3 ; un4 , while scalar fields can attribute degrees of freedom for
higher-order theories of gravity lan1  providing an equivalent description of
modified theories of gravity.

The simplest scalar field model proposed in the literature is the quintessence
model consisting of a minimally coupled single scalar field with positive
energy density and equation of state parameter $w_{Q}$ bounded in the range
$\left|w_{Q}\right|\leq 1$ sf1 . The phantom field is an alternative to
quintessence model in which the scalar fields have a negative kinetic energy,
such that the equation of state parameter $w_{P}$ is bounded as $w_{P}\leq-1$,
which means that $w_{P}$ can cross the phantom divide line $w_{P}=-1$ sf2 .
Scalar fields nonminimally coupled to gravity have been also studied in detail
for example in Brans-Dicke theory sf3 , O’Hanlon gravity sf4 , Hordenski
theory sf5  and others sf6 ; sf7 ; sf8 . Multifield models have been widely
studied in literature; the additional degrees of freedom that multifield
models provide overpass various problems of the single scalar field models,
providing a richer cosmological evolution mf1 ; mf2 ; mf3 ; mf4 .

In this work we are interested in multifield cosmological models in a
spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) background space
consisting of two scalar fields minimally coupled to gravity which are defined
in a hyperbolic field space. Additionally, we consider the contribution to the
cosmic fluid of a perfect fluid of constant equation of state parameter, that
is, an ideal gas. This specific multifield model has been widely studied to
describe the inflationary epoch as alternative to the inflation mechanism ch2
; ch3 ; ch4 ; ch5 ; ch6 ; ch7 . In addition, it can provide a cosmological
history which explains the transition from the inflationary epoch to the
matter era and then to the late-time acceleration phase of the universe an1 ;
an2 . For this cosmological model we investigate the existence of conservation
laws and the integrability properties of the field equations. The equations of
motion for the scale factor and the scalar fields are of second-order and form
a Hamiltonian system described by a point-like Lagrangian. We use that
property in order to find all the functional forms for the potential functions
of the scalar fields such that variational point symmetries exist ns1 .
Specifically, we apply Noether’s first and second theorem to calculate the
variational point symmetries and write the corresponding conservation laws ns2
. Point symmetries have been widely applied in gravitational physics and in
cosmology sym1 ; sym2 ; sym3 .

The main idea of this work is to use the variational symmetries to perform a
classification scheme of the potential function. This classification of a
given set of differential equations according to the admitted symmetry vectors
has been proposed by Ovsiannikov ovs . The requirement of the unknown
potential function to be constrained by the symmetry conditions is also a
geometric selection rule. Variational symmetries of the field equations are
related to geometric symmetries of the kinematic metric which define the
point-like Lagrangian sym4 ; sym5 . Consequently, the requirement that the two
scalar fields are defined in a hyperbolic field space is directly related to
the existence of variational symmetries. Previous studies for the
determination of exact solutions for the two-scalar field model are presented
in sl1 ; sl2 ; sl3 ; sl4 ; sl5 .

The plan of the paper is as follows: in Section II we present the cosmological
model of our considerations and we derive the field equations. The main
properties and definitions of the variational symmetries are given in Section
III. The classification problem of this work is performed in Section IV where
we present all the functional forms of the scalar field potentials in which
the field equations admit variational point symmetries and conservation laws
linear in the momentum. In Section V we construct analytic and exact solutions
to the cosmological models that are Liouville integrable which are obtained
from the classification scheme. In Section VI we perform a stability analysis
of the scaling solutions using a similar procedure in order to obtain the
scaling solutions in the previous Section. Finally, in Section VII we discuss
our results and our conclusions are drawn.

##  II Field equations

According to the cosmological principle in large scales the universe is
isotropic and homogeneous and its geometry is described by the spatially flat
Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime

$ds^{2}=-N\left(t\right)^{2}dt^{2}+a\left(t\right)^{2}\left(dx^{2}+dy^{2}+dz^{2}\right),$
(1)

where $N\left(t\right)$ is the lapse function and $a\left(t\right)$ is the
scale factor which is the radius of a three-dimensional hypersurface.

For the gravitational model we assume the Action Integral

$S=S_{GR}+S_{\text{multifield}}+S_{m},$ (2)

where $S_{GR}$ is the action integral of General Relativity

$S=\int\sqrt{-g}dx^{4}R,$ (3)

in which $R$ is the Ricci scalar of the background metric tensor $g_{\mu\nu}$.

The Action Integral$~{}S_{\text{multifield}}$ is given by sl1

$S_{\text{multifield}}=-\int\sqrt{-g}\left(\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi+\frac{\varepsilon}{2}g^{\mu\nu}e^{\kappa\phi}\nabla_{\mu}\psi\nabla_{\nu}\psi+V\left(\phi,\psi\right)\right),$
(4)

where the two scalar fields are defined in a hyperbolic field space, $\kappa$
is the coupling parameter and it is related to the curvature of the hyperbolic
space $R_{h}$ as $R_{h}=-\kappa^{2}/2$. Parameter $\varepsilon=\pm 1$
indicates if the second field $\psi\left(x^{\mu}\right)$ is phantom
$\left(\varepsilon=-1\right)$ or not $\left(\varepsilon=+1\right)$. In the
special case where $\kappa=0$ and $\varepsilon=-1$, the quintom model is
recovered, however in this work we assume $\kappa\neq 0.$ Finally, $S_{m}$
corresponds to the action integral for a perfect fluid with energy density
$\rho,~{}$pressure $p$ and constant equation of state parameter
$p=w_{m}\rho~{}$in which $w_{m}\in\left(-1,1\right)$ ss1 ; ss2 .

For the background space (1) and by assuming that scalar fields inherit the
spacetime symmetries we ended with the gravitational field equation

$\displaystyle 3H^{2}$
$\displaystyle=\frac{1}{2N^{2}}\dot{\phi}^{2}+\frac{\varepsilon}{2N^{2}}e^{\kappa\phi}\dot{\psi}^{2}+V\left(\phi\right)+\rho,$
(5) $\displaystyle-\left(2\dot{H}+3H^{2}\right)$
$\displaystyle=\frac{1}{2N^{2}}\dot{\phi}^{2}+\frac{\varepsilon}{2N^{2}}e^{\kappa\phi}\dot{\psi}-V\left(\phi\right)+w_{m}\rho,$
(6)

where dot means derivative with respect to the time variable and
$H\left(t\right)=\frac{1}{N}\frac{d}{dt}\left(\ln a\right)$ is the Hubble
function.

The equations of motion for the fluid components are

$\displaystyle\left(\ddot{\phi}+\frac{\dot{N}}{N}\dot{\phi}+3H\dot{\phi}\right)-\frac{\varepsilon}{2}\kappa
e^{\kappa\phi}\dot{\psi}^{2}+N^{2}V_{,\phi}=0~{},$ (7)
$\displaystyle\ddot{\psi}-\varepsilon\frac{\dot{N}}{N}\dot{\psi}+3H\dot{\psi}+\kappa\dot{\phi}\dot{\psi}+\varepsilon
e^{-\kappa\phi}V_{,\psi}=0~{},$ (8)
$\displaystyle\dot{\rho}+3NH(1+w_{m})\rho=0\,$ (9)

where from this last one, the conservation law follows
$\rho\left(t\right)=\rho_{m0}a\left(t\right)^{-3\left(1+w_{m}\right)}$.
By replacing in (5)-(8) and for arbitrary $N(t)$ we end with a Hamiltonian
dynamical system which is described by the singular point-like Lagrangian

$L\left(N,a,\dot{a},\phi,\dot{\phi},\psi,\dot{\psi}\right)=\frac{1}{2N\left(t\right)}\left(-6a\dot{a}^{2}+a^{3}\left(\dot{\phi}^{2}+\varepsilon
e^{\kappa\phi}\dot{\psi}^{2}\right)\right)-N\left(t\right)\left(a^{3}V\left(\phi,\psi\right)+\rho_{m0}a^{-3w_{m}}\right).$
(10)

In the following we assume the lapse function $N\left(t\right)=a^{3w_{m}}$.
Then, the point-like Lagrangian (10) describes the trajectory
$U\left(t\right)=U\left(a\left(t\right),\phi\left(t\right),\psi\left(t\right)\right)$
of a point particle in the three dimensional geometry

$ds_{\left(3\right)}^{2}=a^{-3w_{m}}\left(-6ada^{2}+a^{3}\left(d\phi^{2}+\varepsilon
e^{\kappa\phi}d\psi^{2}\right)\right),$ (11)

under the action of the effective potential
$V_{eff}=a^{3\left(1+w_{m}\right)}V\left(\phi,\psi\right)$. The integration
constant $\rho_{m0}$ is related to the “energy $h$” of the point-particle,
that is, $h=-\rho_{m0}$, in which $h$ is a conservation law for the equation
of motions.

We continue with the presentation of the basic properties for the variational
symmetries.

##  III Variational symmetries

Variational symmetries can be defined for differential equations of any order
which are deduced from a variational principle. Thus in this article we work
with second-order differential equations with a Lagrangian function
$L=L\left(t,q,\dot{q}\right)$.

Considering the infinitesimal transformation

$\bar{t}=t+\varepsilon\tau\left(t,q\right),\qquad\bar{q}=q+\varepsilon\eta\left(t,q\right),$
(12)

generated by the differential operator
$\Gamma=\tau\partial_{t}+\eta\partial_{q},$ where $\varepsilon$ is an
infinitesimal parameter. Under this transformation, the Action Integral
$A=\int_{t_{0}}^{t_{1}}L\left(t,q,\dot{q}\right)\mbox{\rm d}t$ becomes
$\bar{A}=\int_{\bar{t}_{0}}^{\bar{t}_{1}}L\left(\bar{t},\bar{q},\dot{\bar{q}}\right)\mbox{\rm
d}\bar{t}$ , which up to first order in the infinitesimal $\varepsilon$ is
written as follows

$\displaystyle\bar{A}$
$\displaystyle=\int_{t_{0}}^{t_{1}}\left[L+\varepsilon\left(\tau\displaystyle{\frac{\partial
L}{\partial t}}+\eta\displaystyle{\frac{\partial L}{\partial
q}}+\zeta\displaystyle{\frac{\partial
L}{\partial\dot{q}}}+\dot{\tau}L\right)\right]\mbox{\rm d}t+\varepsilon F,$
$\displaystyle
F:=\tau{t_{1}}L(t_{1},q_{1},\dot{q}_{1})-\tau{t_{0}}L(t_{0},q_{0},\dot{q}_{0}),$
(13)

where now $\zeta=\dot{\eta}-\dot{q}\dot{\tau}$ and
$L(t_{0},q_{0},\dot{q}_{0})$ and $L(t_{1},q_{1},\dot{q}_{1})$ are the values
of $L$ at the endpoints $t_{0}$ and $t_{1}$, respectively. Therefore, it
follows

$\bar{A}=A+\varepsilon\int_{t_{0}}^{t_{1}}\left(\tau\displaystyle{\frac{\partial
L}{\partial t}}+\eta\displaystyle{\frac{\partial L}{\partial
q}}+\zeta\displaystyle{\frac{\partial
L}{\partial\dot{q}}}+\dot{\tau}L\right)\mbox{\rm d}t+\varepsilon F.$ (14)

As $F$ depends only upon the endpoints, we may write it as

$F=-\int_{t_{0}}^{t_{1}}\dot{f}\mbox{\rm d}t.$ (15)

We have to say that the generator $\Gamma$ of the infinitesimal transformation
will be a variational symmetry, that is, a Noether symmetry if $\bar{A}=A$,
i.e.

$\int_{t_{0}}^{t_{1}}\left(\tau\displaystyle{\frac{\partial L}{\partial
t}}+\eta\displaystyle{\frac{\partial L}{\partial
q}}+\zeta\displaystyle{\frac{\partial
L}{\partial\dot{q}}}+\dot{\tau}L-\dot{f}\right)\mbox{\rm d}t=0,$ (16)

from which it follows the symmetry condition (Noether’s first theorem):

$\dot{f}=\tau\displaystyle{\frac{\partial L}{\partial
t}}+\eta\displaystyle{\frac{\partial L}{\partial
q}}+\zeta\displaystyle{\frac{\partial L}{\partial\dot{q}}}+\dot{\tau}L.$ (17)

Hence, according to Noether’s second theorem if there exists a vector field
$\Gamma$ and a function $f\left(t,q,\dot{q}\right)$ such that condition (17)
is true, then the quantity

$I\left(t,q,\dot{q}\right)=f-\left[\tau
L+\left(\eta-\dot{q}\tau\right)\frac{\partial L}{\partial\dot{q}}\right],$
(18)

is a conservation law for the Euler-Lagrange equations with Lagrangian
function $L\left(t,q,\dot{q}\right)$. For a recent discussion on Noether’s
theorem and for extensions and generalizations we refer the reader to ns1 .

##  IV Symmetry classification

For Lagrangian functions of the form

$L\left(t,q,\dot{q}\right)=\frac{1}{2}\gamma_{\alpha\beta}\left(q\right)\dot{q}^{\alpha}\dot{q}^{\beta}-V\left(q\right),$
(19)

variational symmetries are constructed by the elements of the Homothetic
algebra of the metric tensor $\gamma_{\alpha\beta}$. We omit the presentation
of the details for the derivation of the variational symmetry vectors and of
the symmetry conditions which constraint the effective potential function for
Lagrangian density of the form (19). They are presented into a theorem in sym7
with applications which demonstrate the main theorem.

The minisuperspace line element (11) is conformally flat and admits the vector
field as elements of the Homothetic algebra

$\displaystyle K^{1}$
$\displaystyle=\psi\partial_{\phi}+\left(\frac{e^{-\kappa\phi}}{\kappa\varepsilon}-\frac{\kappa}{4}\psi^{2}\right)\partial_{\psi}~{},~{}$
$\displaystyle K^{2}$
$\displaystyle=\partial_{\phi}-\frac{\kappa}{2}\psi\partial_{\psi},~{}$
$\displaystyle K^{3}$
$\displaystyle=\partial_{\psi},~{}H=\frac{2}{3\left(1-w_{m}\right)}a\partial_{a},$

in which $K^{1},~{}K^{2}$ and $K^{3}$ are Killing symmetries and $H$ is the
proper Homothetic vector field. Because variational symmetries are generated
by the vector fields $\left\\{K^{1},K^{2},K^{3},H\right\\}$ or linear
combination of them, we should derive the one-dimensional optimal system of
the Lie algebra.

Table 1: Commutator of the four-dimensional Homothetic algebra $\left[,\right]$ | $\mathbf{K}^{1}$ | $\mathbf{K}^{2}$ | $\mathbf{K}^{3}$ | $\mathbf{H}$
---|---|---|---|---
$\mathbf{K}^{1}$ | $0$ | $\frac{\kappa}{2}K^{1}$ | $-K^{2}$ | $0$
$\mathbf{K}^{2}$ | $\frac{\kappa}{2}K^{1}$ | $0$ | $\frac{\kappa}{2}K^{3}$ | $0$
$\mathbf{K}^{3}$ | $K^{2}$ | $-\frac{\kappa}{2}K^{3}$ | $0$ |  $0$
$\mathbf{H}$ | $0$ | $0$ | $0$ | $0$
Table 2: Adjoint representation for the four-dimensional Homothetic Lie algebra $Ad\left(e^{\left(\mu\mathbf{\Gamma}_{i}\right)}\right)\mathbf{\Gamma}_{j}$ | $\mathbf{K}^{1}$ | $\mathbf{K}^{2}$ | $\mathbf{K}^{3}$ | $\mathbf{H}$
---|---|---|---|---
$\mathbf{K}^{1}$ | $K^{1}$ | $-\frac{\kappa}{2}\mu K^{1}+K^{2}$ | $-\frac{\kappa}{4}\mu^{2}K^{1}+\mu K^{2}+K^{3}$ | $H$
$\mathbf{K}^{2}$ | $e^{\frac{\kappa}{2}\mu}$ | $K^{2}$ | $e^{-\frac{\kappa}{2}\mu}K^{3}$ | $H$
$\mathbf{K}^{3}$ | $K^{1}-\mu K^{2}-\frac{\kappa}{4}\mu^{2}K^{3}$ | $K^{2}+\frac{\kappa}{2}\mu K^{3}$ | $K^{3}$ | $H$
$\mathbf{H}$ | $K^{1}$ | $K^{2}$ | $K^{3}$ | $H$

The one-dimensional optimal system provides all the independent dynamical
systems which do not communicate under the adjoint representation. Indeed, we
consider the two generic vector fields

$\displaystyle\mathbf{Z}$
$\displaystyle=\alpha_{1}K^{1}+\alpha_{2}K^{2}+\alpha_{3}K^{3}+\alpha_{0}H,~{}$
(20) $\displaystyle\mathbf{W}$
$\displaystyle=\beta_{1}K^{1}+\beta_{2}K^{2}+\beta_{3}K^{3}+\beta_{0}H,$ (21)

where $\mathbf{\alpha}$ and $\beta$ are constants. The two vector fields
$\mathbf{Z,~{}W}$ are equivalent if and only if

$\mathbf{W}=\sum_{j=i}^{n}Ad\left(\exp\left(\mu_{i}X^{i}\right)\right)\mathbf{Z}$
(22)

or

$\mathbf{W}=c\mathbf{Z}~{},~{}c=const.$ (23)

where the operator $Ad\left(\exp\left(\mu
X_{i}\right)\right)X_{j}=X_{j}-\mu\left[X_{i},X_{j}\right]+\frac{1}{2}\mu^{2}\left[X_{i},\left[X_{i},X_{j}\right]\right]+...~{}$is
called the Adjoint representation olver . In Tables 1 and 2 the commutators
and the Adjoint representation for the Homothetic algebra
$\left\\{K^{1},K^{2},K^{3},H\right\\}$ are presented. Therefore, from the
results of the Tables we calculate the one-dimensional optimal system given by
the one-dimensional Lie algebra
$\left\\{K^{1}\right\\},~{}\left\\{K^{2}\right\\}~{},~{}\left\\{K^{3}\right\\}~{},~{}\left\\{H\right\\}~{},~{}\left\\{H+\alpha
K^{1}\right\\},~{}\left\\{H+aK^{2}\right\\}$ and $\left\\{H+\alpha
K^{3}\right\\}$.

However, in a special case in which
$\kappa=\pm\frac{\sqrt{6}}{2}\left(w_{m}-1\right)$ the admitted Homothetic
algebra is of higher dimension. Let us assume
$\kappa=+\frac{\sqrt{6}}{2}\left(w_{m}-1\right),$ the case with
$\kappa=-\frac{\sqrt{6}}{2}\left(w_{m}-1\right)$ is recovered with the change
of variables $\phi\rightarrow-\phi$. Therefore, for
$\kappa=\pm\frac{\sqrt{6}}{2}\left(w_{m}-1\right)$, the admitted Homothetic
vector fields by the minisuperspace are

$\displaystyle\bar{K}^{1}$
$\displaystyle=\left(a^{\frac{3}{2}}e^{\frac{\sqrt{6}}{4}\phi}\right)^{\left(w_{m}-1\right)}\left(a\psi\partial_{a}+\sqrt{6}\psi\partial_{\phi}+\frac{4e^{-\frac{\sqrt{6}}{2}\left(w_{m}-1\right)\phi}}{\varepsilon\left(w_{m}-1\right)}\right)~{},$
$\displaystyle\bar{K}^{2}$
$\displaystyle=\left(a^{\frac{3}{2}}e^{\frac{\sqrt{6}}{4}\phi}\right)^{\left(w_{m}-1\right)}\left(a\partial_{a}+\sqrt{6}\partial_{\phi}\right)~{},$
$\displaystyle\bar{K}^{3}$
$\displaystyle=\frac{1}{8}a^{\frac{3}{2}w_{m}-\frac{1}{2}}\left(3\varepsilon\left(w_{m}-1\right)^{2}\psi^{2}e^{\frac{\sqrt{6}}{4}\left(w_{m}-1\right)\phi}+8e^{-\frac{\sqrt{6}}{4}\left(w_{m}-1\right)\phi}\right)\partial_{a}+$
$\displaystyle+\frac{1}{8}a^{\frac{3}{2}\left(w_{m}-1\right)}\left(3\varepsilon\left(w_{m}-1\right)^{2}\psi^{2}e^{\frac{\sqrt{6}}{4}\left(w_{m}-1\right)\phi}-8e^{-\frac{\sqrt{6}}{4}\left(w_{m}-1\right)\phi}\right)\partial_{\phi}+$
$\displaystyle+3\left(w_{m}-1\right)a^{\frac{3}{2}\left(w_{m}-1\right)}\psi
e^{-\frac{\sqrt{6}}{4}\left(w_{m}-1\right)\phi}\partial_{\psi}~{},~{}$
$\bar{K}^{4}=\psi\partial_{\psi}+\left(\frac{8\sqrt{6}}{\varepsilon\left(w_{m}-1\right)}e^{-\frac{\sqrt{6}}{2}\left(w_{m}-1\right)\phi}-3\left(w_{m}-1\right)\sqrt{6}\psi^{2}\right)\partial_{\phi}~{},~{}$
$\bar{K}^{5}=\partial_{\phi}-\frac{\sqrt{6}}{4}\left(w_{m}-1\right)\psi\partial_{\psi}~{},~{}\bar{K}^{6}=\partial_{\psi}~{}\
,~{}H=\frac{2}{3\left(1-w_{m}\right)}a\partial_{a}~{},$

where $H$ is the proper Homothetic vector field and
$\left\\{\bar{K}^{1},\bar{K}^{2},\bar{K}^{3},\bar{K}^{4},\bar{K}^{5},\bar{K}^{6}\right\\}$
form a six dimensional Killing algebra. Consequently, when
$\kappa=+\frac{\sqrt{6}}{2}\left(w_{m}-1\right)$ the minisuperspace is
maximally symmetric. Moreover, we observe that the vector fields
$\left\\{\bar{K}^{1},\bar{K}^{2},\bar{K}^{3}\right\\}$ are gradient from which
we infer that the minisuperspace is the flat space. Therefore,
$\left\\{\bar{K}^{1},\bar{K}^{2},\bar{K}^{3}\right\\}$ correspond to the three
translation symmetries of the three dimensional flat space and
$\left\\{\bar{K}^{4},\bar{K}^{5},\bar{K}^{6}\right\\}$ form the
$SO\left(3\right).~{}$The commutators and the Adjoint representation of the
seven-dimensional Homothetic algebra are presented in Tables 3 and 4.

Table 3: Commutator of the seven-dimensional Homothetic algebra $\left[,\right]$ | $\mathbf{\bar{K}}^{1}$ | $\mathbf{\bar{K}}^{2}$ | $\mathbf{\bar{K}}^{3}$ | $\mathbf{\bar{K}}^{4}$ | $\mathbf{\bar{K}}^{5}$ | $\mathbf{\bar{K}}^{6}$ | $\mathbf{H}$
---|---|---|---|---|---|---|---
$\mathbf{K}^{1}$ | $0$ | $0$ | $0$ | $-\frac{\sqrt{6}}{3\varepsilon\left(w_{m}-1\right)}\bar{K}^{3}$ | $0$ | $-\bar{K}^{2}$ | $0$
$\mathbf{K}^{2}$ | $0$ | $0$ | $0$ | $-\frac{\sqrt{6}\left(w_{m}-1\right)}{4}\bar{K}^{1}$ | $-\frac{\sqrt{6}}{4}\bar{K}^{2}$ | $0$ | $0$
$\mathbf{K}^{3}$ |  $0$ |  $0$ |  $0$ | $0$ | $\frac{\sqrt{6}\left(w_{m}-1\right)}{4}\bar{K}^{3}$ | $-\frac{3\left(w_{m}-1\right)^{2}}{4}\bar{K}^{1}$ |  $0$
$\mathbf{\bar{K}}^{4}$ | $\frac{\sqrt{6}}{3\varepsilon\left(w_{m}-1\right)}\bar{K}^{3}$ | $\frac{\sqrt{6}\left(w_{m}-1\right)}{4}\bar{K}^{1}$ | $0$ | $0$ | $\frac{\sqrt{6}\left(w_{m}-1\right)}{4}\bar{K}^{4}$ | $-\bar{K}^{5}$ | $0$
$\mathbf{\bar{K}}^{5}$ | $0$ | $\frac{\sqrt{6}}{4}\bar{K}^{2}$ | $-\frac{\sqrt{6}\left(w_{m}-1\right)}{4}\bar{K}^{3}$ | $-\frac{\sqrt{6}\left(w_{m}-1\right)}{4}\bar{K}^{4}$ | $0$ | $\frac{\sqrt{6}\left(w_{m}-1\right)}{4}\bar{K}^{6}$ | $0$
$\mathbf{\bar{K}}^{6}$ | $\bar{K}^{2}$ | $0$ | $\frac{3\left(w_{m}-1\right)^{2}}{4}\bar{K}^{1}$ | $\bar{K}^{5}$ | $-\frac{\sqrt{6}\left(w_{m}-1\right)}{4}\bar{K}^{6}$ | $0$ |  $0$
$\mathbf{H}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 4: Adjoint representation for the seven-dimensional Homothetic Lie
algebra, in which $W=w_{m}-1$

${\small Ad}$ | $\mathbf{\bar{K}}^{1}$ | $\mathbf{\bar{K}}^{2}$ | $\mathbf{\bar{K}}^{3}$ | $\mathbf{\bar{K}}^{4}$ | $\mathbf{\bar{K}}^{5}$ | $\mathbf{\bar{K}}^{6}$ | $\mathbf{H}$
---|---|---|---|---|---|---|---
$\mathbf{K}^{1}$ | ${\small\bar{K}}^{1}$ | ${\small\bar{K}}^{2}$ | ${\small\bar{K}}^{3}$ | $\frac{\sqrt{6}\varepsilon\mu}{3\left(w-1\right)}{\small\bar{K}}^{3}{\small+\bar{K}}^{4}$ | ${\small\bar{K}}^{5}$ | ${\small\mu\bar{K}}^{2}{\small+\bar{K}}^{6}$ | ${\small H-\mu}\bar{K}^{1}$
$\mathbf{K}^{2}$ | ${\small\bar{K}}^{1}$ | ${\small\bar{K}}^{2}$ | ${\small\bar{K}}^{3}$ | $\frac{\sqrt{6}\left(w-1\right){\small\mu}}{4}{\small\bar{K}}^{1}{\small+\bar{K}}^{4}$ | $\frac{\sqrt{6}W{\small\mu}}{4}{\small\bar{K}}^{2}{\small+\bar{K}}^{5}$ | ${\small\bar{K}}^{6}$ | ${\small H-\mu}\bar{K}^{2}$
$\mathbf{K}^{3}$ | ${\small\bar{K}}^{1}$ | ${\small\bar{K}}^{2}$ | ${\small\bar{K}}^{3}$ | ${\small\bar{K}}^{4}$ | ${\small-}\frac{\sqrt{6}W{\small\mu}}{4}{\small\bar{K}}^{3}{\small+\bar{K}}^{5}$ | $\frac{3\varepsilon W^{2}{\small\mu}}{4}{\small\bar{K}}^{1}{\small+\bar{K}}^{6}$ | ${\small H-\mu}\bar{K}^{3}$
$\mathbf{\bar{K}}^{4}$ | ${\small\bar{K}}^{1}{\small-}\frac{\sqrt{6}\varepsilon\mu}{3W}{\small K}^{3}$ | ${\small\bar{K}}^{2}{\small+}\frac{\mu^{2}}{4\varepsilon}{\small\bar{K}}^{3}{\small-}\frac{\sqrt{6}W}{4}{\small\mu\bar{K}}^{1}$ | ${\small\bar{K}}_{3}$ | ${\small\bar{K}}^{4}$ | ${\small-}\frac{\sqrt{6}W{\small\mu}}{4}{\small\bar{K}}^{4}{\small+\bar{K}}^{5}$ | ${\small\bar{K}}^{6}{\small-\frac{\sqrt{6}W}{8}\mu^{2}\bar{K}^{4}+\mu\bar{K}}^{5}$ | ${\small H}$
$\mathbf{\bar{K}}^{5}$ | ${\small\bar{K}}^{1}$ | ${\small e}^{-\frac{\sqrt{6}W}{4}\mu}{\small\bar{K}}^{2}$ | ${\small e}^{\frac{\sqrt{6}W}{4}\mu}{\small\bar{K}}^{3}$ | ${\small e}^{\frac{\sqrt{6}\left(w-1\right)}{4}\mu}{\small\bar{K}}^{4}$ | ${\small\bar{K}}^{5}$ | ${\small e}^{-\frac{\sqrt{6}W}{4}\mu}{\small\bar{K}}^{6}$ | ${\small H}$
$\mathbf{\bar{K}}^{6}$ | ${\small\bar{K}}^{1}{\small-\mu\bar{K}}^{2}$ | ${\small\bar{K}}^{2}$ | ${\small\bar{K}}^{3}{\small-}\frac{3\varepsilon W^{2}{\small\mu}\left(\bar{K}^{1}-\mu\bar{K}^{2}\right)}{4}$ | ${\small\bar{K}}^{4}{\small-\mu\bar{K}}^{5}{\small+}\frac{\sqrt{6}W{\small\mu}}{4}{\small\bar{K}}^{6}$ | ${\small\bar{K}}^{5}{\small+}\frac{\sqrt{6}W{\small\mu}}{4}{\small\bar{K}}^{6}$ | ${\small\bar{K}}^{6}$ | ${\small H}$
$\mathbf{H}$ | ${\small\bar{K}}^{1}$ | ${\small\bar{K}}^{2}$ | ${\small\bar{K}}^{3}$ | ${\small\bar{K}}^{4}$ | ${\small\bar{K}}^{5}$ | ${\small\bar{K}}^{6}$ | ${\small H}$

Consequently, from Tables 3 and 4 we calculate the one-dimensional system
given by the one-dimensional Lie algebras $\left\\{\bar{K}^{A}\right\\}$ ,
$\left\\{\bar{K}^{A}+\beta K^{B}\right\\}$ ,
$\left\\{\bar{K}^{1}+\alpha\bar{K}^{2}+\beta\bar{K}^{3}\right\\}$,
$\left\\{\bar{K}^{J}\right\\}$,
$\left\\{\bar{K}^{J}+\alpha\bar{K}^{A}\right\\}$, $\left\\{H\right\\}$,
$\left\\{\bar{K}^{J}+\gamma H\right\\}$; in which $A,B=1,2,3$ and $J=4,5,6$.

Until now we have assumed that $w_{m}-1\neq 0$. The case where $w_{m}=1$ will
be studied elsewhere.

###  IV.1 Classification of $V\left(\phi,\psi\right)$

We continue with the presentation of the special potential forms
$V\left(\phi,\psi\right)$ where the cosmological field equations (5)-(9) admit
conservation laws. We omit the calculations and for each potential function
which admit a variational symmetry we present the conservation law.

####  IV.1.1 Arbitrary $\kappa$

For potential
$V_{A}\left(\phi,\psi\right)=V\left(G_{A}\left(\phi,\psi\right)\right)~{},~{}G_{A}\left(\phi,\psi\right)=e^{-\frac{\kappa}{2}\phi}\left(4+\varepsilon\kappa^{2}\psi^{2}e^{\kappa\phi}\right)$,
the vector field $K^{1}$ is a variational symmetry with conservation law
function
$I^{1}\left(\mathbf{q,p}\right)=K^{1}\left(\mathbf{q}\right)\mathbf{p}$, where
$\mathbf{p}=\left(p_{a},p_{\phi},p_{\psi}\right)$; that is
$I^{1}\left(a,\phi,\psi,p_{a},p_{\phi,}p_{\psi}\right)=\psi
p_{\phi}+\left(\frac{e^{-\kappa\phi}}{\kappa\varepsilon}-\frac{\kappa}{4}\psi^{2}\right)p_{\psi}$.
Functions $\mathbf{p}$ are the momentum defined as $p_{\alpha}=\frac{\partial
L}{\partial q^{\alpha}}$, i.e.
$p_{a}=-6a^{1-3w_{m}}\dot{a}~{},~{}p_{\phi}=a^{3\left(1-w_{m}\right)}\dot{\phi}$
and $p_{\psi}=a^{3\left(1-w_{m}\right)}\varepsilon e^{\kappa\phi}\dot{\psi}$.

For
$V_{B}\left(\phi,\psi\right)=V\left(G_{B}\left(\phi,\psi\right)\right)~{},~{}G_{B}=\psi
e^{\frac{\kappa}{2}\phi}$, the vector field $K^{2}$ is a variational symmetry
and the corresponding Noetherian conservation law is
$I^{2}\left(\mathbf{q,p}\right)=K^{2}\left(\mathbf{q}\right)\mathbf{p}$.

When $V_{C}\left(\phi,\psi\right)=V\left(\phi\right)$ the field equations
admit the variational symmetry $K^{3}$ and the conservation law
$I^{3}\left(\mathbf{q,p}\right)=K^{3}\left(\mathbf{q}\right)\mathbf{p}$.

Finally, when $V_{D}\left(\phi,\psi\right)=V_{0}$ the field equations admit
three variational symmetries, the fields $K^{1},~{}K^{2}$ and $K^{3}$ with the
corresponding conservation laws
$I^{1}\left(\mathbf{q,p}\right),~{}I^{2}\left(\mathbf{q,p}\right)$ and
$I^{3}\left(\mathbf{q,p}\right)$.

####  IV.1.2 Case $\kappa=\frac{\sqrt{6}}{2}\left(w_{m}-1\right)$

For the potential function
$\bar{V}_{A}\left(\phi,\psi\right)=V\left(\bar{G}_{A}\left(\phi,\psi\right)\right)e^{-\frac{\sqrt{6}}{2}\left(w_{m}+1\right)\phi}$
with
$\bar{G}_{A}\left(\phi,\psi\right)=\frac{8}{3}e^{-\frac{\sqrt{6}\left(w_{m}-1\right)}{2}\phi}+\varepsilon\left(w_{m}-1\right)^{2}\psi^{2}$
the cosmological field equations admit the conservation law
$\bar{I}^{1}\left(\mathbf{q,p}\right)=\bar{K}^{1}\left(\mathbf{q}\right)\mathbf{p}$
where the corresponding symmetry vector is the isometry $\bar{K}^{1}$.

For
$\bar{V}_{B}\left(\phi,\psi\right)=V\left(\psi\right)e^{-\frac{\sqrt{6}}{2}\left(w_{m}+1\right)\phi}$,
vector field $\bar{K}^{2}$ is a variational symmetry and the corresponding
conservation law is
$\bar{I}^{2}\left(\mathbf{q,p}\right)=\bar{K}^{2}\left(\mathbf{q}\right)\mathbf{p}$.

Similarly, for
$\bar{V}_{C}\left(\phi,\psi\right)=V\left(\bar{G}_{C}\left(\phi,\psi\right)\right)\,\
$with
$\bar{G}_{C}\left(\phi,\psi\right)=\left(\frac{8}{3}e^{\frac{\sqrt{6}}{2}\phi}+\varepsilon\left(w_{m}-1\right)^{2}\psi^{2}\right)e^{-\frac{\sqrt{6}}{4}\left(w_{m}+1\right)\phi}$
the symmetry vector is $\bar{K}^{4}$ with
$\bar{I}^{4}\left(\mathbf{q,p}\right)=\bar{K}^{4}\left(\mathbf{q}\right)\mathbf{p\,.}$

When potential
$\bar{V}_{D}\left(\phi,\psi\right)=V\left(\bar{G}_{D}\left(\phi,\psi\right)\right),~{}\bar{G}_{D}\left(\phi,\psi\right)=\psi
e^{\frac{\sqrt{6}}{4}\left(w_{m}+1\right)\phi}$ the system admits the
variational symmetry $\bar{K}^{5}$ with conservation law
$\bar{I}^{5}\left(\mathbf{q,p}\right)=\bar{K}^{5}\left(\mathbf{q}\right)\mathbf{p}$.

For $\bar{V}_{E}\left(\phi,\psi\right)=V\left(\phi\right)$ the field equations
admit the conservation law
$\bar{I}^{6}\left(\mathbf{q,p}\right)=\bar{K}^{6}\left(\mathbf{q}\right)\mathbf{p}$
generated by the symmetry vector $\bar{K}^{6}$.

Moreover, for the potential function
$\bar{V}_{F}\left(\phi,\psi\right)=V\left(\bar{G}_{F}\left(\phi,\psi\right)\right)e^{-\frac{\sqrt{6}}{2}\left(w_{m}+1\right)\phi},~{}\bar{G}_{F}\left(\phi,\psi\right)=\frac{8}{3}e^{-\frac{\sqrt{6}\left(w_{m}-1\right)}{2}\phi}+\varepsilon\left(w_{m}-1\right)^{2}\psi\left(\psi+2\beta\right)$,
the vector field $\bar{K}^{1}+\bar{K}^{2}$ is a variational symmetry and the
corresponding Noetherian conservation law is
$\bar{I}_{\beta}^{12}\left(\mathbf{q,p}\right)=\left(\bar{K}^{1}\left(\mathbf{q}\right)+\beta\bar{K}^{2}\left(\mathbf{q}\right)\right)\mathbf{p}$.

The vector field $K^{3}$ is a variational symmetry for the field equations
with conservation law
$\bar{I}^{3}\left(\mathbf{q,p}\right)=\bar{K}^{3}\left(\mathbf{q}\right)\mathbf{p}$
if the potential function
$\bar{V}_{G}\left(\phi,\psi\right)=V\left(\phi,\psi\right)$ satisfies the
differential equation $F^{3}=0$ in which

$\displaystyle F^{3}$
$\displaystyle\equiv-e^{-\frac{\sqrt{6}}{4}\left(w_{m}+1\right)\phi}\left(\sqrt{6}V_{,\phi}-3\left(w_{m}-1\right)\psi
V_{,\psi}-3\left(w_{m}+1\right)V\right)+$
$\displaystyle+3\left(w_{m}-1\right)^{2}\varepsilon\psi^{2}e^{-\frac{\sqrt{6}}{4}\left(w_{m}+1\right)\phi}\left(3\left(w_{m}+1\right)V+\sqrt{6}V_{,\phi}\right).$
(24)

Similarly for the linear combinations $K^{1}+\beta K^{3},~{}K^{2}+\beta K^{3}$
and $K^{1}+\beta K^{2}+\gamma K^{3}$ the symmetry conditions are

$0=\psi
e^{\frac{\sqrt{6}}{4}\left(w_{m}-1\right)\phi}\varepsilon\left(w_{m}-1\right)\left(3\left(w_{m}+1\right)V+\sqrt{6}V_{,\phi}\right)+4e^{\frac{\sqrt{6}}{4}\left(w_{m}-1\right)\phi}V_{,\phi}+\varepsilon\left(w_{m}-1\right)\frac{\beta}{8}F^{3}~{},~{}$
(25)
$0=e^{\frac{\sqrt{6}}{4}\left(w_{m}-1\right)\phi}\left(3\left(w_{m}+1\right)V+\sqrt{6}V_{,\phi}\right)+\frac{\beta}{8}F^{3}~{},$
$\displaystyle 0$ $\displaystyle=\psi
e^{\frac{\sqrt{6}}{4}\left(w_{m}-1\right)\phi}\varepsilon\left(w_{m}-1\right)\left(3\left(w_{m}+1\right)V+\sqrt{6}V_{,\phi}\right)+4e^{\frac{\sqrt{6}}{4}\left(w_{m}-1\right)\phi}V_{,\phi}+$
$\displaystyle+\beta\varepsilon\left(w_{m}-1\right)e^{\frac{\sqrt{6}}{4}\left(w_{m}-1\right)\phi}\left(3\left(w_{m}+1\right)V+\sqrt{6}V_{,\phi}\right)+\varepsilon\left(w_{m}-1\right)\frac{\gamma}{8}F^{3},$

respectively, while the corresponding conservation laws are
$\bar{I}_{\beta}^{13}\left(\mathbf{q,p}\right)=\left(\bar{K}^{1}\left(\mathbf{q}\right)+\beta\bar{K}^{3}\left(\mathbf{q}\right)\right)\mathbf{p}$
,
$\bar{I}_{\beta}^{23}\left(\mathbf{q,p}\right)=\left(\bar{K}^{2}\left(\mathbf{q}\right)+\beta\bar{K}^{3}\left(\mathbf{q}\right)\right)\mathbf{p}$
, and
$\bar{I}_{\beta\gamma}^{123}\left(\mathbf{q,p}\right)=\left(\bar{K}^{1}\left(\mathbf{q}\right)+\beta\bar{K}^{2}\left(\mathbf{q}\right)+\gamma\bar{K}^{2}\left(\mathbf{q}\right)\right)\mathbf{p}$.
The rest of the linear combinations do not provide any other potential
function which satisfies a conservation law.

We have found all the possible cases for the scalar field potential where the
cosmological field equations admit at least a variational symmetry. The field
equations form a dynamical system of three dimensions, therefore in order to
infer about the integrability by point symmetries we need at least three
conservation laws. By definition an autonomous Hamiltonian system
$\mathcal{H}$ of dimension $n~{}$is characterized as Liouville integrable arn1
if there exist at least $n-1$ independent conservation laws
$I_{J},~{}J=1,2,3,...,n-1,$ i.e. $\left\\{I_{J},H\right\\}=0$, which are in
involution, that is $\left\\{I_{J},I_{\bar{J}}\right\\}=0$.

For the scalar field potential

$V_{I}\left(\phi,\psi\right)=V_{0}e^{-\frac{\sqrt{6}\left(w_{m}+1\right)}{2}\phi},$
(26)

we find that the cosmological field equations admit the conservation laws
$\bar{I}^{1}\left(\mathbf{q,p}\right),~{}\bar{I}^{2}\left(\mathbf{q,p}\right)$
and
$\bar{I}^{6}\left(\mathbf{q,p}\right)=\bar{K}^{6}\left(\mathbf{q}\right)\mathbf{p}$.
Potential $V_{I}\left(\phi,\psi\right)$ admits as asymptotic solution the so-
called hyperinflation model where the two scalar fields drive the dynamics. We
refer the reader to sl5  and references therein for more details.

The potential function

$V_{II}\left(\phi,\psi\right)=V_{0}e^{\frac{\sqrt{6}}{2}\frac{\left(w_{m}+1\right)^{2}}{w_{m-1}}\phi}\left(\frac{3}{8}\psi^{2}\left(w_{m}-1\right)^{2}e^{\frac{\sqrt{6}w_{m}\phi}{2}}+8e^{\frac{\sqrt{6}\phi}{2}}\right)^{-2\frac{w_{m}+1}{w_{m}-1}}$
(27)

admits the variational symmetries $\bar{K}^{1}~{},~{}\bar{K}^{3}$ and
$\bar{K}^{4}$ with conservation laws
$\bar{I}^{1}\left(\mathbf{q,p}\right),~{}\bar{I}^{3}\left(\mathbf{q,p}\right)$
and
$\bar{I}^{6}\left(\mathbf{q,p}\right)=\bar{K}^{4}\left(\mathbf{q}\right)\mathbf{p}$.

For the scalar field potential

$V_{III}\left(\phi,\psi\right)=V_{0}e^{-\frac{\sqrt{6}\left(w_{m}+1\right)}{2}\phi}\psi^{-2\frac{w_{m}+1}{w_{m}-1}},$
(28)

the Hamiltonian system admits the conservation laws
$\bar{I}^{2}\left(\mathbf{q,p}\right),~{}\bar{I}^{3}\left(\mathbf{q,p}\right)$
and
$\bar{I}^{6}\left(\mathbf{q,p}\right)=\bar{K}^{4}\left(\mathbf{q}\right)\mathbf{p}$
generated by the Noether symmetries $\bar{K}^{2},~{}\bar{K}^{3}$ and
$\bar{K}^{5}$.

Furthermore, when

$V_{IV}\left(\phi,\psi\right)=V_{0}e^{\frac{\sqrt{6}}{2}\frac{\left(w_{m}+1\right)^{2}}{w_{m-1}}\phi}\left(3\psi\left(2\beta+\psi\right)\left(w_{m}-1\right)^{2}e^{\frac{\sqrt{6}w_{m}\phi}{2}}+8e^{\frac{\sqrt{6}\phi}{2}}\right)^{-2\frac{w_{m}+1}{w_{m}-1}},$
(29)

the conservation laws are
$\bar{I}^{12}\left(\mathbf{q,p}\right),~{}\bar{I}^{3}\left(\mathbf{q,p}\right)$
and
$\bar{I}^{45}\left(\mathbf{q,p}\right)=\left(\bar{K}^{4}\left(\mathbf{q}\right)+\beta\bar{K}^{5}\left(\mathbf{q}\right)\right)\mathbf{p}$.

The potential

$V_{V}\left(\phi,\psi\right)=V_{0}e^{-\frac{\sqrt{6}\left(w_{m}+1\right)}{2}\phi}\left(3\beta\psi\left(w_{m}-1\right)^{2}\varepsilon+4\right)^{-2\frac{w_{m}+1}{w_{m}-1}},$
(30)

admits the conservation laws
$\bar{I}^{13}\left(\mathbf{q,p}\right),~{}\bar{I}^{2}\left(\mathbf{q,p}\right)$
and
$\bar{I}^{56}\left(\mathbf{q,p}\right)=\left(\bar{K}^{5}\left(\mathbf{q}\right)-\frac{\sqrt{6}}{3\beta\varepsilon\left(w_{m}-1\right)}\bar{K}^{6}\left(\mathbf{q}\right)\right)\mathbf{p}$.

For the potential function

$V_{VI}\left(\phi,\psi\right)=V_{0}e^{\frac{\sqrt{6}}{2}\frac{\left(w_{m}+1\right)^{2}}{w_{m-1}}\phi}\left(\left(3\gamma\psi\left(2\beta+\psi\right)\left(w_{m}-1\right)^{2}+8\beta\right)e^{\frac{\sqrt{6}w_{m}\phi}{2}}+8\gamma
e^{\frac{\sqrt{6}\phi}{2}}\right),$ (31)

the Noetherian conservation laws are
$\bar{I}^{12}\left(\mathbf{q,p}\right)=\left(\bar{K}^{1}\left(\mathbf{q}\right)+\beta\bar{K}^{2}\left(\mathbf{q}\right)\right)\mathbf{p}$
, $\bar{I}^{\prime
12}\left(\mathbf{q,p}\right)=\left(\bar{K}^{1}\left(\mathbf{q}\right)+\gamma\bar{K}^{2}\left(\mathbf{q}\right)\right)\mathbf{p~{}}$and
$\bar{I}^{56}\left(\mathbf{q,p}\right)=\left(\bar{K}^{5}\left(\mathbf{q}\right)-\frac{\sqrt{6}}{3\beta\varepsilon\left(w_{m}-1\right)}\bar{K}^{6}\left(\mathbf{q}\right)\right)\mathbf{p}$,
with $\beta\neq\gamma$.

For the special case where $w_{m}=0$, there exist additional scalar field
potentials which admit conservation laws. These potentials are presented in
sym10 . The field equations for the six scalar field potentials
$V_{I-VI}\left(\phi,\psi\right)$ are Liouville integrable and specifically
they are superintegrable because they admit more than three indepedent
conservation laws. In the following, we construct the analytic solutions for
these integrable dynamical systems.

##  V Analytic solutions

In order to calculate the analytic solutions for the superintegrable scalar
field potentials $V_{I-VI}\left(\phi,\psi\right)~{}$we prefer to work with
normal coordinates. We consider the change of variables

$\displaystyle a\left(x,y,z\right)$
$\displaystyle=\left(\frac{2}{\left|w_{m}-1\right|}\right)^{\frac{2}{3\left(w_{m}-1\right)}}\left(4yz-\frac{3}{2}\varepsilon
x^{2}\right)^{-\frac{1}{3\left(w_{m}-1\right)}}~{},~{}$ (32)
$\displaystyle\phi\left(x,y,z\right)$
$\displaystyle=-\frac{\sqrt{6}}{3\left(w_{m}-1\right)}\ln\left(\frac{4yz-\frac{3}{2}\varepsilon
x}{y^{2}}\left(w_{m}-1\right)^{2}\right)~{},$ (33)
$\displaystyle\psi\left(x,y,z\right)$ $\displaystyle=\frac{x}{y},$ (34)

in which the point-like Lagrangian for the cosmological field equations
becomes

$L\left(x,\dot{x},y,\dot{y},z,\dot{z}\right)=\frac{1}{2}\varepsilon\dot{x}^{2}-\frac{4}{3}\dot{y}\dot{z}-U_{eff}\left(x,y,z\right)-\rho_{m0},$
(35)

where
$U_{eff}\left(x,y,z\right)=a^{3\left(1+w_{m}\right)}V\left(\phi,\psi\right)$.
In the new variables the second-order field equations become

$\displaystyle\varepsilon\ddot{x}+\left(U_{eff}\right)_{,x}$
$\displaystyle=0,$ (36)
$\displaystyle\ddot{y}-\frac{3}{4}\left(U_{eff}\right)_{,z}$
$\displaystyle=0,$ (37)
$\displaystyle\ddot{z}-\frac{3}{4}\left(U_{eff}\right)_{,y}$
$\displaystyle=0,$ (38)

while the constraint equation becomes

$\frac{1}{2}\varepsilon\dot{x}^{2}-\frac{4}{3}\dot{y}\dot{z}+U_{eff}\left(x,y,z\right)+\rho_{m0}=0.$
(39)

###  V.1 Scalar field potential $V_{I}\left(\phi,\psi\right)$

The exact solution for potential $V_{I}\left(\phi,\psi\right)$ was derived for
the first time in sl5 . However, we present the analytic solution for
completeness. For the scalar field potential $V_{I}\left(\phi,\psi\right)$ the
effective potential in the variables $\left\\{x,y,z\right\\}$ of point-like
Lagrangian (35) becomes
$U_{eff}^{I}\left(x,y,z\right)=V_{0}y^{-2\frac{w_{m}+1}{w_{m}-1}}$.

Thus, the field equations (36)-(38) become

$\varepsilon\ddot{x}=0~{},~{}\ddot{y}=0,$ (40)
$\ddot{z}+\frac{3}{2}\frac{w_{m}+1}{w_{m}-1}V_{0}y^{-\frac{3w_{m}+1}{w_{m}-1}}=0.$
(41)

Therefore, the analytic solution is

$x\left(t\right)=x_{0}\left(t-t_{1}\right)~{},~{}y\left(t\right)=y_{0}\left(t-t_{2}\right),$
(42)
$z\left(t\right)=-\frac{3V_{0}(w_{m}-1)y_{0}^{-\frac{3w_{m}+1}{w_{m}-1}}(t-t_{2})^{\frac{w_{m}+3}{1-w_{m}}}}{4(w_{m}+3)}+z_{0}\left(t-t_{3}\right),$
(43)

with constraint condition

$\rho_{m0}=\left(\frac{4}{3}y_{0}z_{0}-\frac{\varepsilon}{2}\left(x_{0}\right)^{2}\right).$
(44)

In the special case where $w_{m}=-\frac{1}{3}$, the closed-form solution is

$x\left(t\right)=x_{0}\left(t-t_{1}\right)~{},~{}y\left(t\right)=y_{0}\left(t-t_{2}\right),~{}$
(45)
$z\left(t\right)=\frac{3}{8}V_{0}(t-t_{2})^{2}+z_{0}\left(t-t_{3}\right),$
(46)

with the same constraint condition. However, when $w_{m}=-\frac{1}{3}$ the
additional matter source can play the role of the spatial curvature $k$, where
$\rho_{m0}=k$, and in such case we recover the closed-form solution for the
multifield model in a nonflat FLRW background space and spatial curvature $k$.

Thus, the scale factor is

$\left(a\left(t\right)\right)^{-3\left(w_{m}-1\right)}\simeq\left(4\left(\frac{3V_{0}\left(1-w_{m}\right)}{4\left(w_{m}+3\right)}\left(y_{0}\left(t-t_{2}\right)\right)^{-\frac{w_{m}+3}{w_{m}-1}+1}+z_{0}y_{0}\left(t-t_{2}\right)\left(t-t_{3}\right)\right)-\frac{3}{2}\varepsilon
x_{0}\left(t-t_{1}\right)\right),$ (47)

when the term $\left(t-t_{2}\right)^{-\frac{w_{m}+3}{w_{m}-1}+1}$ dominates,
the scale factor is approximated as
$a\left(t\right)\simeq\left(t-t_{2}\right)^{\frac{4}{3\left(w_{m}-1\right)^{2}}}$.

Thus, the line element of the background FLRW space is approximated as

$ds^{2}=-\left(t-t_{2}\right)^{\frac{8w_{m}}{\left(w_{m}-1\right)^{2}}}dt^{2}+\left(t-t_{2}\right)^{\frac{8}{3\left(w_{m}-1\right)^{2}}}\left(dx^{2}+dy^{2}+dz^{2}\right),$
(48)

or, equivalently

$ds^{2}=-d\tau^{2}+\tau^{\frac{8}{{}^{3\left(\left(w_{m}-1\right)^{2}+w_{m}\right)}}}\left(dx^{2}+dy^{2}+dz^{2}\right).$
(49)

For $w_{m}$ the closed-form solution can be found in an1 .

###  V.2 Scalar field potential $V_{II}\left(\phi,\psi\right)$

For the scalar field potential $V_{II}\left(\phi,\psi\right)~{}$we derive
$U_{eff}^{II}\left(x,y,z\right)=V_{0}z^{-2\frac{w_{m}+1}{w_{m}-1}}$. We
observe that changing variables $\left(y,z\right)\rightarrow\left(z,y\right)$
we end with the dynamical system of potential $V_{I}\left(\phi,\psi\right)$.
The field equations (36)-(38) become

$\varepsilon\ddot{x}=0~{},~{}\ddot{z}=0,$ (50)
$\ddot{y}+\frac{3}{2}\frac{w_{m}+1}{w_{m}-1}V_{0}z^{-\frac{3w_{m}+1}{w_{m}-1}}=0.$
(51)

Hence, the analytic solution is

$x\left(t\right)=x_{0}\left(t-t_{1}\right)~{},~{}z\left(t\right)=z_{0}\left(t-t_{2}\right),\;y\left(t\right)=-\frac{3V_{0}(w_{m}-1)z_{0}^{-\frac{3w_{m}+1}{w_{m}-1}}(t-t_{2})^{\frac{w_{m}+3}{1-w_{m}}}}{4(w_{m}+3)}+y_{0}\left(t-t_{3}\right),$
(52)

with
$\rho_{m0}=\left(\frac{4}{3}y_{0}z_{0}-\frac{\varepsilon}{2}\left(x_{0}\right)^{2}\right)$.
At the limit where $t\rightarrow t_{2}$ we have a similar behaviour as for the
previous potential.

###  V.3 Scalar field potential $V_{III}\left(\phi,\psi\right)$

For the third potential of our analysis $V_{III}\left(\phi,\psi\right)$ in the
new variables it follows
$U_{eff}^{III}\left(x,y,z\right)=V_{0}x^{-2\frac{w_{m}+1}{w_{m}-1}}$.

The field equations (36)-(38) become

$\displaystyle\ddot{x}-2V_{0}\varepsilon\frac{w_{m}+1}{w_{m}-1}x^{-\frac{3w_{m}+1}{w_{m}-1}}=0,$
(53) $\displaystyle\ddot{y}=0,\quad\ddot{z}=0.$ (54)

Furthermore, from the Friedmann equation it follows

$\frac{1}{2}\varepsilon\dot{x}^{2}+V_{0}x^{-2\frac{w_{m}+1}{w_{m}-1}}-\frac{4}{3}y_{0}z_{0}+\rho_{m0}=0,$
(55)

that is

$\bigints\frac{dx}{\sqrt{2\varepsilon\left(\frac{4}{3}y_{0}z_{0}-\rho_{m0}-V_{0}x^{-2\frac{w_{m}+1}{w_{m}-1}}\right)}}=t-t_{0}.$
(56)

Firstly, we solve (53). The general solution is given in terms of a
hypergeometric function ${}_{2}F_{1}(a,b;c;u)$:

$\displaystyle\frac{x^{2}\left(1-\frac{2V_{0}\varepsilon
x^{-\frac{2(w_{m}+1)}{w_{m}-1}}}{c_{1}}\right)\,_{2}F_{1}\left(\frac{1}{2},\frac{1-w_{m}}{2w_{m}+2};\frac{w_{m}+3}{2w_{m}+2};\frac{2V_{0}\varepsilon
x^{-\frac{2(w_{m}+1)}{w_{m}-1}}}{c_{1}}\right){}^{2}}{c_{1}-2V_{0}\varepsilon
x^{-\frac{2(w_{m}+1)}{w_{m}-1}}}=\left(c_{2}+t\right){}^{2}.$ (57)

Given the complexity in solving this implicit equation, we investigate
powerlaw solutions of the system.

In the special case where $\frac{4}{3}y_{0}z_{0}-\rho_{m0}=0$, the closed form
solution is
$x\left(t\right)\simeq\left(t-t_{0}\right)^{\frac{w_{m}-1}{2w_{m}}}\,$, where
for $w_{m}=-\frac{1}{3}$, $x\left(t\right)\simeq\left(t-t_{0}\right)^{2}$.

For $w_{m}\neq 0$ we can obtain a power-law solution as follows. We propose
the ansatz $x(t)=x_{0}(t-t_{1})^{p}$. Then we obtain the equations:

$\displaystyle(p-1)px_{0}(t-t_{1})^{p-2}-\frac{2V_{0}(w_{m}+1)\varepsilon
x_{0}^{-\frac{4}{w_{m}-1}-3}(t-t_{1})^{\frac{3pw_{m}+p}{1-w_{m}}}}{w_{m}-1}=0,$
(58)
$\displaystyle\frac{1}{2}p^{2}x_{0}^{2}\varepsilon(t-t_{1})^{2p-2}+V_{0}x_{0}^{-\frac{2(w_{m}+1)}{w_{m}-1}}(t-t_{1})^{-\frac{2p(w_{m}+1)}{w_{m}-1}}+\rho_{m0}-\frac{4y_{0}z_{0}}{3}=0.$
(59)

The first equation must be valid for all $t$. Then, balancing the powers we
obtain $p=\frac{w_{m}-1}{2w_{m}}$. Substituting back in the second equation we
obtain

$\rho_{m0}+\frac{x_{0}^{2}(t-t_{1})^{-\frac{w_{m}+1}{w_{m}}}\left(8V_{0}w_{m}^{2}x_{0}^{-\frac{4w_{m}}{w_{m}-1}}+(w_{m}-1)^{2}\varepsilon\right)}{8w_{m}^{2}}-\frac{4y_{0}z_{0}}{3}=0.$
(60)

This expression must be valid for all $t$. Therefore, we have the additional
restrictions in the parameters:

$V_{0}=-\frac{(w_{m}-1)^{2}\varepsilon
x_{0}^{\frac{4w_{m}}{w_{m}-1}}}{8w_{m}^{2}},\quad\rho_{m0}-\frac{4y_{0}z_{0}}{3}=0.$
(61)

Finally, we have the powerlaw analytic solution

$x(t)=x_{0}(t-t_{1})^{\frac{w_{m}-1}{2w_{m}}},\;y\left(t\right)=y_{0}\left(t-t_{2}\right)~{},~{}z\left(t\right)=z_{0}\left(t-t_{3}\right).$
(62)

Consider now the limit where
$x\left(t\right)\simeq\left(t-t_{1}\right)^{\frac{w_{m}-1}{2w_{m}}}$ then for
$w_{m}>0$, because $\frac{w_{m}-1}{2w_{m}}<0$, the scale factor at the limit
$t\rightarrow t_{1}$ is approximated as
$a\left(t\right)\simeq\left(t-t_{1}\right)^{-\frac{1}{6w_{m}}}$. Therefore,
the line element for the background space becomes

$ds^{2}=-\left(t-t_{1}\right)^{-1}dt^{2}+\left(t-t_{1}\right)^{-\frac{1}{3w_{m}}}\left(dx^{2}+dy^{2}+dz^{2}\right),$
(63)

or

$ds^{2}=-d\tau^{2}+e^{-\frac{1}{3w_{m}}\tau}\left(dx^{2}+dy^{2}+dz^{2}\right),$
(64)

which describes a de Sitter universe.

On the other hand when $w_{m}<0$, when $t\rightarrow t_{1}$ the scale factor
is approximated as $a\left(t\right)\simeq
t^{-\frac{2}{3\left(w_{m}-1\right)}}$. Therefore, the line element for
background space is simplified as

$ds^{2}=-t^{-\frac{4w_{m}}{\left(w_{m}-1\right)}}dt^{2}+t^{-\frac{4}{3\left(w_{m}-1\right)}}\left(dx^{2}+dy^{2}+dz^{2}\right),$
(65)

that is,

$ds^{2}=d\tau+\tau^{\frac{2}{3\left(1+w_{m}\right)}}\left(dx^{2}+dy^{2}+dz^{2}\right),$
(66)

which describes an accelerated universe for $w_{m}<-\frac{1}{3}$.

###  V.4 Scalar field potential $V_{IV}\left(\phi,\psi\right)$

The effective potential which corresponds to $V_{IV}\left(\phi,\psi\right)$ is
derived $U^{IV}\left(x,y,z\right)=\bar{V}_{0}\left(3\beta\varepsilon
x+4z\right)^{-2\frac{w_{m}+1}{w_{m}-1}}$ with
$\bar{V}_{0}=V_{0}\left(2\left(w-1\right)^{2}\right)^{-2\frac{w_{m}+1}{w_{m}-1}}$.
The field equations become

$\displaystyle\ddot{x}-6\bar{V}_{0}\frac{w_{m}+1}{w_{m}-1}\beta\varepsilon
Z^{-\frac{3w_{m}+1}{w_{m}-1}}$ $\displaystyle=0,$ (67)
$\displaystyle\ddot{y}+6\bar{V}_{0}\frac{w_{m}+1}{w_{m}-1}Z^{-\frac{3w_{m}+1}{w_{m}-1}}$
$\displaystyle=0,$ (68)
$\displaystyle\ddot{Z}-18\bar{V}_{0}\frac{w_{m}+1}{w_{m}-1}\beta^{2}\varepsilon
Z^{-\frac{3w_{m}+1}{w_{m}-1}}$ $\displaystyle=0,$ (69)

where $Z=3\beta\varepsilon x+4z$. The analytic solution of (69) is expressed
in terms of a hypergeometric function

$\displaystyle\frac{Z^{2}\left(1-\frac{18\beta^{3}V_{0}\varepsilon
Z^{-\frac{2(w_{m}+1)}{w_{m}-1}}}{c_{1}}\right)\,_{2}F_{1}\left(\frac{1}{2},\frac{1-w_{m}}{2w_{m}+2};\frac{w_{m}+3}{2w_{m}+2};\frac{18\bar{V}_{0}\beta^{3}\varepsilon
Z^{-\frac{2(w_{m}+1)}{w_{m}-1}}}{c_{1}}\right){}^{2}}{c_{1}-18\beta^{3}\bar{V}_{0}\varepsilon
Z^{-\frac{2(w_{m}+1)}{w_{m}-1}}}=\left(c_{2}+t\right){}^{2}.$ (70)

For $w_{m}\neq 0$ there exists the exact solution
$Z\left(t\right)=Z_{0}\left(t-t_{1}\right)^{\frac{w_{m}-1}{2w_{m}}}$, where
$Z_{0}$ is a solution of
$\left(w_{m}-1\right)^{2}+72Z_{0}^{-\frac{4}{w_{m}-1}-1}\bar{V}_{0}w^{2}\beta^{2}\varepsilon=0$.
Hence, we have the powerlaw solution

$\displaystyle
x\left(t\right)=\frac{Z_{0}(t-t_{1})^{\frac{w_{m}-1}{2w_{m}}}}{3\beta}+x_{0}\left(t-t_{2}\right),$
(71) $\displaystyle
y\left(t\right)=-\frac{Z_{0}(t-t_{1})^{\frac{w_{m}-1}{2w_{m}}}}{3\beta^{2}}\varepsilon+y_{0}\left(t-t_{3}\right),$
(72) $\displaystyle
Z\left(t\right)=Z_{0}\left(t-t_{1}\right)^{\frac{w_{m}-1}{2w_{m}}}.$ (73)

The restriction
$\rho_{m0}+\bar{V}_{0}Z^{-\frac{2(w_{m}+1)}{w_{m}-1}}-\frac{1}{3}\dot{y}\left(\dot{Z}-3\beta\dot{x}\right)+\frac{1}{2}\varepsilon{\dot{x}}^{2}=0$
becomes $\rho_{m0}+\frac{x_{0}^{2}\varepsilon}{2}+\beta x_{0}y_{0}=0$.

Thus for $w_{m}>0$ and in the limit $t\rightarrow t_{1}$ the scale factor
becomes
$a\left(t\right)\simeq\left(t-t_{1}\right)^{-\frac{3w_{m}-1}{6w_{m}\left(w_{m}-1\right)}}$
for $\frac{w_{m}-1}{2w_{m}}+1<0\,$. Hence the background space becomes

$ds^{2}=-\left(t-t_{1}\right)^{-\frac{3w_{m}-1}{\left(w_{m}-1\right)}}dt^{2}+\left(t-t_{1}\right)^{-\frac{3w_{m}-1}{3w_{m}\left(w_{m}-1\right)}}\left(dx^{2}+dy^{2}+dz^{2}\right),$
(74)

or equivalently

$ds^{2}=-d\tau^{2}+\tau^{\frac{1}{3w_{m}}}\left(dx^{2}+dy^{2}+dz^{2}\right),$
(75)

which describes an accelerated universe for
$w_{m}\in\left(0,\frac{1}{3}\right)$.

On the other hand, for $w_{m}<0$ for large values of $t$, the scale factor is
approximated as $a\left(t\right)\simeq t^{-\frac{2}{3\left(w_{m}-1\right)}}$,
which leads to the line element (65).

###  V.5 Scalar field potential $V_{V}\left(\phi,\psi\right)$

From the potential function $V_{V}\left(\phi,\psi\right)$ we calculate
$U_{eff}^{V}\left(x,y,z\right)=V_{0}\left(\left(w-1\right)^{2}\beta\varepsilon
x+4y\right)^{-2\frac{w_{m}+1}{w_{m}-1}}$. We define the new variable
$Y=\left(w-1\right)^{2}\beta\varepsilon x+4y$ where the field equations are
written as

$\displaystyle\ddot{Y}-2V_{0}\left(w_{m}-1\right)^{3}(w_{m}+1)\varepsilon\beta^{2}Y^{-\frac{3w_{m}+1}{w_{m}-1}}~{}$
$\displaystyle=0,$ (76)
$\displaystyle\ddot{x}-2V_{0}\left(w_{m}^{2}-1\right)\beta
Y^{-\frac{3w_{m}+1}{w_{m}-1}}$ $\displaystyle=0,$ (77)
$\displaystyle\ddot{z}+\frac{6V_{0}\left(w_{m}+1\right)}{w_{m}-1}Y^{-\frac{3w_{m}+1}{w_{m}-1}}$
$\displaystyle=0,$ (78)

with constraint

$\displaystyle\rho_{m0}+V_{0}Y^{-\frac{2(w_{m}+1)}{w_{m}-1}}-\frac{1}{3}\dot{z}\left(\dot{Y}-\beta(w_{m}-1)^{2}\varepsilon\dot{x}\right)+\frac{1}{2}\varepsilon{\dot{x}}^{2}=0.$
(79)

The closed-form solution of equation (76) is expressed in terms of
hypergeometric function

$\displaystyle\frac{Y^{2}\left(1-\frac{2\beta^{2}V_{0}(w_{m}-1)^{4}\varepsilon
Y^{-\frac{2(w_{m}+1)}{w_{m}-1}}}{c_{1}}\right)\,_{2}F_{1}\left(\frac{1}{2},\frac{1-w_{m}}{2w_{m}+2};\frac{w_{m}+3}{2w_{m}+2};\frac{2V_{0}(w_{m}-1)^{4}\beta^{2}\varepsilon
Y^{-\frac{2(w_{m}+1)}{w_{m}-1}}}{c_{1}}\right){}^{2}}{c_{1}-2\beta^{2}V_{0}(w_{m}-1)^{4}\varepsilon
Y^{-\frac{2(w_{m}+1)}{w_{m}-1}}}=\left(c_{2}+t\right){}^{2}.$ (80)

For $w_{m}=-\frac{1}{3}$ we recover the closed-form solution

$\displaystyle x\left(t\right)=-\frac{8}{9}\beta
V_{0}(t-t_{1})^{2}+x_{0}(t-t_{2}),\;Y\left(t\right)=Y_{0}(t-t_{1})^{2},\;z\left(t\right)=\frac{3}{2}V_{0}(t-t_{1})^{2}+z_{0}(t-t_{3}),$
(81)

with $V_{0}=-\frac{81Y_{0}}{128\beta^{2}\varepsilon},\quad
6\rho_{m0}+x_{0}\varepsilon\left(3x_{0}+\frac{32\beta z_{0}}{9}\right)=0$.

The special exact solutions for arbitrary value of $w_{m}\neq 0$ exist and for
this potential in a similar way as we calculated them for potential
$V_{IV}\left(\phi,\psi\right)$ where we have to perform the change of
variables $\left(y,z\right)\rightarrow\left(z,y\right)$. We can obtain a
power-law solution

$\displaystyle Y(t)=Y_{0}(t-t_{1})^{\frac{w_{m}-1}{2w_{m}}},$ (82)
$\displaystyle x(t)=\frac{\varepsilon
Y_{0}(t-t_{1})^{\frac{w_{m}-1}{2w_{m}}}}{\beta(w_{m}-1)^{2}}+x_{0}(t-t_{2}),$
(83) $\displaystyle z(t)=-\frac{3\varepsilon
Y_{0}(t-t_{1})^{\frac{w_{m}-1}{2w_{m}}}}{\beta^{2}(w_{m}-1)^{4}}+z_{0}(t-t_{3}),$
(84)

where $8\beta^{2}V_{0}(w_{m}-1)^{3}w_{m}^{2}(w_{m}+1)\varepsilon
Y_{0}^{-\frac{4w_{m}}{w_{m}-1}}+w_{m}^{2}-1=0$ gives $V_{0}$ and
$6\rho_{m0}+x_{0}\varepsilon\left(3x_{0}+2\beta(w_{m}-1)^{2}z_{0}\right)=0$.

###  V.6 Scalar field potential $V_{VI}\left(\phi,\psi\right)$

Finally, from $V_{VI}\left(\phi,\psi\right)$ it follows
$U_{eff}^{VI}\left(x,y,z\right)=V_{0}\left(6\left(w_{m}-1\right)^{2}\beta\varepsilon
x+8\left(\beta
y+\left(w-1\right)^{2}\right)z\right)^{-2\frac{w_{m}+1}{w_{m}-1}}$. With the
use of the new variable $U=6\left(w_{m}-1\right)^{2}\beta\varepsilon
x\gamma+8\left(\beta y+\left(w-1\right)^{2}\gamma z\right)$, the field
equations become

$\displaystyle\ddot{U}-24\gamma
V_{0}\left(w_{m}^{2}-1\right)\beta\left(\varepsilon\beta\left(w_{m}-1\right)^{2}-8\right)U^{-\frac{3w_{m}+1}{w_{m}-1}}~{}$
$\displaystyle=0,$ (85)
$\displaystyle\ddot{x}-12V_{0}\gamma\left(w_{m}^{2}-1\right)\beta
U^{-\frac{3w_{m}+1}{w_{m}-1}}$ $\displaystyle=0,$ (86)
$\displaystyle\ddot{z}+\frac{12V_{0}\left(w_{m}+1\right)}{w_{m}-1}\beta
U^{-\frac{3w_{m}+1}{w_{m}-1}}$ $\displaystyle=0.$ (87)

with constraint

$\displaystyle\rho_{m0}-\frac{\dot{U}\dot{z}}{6\beta}+V_{0}U^{-\frac{2(w+1)}{w-1}}+\gamma(w-1)^{2}\varepsilon\dot{x}\dot{z}+\frac{4\gamma(w-1)^{2}\dot{z}^{2}}{3\beta}+\frac{1}{2}\varepsilon\dot{x}^{2}=0.$
(88)

The results from the previous two potentials follow: the behaviour of the
solution at the limits has similar properties as before.

Firstly, we solve (85). The general solution is given in terms of a
hypergeometric function:

$\displaystyle\frac{U^{2}\left(1-\frac{24\beta\gamma
V_{0}(w_{m}-1)^{2}U^{-\frac{2(w_{m}+1)}{w_{m}-1}}\left(\beta(w_{m}-1)^{2}\varepsilon-8\right)}{c_{1}}\right)\,_{2}F_{1}\left(\frac{1}{2},\frac{1-w_{m}}{2w_{m}+2};\frac{w_{m}+3}{2w_{m}+2};\frac{24V_{0}(w_{m}-1)^{2}\beta\gamma\left((w_{m}-1)^{2}\beta\varepsilon-8\right)U^{-\frac{2(w_{m}+1)}{w_{m}-1}}}{c_{1}}\right){}^{2}}{c_{1}-24\beta\gamma
V_{0}(w_{m}-1)^{2}U^{-\frac{2(w_{m}+1)}{w_{m}-1}}\left(\beta(w_{m}-1)^{2}\varepsilon-8\right)}$
$\displaystyle=\left(c_{2}+t\right){}^{2}.$ (89)

For $w_{m}\neq 0$ the system (85), (86), (87) admits the following power-law
solution

$\displaystyle U(t)=U_{0}(t-t_{1})^{\frac{w_{m}-1}{2w_{m}}},$ (90)
$\displaystyle
x(t)=\frac{U_{0}(t-t_{1})^{\frac{w_{m}-1}{2w_{m}}}}{6\beta\gamma(w_{m}-1)^{2}\varepsilon-16}+x_{0}(t-t_{2}),$
(91) $\displaystyle
z(t)=-\frac{U_{0}(t-t_{1})^{\frac{w_{m}-1}{2w_{m}}}}{2\gamma(w_{m}-1)^{2}\left(3\beta\gamma(w_{m}-1)^{2}\varepsilon-8\right)}+z_{0}(t-t_{3}),$
(92)

where $U_{0}^{\frac{4w_{m}}{w_{m}-1}}+96\beta\gamma
V_{0}w_{m}^{2}\left(3\beta\gamma(w_{m}-1)^{2}\varepsilon-8\right)=0$ gives
$V_{0}$ and $6\beta\rho_{m0}+2\gamma(w-1)^{2}z_{0}(3\beta
x_{0}\varepsilon+4z_{0})+3\beta x_{0}^{2}\varepsilon=0$ gives $\rho_{m0}$.

##  VI Stability analysis of the scaling solutions

According to the methods in Ratra:1987rm ; Liddle:1998xm ; Uzan:1999ch  let be

$F(\ddot{\phi},\dot{\phi},\phi)=0,$ (93)

a second order differential equation in one dimension which admits a singular
powerlaw solution

$\phi_{c}(t)=\phi_{0}t^{\beta}.$ (94)

To examine the stability of the solution $\phi_{c}$, the logarithmic time
$\tau$ through $t=e^{\tau}$ is introduced, such that $t\rightarrow 0$ as
$\tau\rightarrow-\infty$ and $t\rightarrow+\infty$ as
$\tau\rightarrow+\infty$. We use $\phi^{\prime}\equiv\frac{d\phi}{d\tau}$ in
the following discussion.

The following dimensionless function is introduced

$u(\tau)=\frac{\phi(\tau)}{\phi_{c}(\tau)},$ (95)

and the stability analysis in translated into the analysis of the stability of
$u=1$ of a transformed dynamical system. To construct the aforementioned
system the following relations are useful:

$\dot{\phi}=e^{-\tau}\phi^{\prime},\quad\ddot{\phi}=e^{-2\tau}(\phi^{\prime\prime}-\phi^{\prime}),\quad\text{and}\quad\frac{\phi_{c}^{\prime}}{\phi_{c}}=\beta\quad\text{if}\quad\phi_{c}(t)=\phi_{0}t^{\beta}.$
(96)

In this section we use a similar procedure for analyzing stability of the
scaling solutions that are obtained in Section V.
Due to the complexity of the dynamics, we have supported our analytical
results of this section in numerical simulations by using initial conditions
within the constraint surfaces:
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=1$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=1$. The auxiliary variables
$\phi_{i},\Phi_{i}$ are dummy variables that are defined for each potential,
and they are used to classify the stability of the powerlaw solutions obtained
for each potential. The forward integration is based in two simple basis: i)
if the point is an attractor, the boundary surfaces act as trapping surfaces
due to all the orbits are attracted by the basing of attraction of the
(1,0,1,0,1,0); ii) if the point (1,0,1,0,1,0) have saddle behavior, some
orbits would abandon this region passing through the boundary surfaces to
higher radii, say, having to the past and to the future that
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}>1$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}>1$.

###  VI.1 Scalar field potential $V_{I}\left(\phi,\psi\right)$

In this section we analyze the stability of the analytic solution (42), (43)
of equations (40) (41). We set for simplicity the integration constants
$t_{1},t_{2},t_{3}$ to zero because they are not relevant as
$t\rightarrow\infty$.

With the time variable $\tau=\ln(t)$, and defining the new variables
$Y=V_{0}e^{2\tau}y^{-\frac{3w_{m}+1}{w_{m}-1}}$ and $Z=z(\tau)-z_{0}e^{\tau}$
the equations (40) and (41) become

$\displaystyle x^{\prime\prime}(\tau)-x^{\prime}(\tau)=0,\quad
Z^{\prime\prime}(\tau)-Z^{\prime}(\tau)+\frac{3}{2}\frac{w_{m}+1}{w_{m}-1}Y(\tau)=0,$
$\displaystyle-4w_{m}Y^{\prime}(\tau)^{2}+Y(\tau)\left((3w_{m}+1)Y^{\prime\prime}(\tau)+(w_{m}-5)Y^{\prime}(\tau)\right)+2(w_{m}+3)Y(\tau)^{2}=0.$

The analytic solution (42), (43) becomes:

$x_{c}\left(\tau\right)=x_{0}e^{\tau},~{}Y_{c}\left(\tau\right)=Y_{0}e^{-\frac{\tau(w_{m}+3)}{w_{m}-1}},\quad
Y_{0}=V_{0}y_{0}^{-\frac{3w_{m}+1}{w_{m}-1}}\;Z_{c}\left(\tau\right)=z_{1}e^{-\frac{\tau(w_{m}+3)}{w_{m}-1}},\quad
z_{1}=-\frac{3(w-1)Y_{0}}{4(w+3)}.$

Defining the dimensionless variables

$\phi_{1}=\frac{x}{x_{c}},\quad\phi_{2}=\frac{Y}{Y_{c}},\quad\phi_{3}=\frac{Z}{Z_{c}},$
(97)

we obtain the dynamical system

$\displaystyle\phi_{1}^{\prime}=\Phi_{1},\quad\Phi_{1}^{\prime}=-\Phi_{1},$
(98)
$\displaystyle\phi_{2}^{\prime}=\Phi_{2},\quad\Phi_{2}^{\prime}=-\Phi_{2}+\frac{4w_{m}\Phi_{2}^{2}}{\phi_{2}(1+3w_{m})},$
(99)
$\displaystyle\phi_{3}^{\prime}=\Phi_{3},\quad\Phi_{3}^{\prime}=\frac{(3w_{m}+5)\Phi_{3}}{w_{m}-1}+\frac{2(w_{m}+1)(w_{m}+3)(\phi_{2}-\phi_{3})}{(w_{m}-1)^{2}}.$
(100)

Now we analyze the stability of the solution
$P:=(\phi_{1},\Phi_{1},\phi_{2},\Phi_{2},\phi_{3},\Phi_{3})=(1,0,1,0,1,0)$.
The subsystems for $(\phi_{1},\Phi_{1})$, $(\phi_{2},\Phi_{2})$ and
$(\phi_{2},\Phi_{2},\phi_{3},\Phi_{3})$ are decoupled. The Jacobian matrix of
the full system is

$J:=\left(\begin{array}[]{cccccc}0&1&0&0&0&0\\\ 0&-1&0&0&0&0\\\ 0&0&0&1&0&0\\\
0&0&-\frac{4w_{m}\Phi_{2}^{2}}{(3w_{m}+1)\phi_{2}^{2}}&\frac{8w_{m}\Phi_{2}}{3w_{m}\phi_{2}+\phi_{2}}-1&0&0\\\
0&0&0&0&0&1\\\
0&0&\frac{2(w_{m}+1)(w_{m}+3)}{(w_{m}-1)^{2}}&0&-\frac{2(w_{m}+1)(w_{m}+3)}{(w_{m}-1)^{2}}&3+\frac{8}{w_{m}-1}\\\
\end{array}\right).$ (101)

Evaluating $J$ at the fixed point $P$ the eigenvalues
$\left\\{0,0,-1,-1,-\frac{2(w_{m}+1)}{1-w_{m}},-\frac{w_{m}+3}{1-w_{m}}\right\\}$
are obtained. The stable manifold is 4D if $-1<w_{m}<1$. If the analysis is
restricted to the subspace $(\phi_{2},\Phi_{2},\phi_{3},\Phi_{3})$ the
eigenvalues are
$\left\\{0,-1,-\frac{2(w_{m}+1)}{1-w_{m}},-\frac{w_{m}+3}{1-w_{m}}\right\\}$.
The stable manifold in this subspace is 3D if $-1<w_{m}<1$.

Introducing the new variables

$\displaystyle
u_{1}=\Phi_{2}+\phi_{2}-1,\;u_{2}=\Phi_{1}+\phi_{1}-1,\;v_{1}=\frac{\Phi_{2}(w_{m}+3)}{3w_{m}+1},\;v_{2}=\Phi_{1},$
(102) $\displaystyle
v_{3}=\frac{2(w_{m}+1)(\Phi_{3}(w_{m}-1)(3w_{m}+1)+(w_{m}+3)(\Phi_{2}(w_{m}-1)+3w_{m}\phi_{2}-(3w_{m}+1)\phi_{3}+\phi_{2}))}{(w_{m}-1)^{2}(3w_{m}+1)},$
(103) $\displaystyle
v_{4}=-\frac{(w_{m}+3)(\Phi_{2}(w_{m}-1)+\Phi_{3}(w_{m}-1)+2(w_{m}+1)\phi_{2}-2(w_{m}+1)\phi_{3})}{(w_{m}-1)^{2}},$
(104)

(a) Projections in the space ($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) (left) and
($\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$) (right). We have represented the spheres
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=r^{2}$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=r^{2}$, $r\in\\{1,\sqrt{2}\\}$. (b)
From left to right, from top to bottom: Projections in the planes
($u_{1},u_{2}$), ($v_{1},v_{2}$), ($v_{1},v_{3}$), ($v_{1},v_{4}$),
($v_{2},v_{3}$), ($v_{2},v_{4}$), ($v_{3},v_{4}$), ($u_{1},v_{1}$) and
($u_{2},v_{2}$). The red line in fig 8 (resp. in fig 9) of the bottom array
denotes the invariant set $v_{1}=0$ (resp. $v_{2}=0$) in the projection
$u_{1}$ vs $v_{1}$ (resp. $u_{2}$ vs $v_{2}$).

Figure 1: Some solutions of (a) the system (98)-(100) and (b) the system (105)
for the potential $V_{I}(\phi,\psi)$ when $\omega_{m}=-1$. Notice in the
projection $u_{1}$ vs $u_{2}$ (which contains the center manifold) the origin
behaves as a saddle point. The orbits for the left along the $u_{1}$-axis tend
to the origin, but for the right the orbits depart from the origin.

(a) Projections in the space ($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) (left) and
($\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$) (right). We have represented the spheres
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=r^{2}$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=r^{2}$, $r\in\\{1,\sqrt{2}\\}$. (b)
From left to right, from top to bottom: Projections in the planes
($u_{1},u_{2}$), ($v_{1},v_{2}$), ($v_{1},v_{3}$), ($v_{1},v_{4}$),
($v_{2},v_{3}$), ($v_{2},v_{4}$), ($v_{3},v_{4}$), ($u_{1},v_{1}$) and
($u_{2},v_{2}$). The red line in fig 8 (resp. in fig 9) of the bottom array
denotes the invariant set $v_{1}=0$ (resp. $v_{2}=0$) in the projection
$u_{1}$ vs $v_{1}$ (resp. $u_{2}$ vs $v_{2}$).

Figure 2: Some solutions of (a) the system (98)-(100) and (b) the system (105)
for the potential $V_{I}(\phi,\psi)$ when $\omega_{m}=0$. Notice in the
projection $u_{1}$ vs $u_{2}$ (which contains the center manifold) the origin
is stable (but not asymptotically stable) since any $\epsilon$-neighborhood of
the origin will contain a $\delta$-neighborhood of origin with other points
apart of the origin with
$(u_{1}^{\prime},u_{2}^{\prime})|_{u_{1}=u_{1}^{*},u_{2}=u_{2}^{*}}=(0,0)$.
Therefore, they remain in $\delta$-neighborhood of origin.

(a) Projections in the space ($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) (left) and
($\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$) (right). We have represented the spheres
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=r^{2}$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=r^{2}$, $r\in\\{1,\sqrt{2}\\}$. (b)
From left to right, from top to bottom: Projections in the planes
($u_{1},u_{2}$), ($v_{1},v_{2}$), ($v_{1},v_{3}$), ($v_{1},v_{4}$),
($v_{2},v_{3}$), ($v_{2},v_{4}$), ($v_{3},v_{4}$), ($u_{1},v_{1}$) and
($u_{2},v_{2}$). The red line in fig 8 (resp. in fig 9) of the bottom array
denotes the invariant set $v_{1}=0$ (resp. $v_{2}=0$) in the projection
$u_{1}$ vs $v_{1}$ (resp. $u_{2}$ vs $v_{2}$).

Figure 3: Some solutions of (a) the system (98)-(100) and (b) the system (105)
for the potential $V_{I}(\phi,\psi)$ when $\omega_{m}=\frac{1}{3}$. Notice in
the projection $u_{1}$ vs $u_{2}$ (which contains the center manifold) the
origin behaves as a saddle point. The orbits for the left along the
$u_{1}$-axis tend to the origin, but for the right the orbits depart from the
origin.

we obtain the system

$\displaystyle
u_{1}^{\prime}=\frac{4v_{1}^{2}w_{m}(3w_{m}+1)}{(w_{m}+3)(u_{1}(w_{m}+3)-v_{1}(3w_{m}+1)+w_{m}+3)},$
(105a) $\displaystyle u_{2}^{\prime}=0,$ (105b) $\displaystyle
v_{1}^{\prime}=\frac{v_{1}(-u_{1}(w_{m}+3)+7v_{1}w_{m}+v_{1}-w_{m}-3)}{u_{1}(w_{m}+3)-v_{1}(3w_{m}+1)+w_{m}+3},$
(105c) $\displaystyle v_{2}^{\prime}=-v_{2},$ (105d) $\displaystyle
v_{3}^{\prime}=\frac{2(w_{m}+1)\left(v_{3}(u_{1}(w_{m}+3)-3v_{1}w_{m}-v_{1}+w_{m}+3)+4v_{1}^{2}w_{m}\right)}{(w_{m}-1)(u_{1}(w_{m}+3)-v_{1}(3w_{m}+1)+w_{m}+3)},$
(105e) $\displaystyle
v_{4}^{\prime}=\frac{v_{4}(w_{m}+3)(u_{1}(w_{m}+3)-3v_{1}w_{m}-v_{1}+w_{m}+3)-4v_{1}^{2}w_{m}(3w_{m}+1)}{(w_{m}-1)(u_{1}(w_{m}+3)-v_{1}(3w_{m}+1)+w_{m}+3)}.$
(105f)

The center manifold of the origin is given by

$\Bigg{\\{}(u_{1},u_{2},v_{1},v_{2},v_{3},v_{4})\in\mathbb{R}^{6}:v_{i}=h_{i}(u_{1},u_{2}),\frac{\partial
h_{i}}{\partial u_{1}}(0,0)=0,\frac{\partial h_{i}}{\partial
u_{2}}(0,0)=0,h_{i}(0,0)=0,i=1\ldots 4\Bigg{\\}}.$ (106)

where $h_{i}(u_{1},u_{2})$ satisfies the system of quasi-linear partial
differential equations

$\displaystyle h_{1}\left(h_{1}\left(4w_{m}(3w_{m}+1)\frac{\partial
h_{1}}{\partial
u_{1}}+(-w_{m}-3)(7w_{m}+1)\right)+(u_{1}+1)(w_{m}+3)^{2}\right)=0,$ (107)
$\displaystyle 4w_{m}(3w_{m}+1)h_{1}^{2}\frac{\partial h_{2}}{\partial
u_{1}}+(w_{m}+3)h_{2}((u_{1}+1)(w_{m}+3)-(3w_{m}+1)h_{1})=0,$ (108)
$\displaystyle 2w_{m}((2-3w_{m})w_{m}+1)h_{1}^{2}\frac{\partial
h_{3}}{\partial
u_{1}}+(w_{m}+1)(w_{m}+3)\left(h_{3}((u_{1}+1)(w_{m}+3)-(3w_{m}+1)h_{1})+4w_{m}h_{1}^{2}\right)=0,$
(109)
$\displaystyle-(u_{1}+1)(w_{m}+3)^{3}h_{4}+(3w_{m}+1)h_{1}\left(4w_{m}h_{1}\left((w_{m}-1)\frac{\partial
h_{4}}{\partial u_{1}}+w_{m}+3\right)+(w_{m}+3)^{2}h_{4}\right)=0.$ (110)

Using the expansion in series

$h_{i}(u_{1},u_{2})=\sum_{n=2}^{N}\sum_{k=0}^{n}a^{[i]}_{nk}u_{1}^{n-k}u_{2}^{k}+\mathcal{O}(\|(u_{1},u_{2})\|^{N+1}),i=1,\ldots
4$ (111)

the zero solution for any given accuracy is found.

In figures 1-3 some solutions of the systems (98)-(100) and (105) for the
potential $V_{I}(\phi,\psi)$ when $\omega_{m}=-1$, $0$ and $\frac{1}{3}$ are
represented. More specific, in the upper panel the solutions are projected in
the space ($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) and ($\Phi_{1}$, $\Phi_{2}$,
$\Phi_{3}$), where we have represented the spheres
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=r^{2}$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=r^{2}$ with $r\in\\{1,\sqrt{2}\\}$. In
the lower panel projections in the spaces ($u_{1},u_{2}$), ($v_{1},v_{2}$),
($v_{1},v_{3}$), ($v_{1},v_{4}$), ($v_{2},v_{3}$), ($v_{2},v_{4}$),
($v_{3},v_{4}$), ($u_{1},v_{1}$) and ($u_{2},v_{2}$) are represented. In these
figures we have depicted a red line in the projections ($u_{1},v_{1}$) and
($u_{2},v_{2}$) which denotes the invariant set $v_{1}=0$ and $v_{2}=0$,
respectively. Both lines are stable in these projections. Notice that in
figures 1 and 3, the projection $u_{1}$ vs $u_{2}$, which contains the center
manifold; the origin behaves as a saddle point. The orbits from the left along
the $u_{1}$-axis tend to the origin, but from the right the orbits depart from
the origin. Then, the solution is unstable (saddle behavior). This behavior is
also represented in the 3D projection $(\phi_{1},\phi_{2},\phi_{3})$ where
some orbits abandon the inner spheres backward and forward in time. On the
other hand, the projection $u_{1}$ vs $u_{2}$ in figure 2; the origin is
stable (but not asymptotically stable) since any $\epsilon$-neighborhood of
the origin will contain a $\delta$-neighborhood of origin with other points
apart of the origin with
$(u_{1}^{\prime},u_{2}^{\prime})|_{u_{1}=u_{1}^{*},u_{2}=u_{2}^{*}}=(0,0)$.
Therefore, they remain in $\delta$-neighborhood of origin.

###  VI.2 Scalar field potential $V_{II}\left(\phi,\psi\right)$

In this section we study the stability of the solution (52) of the system
(50), (51). We set for simplicity the integration constants
$t_{1},t_{2},t_{3}$ to zero because they are not relevant as
$t\rightarrow\infty$. Noticing for the scalar field potential
$V_{II}\left(\phi,\psi\right)~{}$we derive
$U_{eff}^{II}\left(x,y,z\right)=V_{0}z^{-2\frac{w_{m}+1}{w_{m}-1}}$. We
observe that doing the change of variables
$\left(y,z\right)\rightarrow\left(z,y\right)$ we end with the dynamical system
of potential $V_{I}\left(\phi,\psi\right)$. Hence, with the time variable
$\tau=\ln(t)$, and defining the new variables
$Z=V_{0}e^{2\tau}z^{-\frac{3w_{m}+1}{w_{m}-1}}$ and $Y=y(\tau)-y_{0}e^{\tau}$
the field equations (50), (51) become

$\displaystyle x^{\prime\prime}(\tau)-x^{\prime}(\tau)=0,\quad
Y^{\prime\prime}(\tau)-Y^{\prime}(\tau)+\frac{3}{2}\frac{w_{m}+1}{w_{m}-1}Z(\tau)=0,$
$\displaystyle-4w_{m}Z^{\prime}(\tau)^{2}+Z(\tau)\left((3w_{m}+1)Z^{\prime\prime}(\tau)+(w_{m}-5)Z^{\prime}(\tau)\right)+2(w_{m}+3)Z(\tau)^{2}=0.$

The analytical solution of the original system becomes

$x_{c}\left(\tau\right)=x_{0}e^{\tau},~{}Y_{c}\left(\tau\right)=y_{1}e^{-\frac{\tau(w_{m}+3)}{w_{m}-1}},\;y_{1}=-\frac{3(w-1)Z_{0}}{4(w+3)},\;Z_{c}\left(\tau\right)=Z_{0}e^{-\frac{\tau(w_{m}+3)}{w_{m}-1}},\quad
Z_{0}=V_{0}z_{0}^{-\frac{3w_{m}+1}{w_{m}-1}},$

Defining the dimensionless variables

$\phi_{1}=\frac{x}{x_{c}},\quad\phi_{2}=\frac{Y}{Y_{c}},\quad\phi_{3}=\frac{Z}{Z_{c}},$

we obtain the dynamical system

$\displaystyle\phi_{1}^{\prime}=\Phi_{1},\quad\Phi_{1}^{\prime}=-\Phi_{1},$
(112)
$\displaystyle\phi_{2}^{\prime}=\Phi_{2},\quad\Phi_{2}^{\prime}=\frac{(3w_{m}+5)\Phi_{2}}{w_{m}-1}+\frac{2(w_{m}+1)(w_{m}+3)(\phi_{3}-\phi_{2})}{(w_{m}-1)^{2}},$
(113)
$\displaystyle\phi_{3}^{\prime}=\Phi_{3},\quad\Phi_{3}^{\prime}=-\Phi_{3}+\frac{4w_{m}\Phi_{3}^{2}}{\phi_{3}(1+3w_{m})}.$
(114)

Now we analyze the stability of the fixed point
$P:=(\phi_{1},\Phi_{1},\phi_{2},\Phi_{2},\phi_{3},\Phi_{3})=(1,0,1,0,1,0)$.
The subsystems for $(\phi_{1},\Phi_{1})$, $(\phi_{3},\Phi_{3})$ and
$(\phi_{2},\Phi_{2},\phi_{3},\Phi_{3})$ are decoupled.

The Jacobian matrix of the full system is

$J:=\left(\begin{array}[]{cccccc}0&1&0&0&0&0\\\ 0&-1&0&0&0&0\\\ 0&0&0&1&0&0\\\
0&0&-\frac{2(w_{m}+1)(w_{m}+3)}{(w_{m}-1)^{2}}&3+\frac{8}{w_{m}-1}&\frac{2(w_{m}+1)(w_{m}+3)}{(w_{m}-1)^{2}}&0\\\
0&0&0&0&0&1\\\
0&0&0&0&-\frac{4w_{m}\Phi_{3}^{2}}{(3w_{m}+1)\phi_{3}^{2}}&\frac{8w_{m}\Phi_{3}}{(3w_{m}+1)\phi_{3}}-1\\\
\end{array}\right).$ (115)

Evaluating $J$ at the fixed point $P$ the eigenvalues
$\left\\{0,0,-1,-1,-\frac{2(w_{m}+1)}{1-w_{m}},-\frac{w_{m}+3}{1-w_{m}}\right\\}$
are found. The stable manifold is 4D if $-1<w_{m}<1$. If the analysis is
restricted to the subspace $(\phi_{2},\Phi_{2},\phi_{3},\Phi_{3})$ the
eigenvalues are
$\left\\{0,-1,-\frac{2(w_{m}+1)}{1-w_{m}},-\frac{w_{m}+3}{1-w_{m}}\right\\}$.
The stable manifold in this subspace is 3D if $-1<w_{m}<1$.

Introducing the new variables

$\displaystyle
u_{1}=\Phi_{3}+\phi_{3}-1,\;u_{2}=\Phi_{1}+\phi_{1}-1,\;v_{1}=\Phi_{3},\;v_{2}=\Phi_{1},$
$\displaystyle
v_{3}=\frac{2(w_{m}+1)(\Phi_{3}(w_{m}-1)(w_{m}+3)+(-3w_{m}-1)(\Phi_{2}-w_{m}(\Phi_{2}+\phi_{3})+(w_{m}+3)\phi_{2}-3\phi_{3}))}{(w_{m}-1)^{2}(3w_{m}+1)},$
$\displaystyle
v_{4}=\frac{(w_{m}+3)(\Phi_{2}+\Phi_{3}-w_{m}(\Phi_{2}+\Phi_{3}+2\phi_{3})+2(w_{m}+1)\phi_{2}-2\phi_{3})}{(w_{m}-1)^{2}},$

(a) Projections in the space ($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) (left) and
($\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$) (right). We have represented the spheres
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=r^{2}$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=r^{2}$, $r\in\\{1,\sqrt{2}\\}$. (b)
From left to right, from top to bottom: Projections in the planes
($u_{1},u_{2}$), ($v_{1},v_{2}$), ($v_{1},v_{3}$), ($v_{1},v_{4}$),
($v_{2},v_{3}$), ($v_{2},v_{4}$), ($v_{3},v_{4}$), ($u_{1},v_{1}$) and
($u_{2},v_{2}$). The red line in fig 8 (resp. fig 9) of the bottom array
denotes the invariant set $v_{1}=0$ (resp. $v_{2}=0$) in the projection
$u_{1}$ vs $v_{1}$ (resp. $u_{2}$ vs $v_{2}$).

Figure 4: Some solutions of (a) the system (112)-(114) and (b) the system
(116) for the potential $V_{II}(\phi,\psi)$ when $\omega_{m}=-1$. Notice in
the projection $u_{1}$ vs $u_{2}$ (which contains the center manifold) the
origin behaves as a saddle point. The orbits for the left along the
$u_{1}$-axis tend to the origin, but for the right the orbits depart from the
origin.

(a) Projections in the space ($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) (left) and
($\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$) (right). We have represented the spheres
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=r^{2}$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=r^{2}$, $r\in\\{1,\sqrt{2}\\}$. (b)
From left to right, from top to bottom: Projections in the planes
($u_{1},u_{2}$), ($v_{1},v_{2}$), ($v_{1},v_{3}$), ($v_{1},v_{4}$),
($v_{2},v_{3}$), ($v_{2},v_{4}$), ($v_{3},v_{4}$), ($u_{1},v_{1}$) and
($u_{2},v_{2}$). The red line in fig 8 (resp. fig 9) of the bottom array
denotes the invariant set $v_{1}=0$ (resp. $v_{2}=0$) in the projection
$u_{1}$ vs $v_{1}$ (resp. $u_{2}$ vs $v_{2}$).

Figure 5: Some solutions of (a) the system (112)-(114) and (b) the system
(116) for the potential $V_{II}(\phi,\psi)$ when $\omega_{m}=0$. Notice in the
projection $u_{1}$ vs $u_{2}$ (which contains the center manifold) the origin
is stable (but not asymptotically stable) since any $\epsilon$-neighborhood of
the origin will contain a $\delta$-neighborhood of origin with other points
apart of the origin with
$(u_{1}^{\prime},u_{2}^{\prime})|_{u_{1}=u_{1}^{*},u_{2}=u_{2}^{*}}=(0,0)$.
Therefore, they remain in $\delta$-neighborhood of origin.

(a) Projections in the space ($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) (left) and
($\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$) (right). We have represented the spheres
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=r^{2}$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=r^{2}$, $r\in\\{1,\sqrt{2}\\}$. (b)
From left to right, from top to bottom: Projections in the planes
($u_{1},u_{2}$), ($v_{1},v_{2}$), ($v_{1},v_{3}$), ($v_{1},v_{4}$),
($v_{2},v_{3}$), ($v_{2},v_{4}$), ($v_{3},v_{4}$), ($u_{1},v_{1}$) and
($u_{2},v_{2}$). The red line in fig 8 (resp. fig 9) of the bottom array
denotes the invariant set $v_{1}=0$ (resp. $v_{2}=0$) in the projection
$u_{1}$ vs $v_{1}$ (resp. $u_{2}$ vs $v_{2}$).

Figure 6: Some solutions of (a) the system (112)-(114) and (b) the system
(116) for the potential $V_{II}(\phi,\psi)$ when $\omega_{m}=\frac{1}{3}$.
Notice in the projection $u_{1}$ vs $u_{2}$ (which contains the center
manifold) the origin behaves as a saddle point. The orbits for the left along
the $u_{1}$-axis tend to the origin, but for the right the orbits depart from
the origin.

we obtain

$\displaystyle
u_{1}^{\prime}=\frac{4v_{1}^{2}w_{m}}{(3w_{m}+1)(u_{1}-v_{1}+1)},$ (116a)
$\displaystyle u_{2}^{\prime}=0,$ (116b) $\displaystyle
v_{1}^{\prime}=v_{1}\left(\frac{4v_{1}w_{m}}{(3w_{m}+1)(u_{1}-v_{1}+1)}-1\right),$
(116c) $\displaystyle v_{2}^{\prime}=-v_{2},$ (116d) $\displaystyle
v_{3}^{\prime}=\frac{2(w_{m}+1)\left(v_{3}(3w_{m}+1)^{2}(u_{1}-v_{1}+1)+4v_{1}^{2}w_{m}(w_{m}+3)\right)}{(w_{m}-1)(3w_{m}+1)^{2}(u_{1}-v_{1}+1)},$
(116e) $\displaystyle
v_{4}^{\prime}=\frac{(w_{m}+3)\left(v_{4}(3w_{m}+1)(u_{1}-v_{1}+1)-4v_{1}^{2}w_{m}\right)}{(w_{m}-1)(3w_{m}+1)(u_{1}-v_{1}+1)}.$
(116f)

The center manifold of the origin is given by

$\Bigg{\\{}(u_{1},u_{2},v_{1},v_{2},v_{3},v_{4})\in\mathbb{R}^{6}:v_{i}=h_{i}(u_{1},u_{2}),\frac{\partial
h_{i}}{\partial u_{1}}(0,0)=0,\frac{\partial h_{i}}{\partial
u_{2}}(0,0)=0,h_{i}(0,0)=0,i=1\ldots 4\Bigg{\\}}.$ (117)

where $h_{i}(u_{1},u_{2})$ satisfies the system of quasi-linear partial
differential equations

$\displaystyle
h_{1}((7w_{m}+1)h_{1}+(-u_{1}-1)(3w_{m}+1))-4w_{m}h_{1}^{2}\frac{\partial
h_{1}}{\partial u_{1}}=0,$ (118) $\displaystyle 4w_{m}h_{1}^{2}\frac{\partial
h_{2}}{\partial u_{1}}+(3w_{m}+1)(-h_{1}+u_{1}+1)h_{2}=0,$ (119)
$\displaystyle 2w_{m}((2-3w_{m})w_{m}+1)h_{1}^{2}\frac{\partial
h_{3}}{\partial
u_{1}}+(w_{m}+1)\left((3w_{m}+1)^{2}(-h_{1}+u_{1}+1)h_{3}+4w_{m}(w_{m}+3)h_{1}^{2}\right)=0,$
(120)
$\displaystyle(w_{m}+3)\left((3w_{m}+1)(-h_{1}+u_{1}+1)h_{4}-4w_{m}h_{1}^{2}\right)-4(w_{m}-1)w_{m}h_{1}^{2}\frac{\partial
h_{4}}{\partial u_{1}}=0.$ (121)

Using the expansion in series

$h_{i}(u_{1},u_{2})=\sum_{n=2}^{N}\sum_{k=0}^{n}a^{[i]}_{nk}u_{1}^{n-k}u_{2}^{k}+\mathcal{O}(\|(u_{1},u_{2})\|^{N+1}),i=1,\ldots
4$ (122)

the zero solution is found for any given accuracy.

In figures 4-6 some solutions of the systems (112)-(114) and (116) for the
potential $V_{II}(\phi,\psi)$ when $\omega_{m}=-1$, $0$ and $\frac{1}{3}$ are
represented. More specific, in the upper panel the solutions are projected in
the space ($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) and ($\Phi_{1}$, $\Phi_{2}$,
$\Phi_{3}$), where we have represented the spheres
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=r^{2}$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=r^{2}$ with $r\in\\{1,\sqrt{2}\\}$. In
the lower panel projections in the spaces ($u_{1},u_{2}$), ($v_{1},v_{2}$),
($v_{1},v_{3}$), ($v_{1},v_{4}$), ($v_{2},v_{3}$), ($v_{2},v_{4}$),
($v_{3},v_{4}$), ($u_{1},v_{1}$) and ($u_{2},v_{2}$) are represented. In these
figures we have depicted a red line in the projections ($u_{1},v_{1}$) and
($u_{2},v_{2}$) which denotes the invariant set $v_{1}=0$ and $v_{2}=0$,
respectively. Both lines are stable in these projections. Notice that in
figures 4 and 6, the projection $u_{1}$ vs $u_{2}$ the origin behaves as a
saddle point. The orbits from the left along the $u_{1}$-axis tend to the
origin, but from the right the orbits depart from the origin. Then, the
solution is unstable (saddle behavior). This behavior is also represented in
the 3D projection $(\phi_{1},\phi_{2},\phi_{3})$ where some orbits abandon the
inner spheres backward and forward in time. On the other hand, the projection
$u_{1}$ vs $u_{2}$ in figure 5 the origin is stable (but not asymptotically
stable) since any $\epsilon$-neighborhood of the origin will contain a
$\delta$-neighborhood of origin with other points apart of the origin with
$(u_{1}^{\prime},u_{2}^{\prime})|_{u_{1}=u_{1}^{*},u_{2}=u_{2}^{*}}=(0,0)$.
Therefore, they remain in $\delta$-neighborhood of origin.

###  VI.3 Scalar field potential $V_{III}\left(\phi,\psi\right)$

In this section we analyze the stability of the analytic solution (62) of
equations (53), (54). We set for simplicity the integration constants
$t_{1},t_{2},t_{3}$ to zero because they are not relevant as
$t\rightarrow\infty$ and we assume $w_{m}\neq 0,1$.

With the time variable $\tau=\ln(t)$ and defining the new variable
$X=V_{0}\varepsilon e^{2\tau}x^{r}$, with $r=-\frac{4w_{m}}{w_{m}-1}$ to
balance the powers of $t$, the equations (53), (54) become

$\displaystyle-32w_{m}^{2}(w_{m}+1)X(\tau)^{3}-(w_{m}-1)^{2}\left((1-5w_{m})X^{\prime}(\tau)^{2}-4X(\tau)\left(X^{\prime}(\tau)-w_{m}X^{\prime\prime}(\tau)\right)+4(w_{m}+1)X(\tau)^{2}\right)=0,$
$\displaystyle y^{\prime\prime}(\tau)-y^{\prime}(\tau)=0,\quad
z^{\prime\prime}(\tau)-z^{\prime}(\tau)=0.$

The analytical solution of the original system becomes

$X_{c}(t)=-\frac{(w_{m}-1)^{2}}{8w_{m}^{2}},\;y_{c}(\tau)=y_{0}e^{\tau},\;z_{c}(\tau)=z_{0}e^{\tau},$

Defining the dimensionless variables

$\phi_{1}=\frac{X}{X_{c}},\quad\phi_{2}=\frac{y}{y_{c}},\quad\phi_{3}=\frac{z}{z_{c}},$

we obtain for $w_{m}\neq 0,1$ the dynamical system

$\displaystyle\phi_{1}^{\prime}=\Phi_{1},\quad\Phi_{1}^{\prime}=\frac{\Phi_{1}}{w_{m}}+\frac{\Phi_{1}^{2}(5w_{m}-1)}{4w_{m}\phi_{1}}+\frac{(w_{m}+1)(\phi_{1}-1)\phi_{1}}{w_{m}},$
(123)
$\displaystyle\phi_{2}^{\prime}=\Phi_{2},\quad\Phi_{2}^{\prime}=-\Phi_{2},$
(124)
$\displaystyle\phi_{3}^{\prime}=\Phi_{3},\quad\Phi_{3}^{\prime}=-\Phi_{3}.$
(125)

Now we analyze the stability of the fixed point
$P:=(\phi_{1},\Phi_{1},\phi_{2},\Phi_{2},\phi_{3},\Phi_{3})=(1,0,1,0,1,0)$.
The subsystems for $(\phi_{1},\Phi_{1})$, $(\phi_{2},\Phi_{2})$ and
$(\phi_{3},\Phi_{3})$ are decoupled.

The Jacobian matrix of the full system is

$J:=\left(\begin{array}[]{cccccc}0&1&0&0&0&0\\\
\frac{4(w_{m}+1)(2\phi_{1}-1)\phi_{1}^{2}+(1-5w_{m})\Phi_{1}^{2}}{4w_{m}\phi_{1}^{2}}&\frac{\frac{(5w_{m}-1)\Phi_{1}}{\phi_{1}}+2}{2w_{m}}&0&0&0&0\\\
0&0&0&1&0&0\\\ 0&0&0&-1&0&0\\\ 0&0&0&0&0&1\\\ 0&0&0&0&0&-1\\\
\end{array}\right).$ (126)

Evaluating $J$ at the fixed point $P$ the eigenvalues
$\left\\{0,0,-1,-1,-1,\frac{1}{w_{m}}+1\right\\}$ are obtained. Therefore, if
$-1<w_{m}<0$ the stable manifold of $P$ for the full system is 4D. If the
analysis is restricted to the subspace $(\phi_{2},\Phi_{2},\phi_{3},\Phi_{3})$
the eigenvalues are $\left\\{0,0,-1,-1\right\\}$. The stable manifold in this
subspace is 2D for all $w_{m}\neq 0,1$.

Defining the new variables

$\displaystyle
u_{1}=\Phi_{3}+\phi_{3}-1,\;u_{2}=\Phi_{2}+\phi_{2}-1,\;v_{1}=\Phi_{3},\;v_{2}=\Phi_{2},$
$\displaystyle
v_{3}=\frac{\Phi_{1}w_{m}+(-w_{m}-1)(\phi_{1}-1)}{2w_{m}+1},\;v_{4}=\frac{(w_{m}+1)(\Phi_{1}+\phi_{1}-1)}{2w_{m}+1}.$

Figure 7: Some solutions of (a) the system (123)-(125) and (b) the system
(127) for the potential $V_{III}(\phi,\psi)$ when $\omega_{m}=-1$. From left
to right, from top to bottom: Projections in the planes $(\phi_{1},\Phi_{1})$,
$(\phi_{2},\Phi_{2})$, $(\phi_{3},\Phi_{3})$, $(u_{1},u_{2})$,
($v_{1},v_{2}$), ($v_{1},v_{3}$), ($v_{1},v_{4}$), ($v_{2},v_{3}$),
($v_{2},v_{4}$), ($v_{3},v_{4}$), ($u_{1},v_{1}$) and ($u_{2},v_{2}$). The red
line in fig 11 (resp. fig 12) of the bottom array denotes the invariant set
$v_{1}=0$ (resp. $v_{2}=0$) in the projections $u_{1}$ vs $v_{1}$ (resp.
$u_{2}$ vs $v_{2}$). Notice in the projection $u_{1}$ vs $u_{2}$ (which
contains the center manifold) the origin is stable (but not asymptotically
stable) since any $\epsilon$-neighborhood of the origin will contain a
$\delta$-neighborhood of origin with other points apart of the origin with
$(u_{1}^{\prime},u_{2}^{\prime})|_{u_{1}=u_{1}^{*},u_{2}=u_{2}^{*}}=(0,0)$.
Therefore, they remain in $\delta$-neighborhood of origin. Figure 8: Some
solutions of (a) the system (123)-(125) and (b) the system (127) for the
potential $V_{III}(\phi,\psi)$ when $\omega_{m}=-\frac{1}{3}$. From left to
right, from top to bottom: Projections in the planes $(\phi_{1},\Phi_{1})$,
$(\phi_{2},\Phi_{2})$, $(\phi_{3},\Phi_{3})$, $(u_{1},u_{2})$,
($v_{1},v_{2}$), ($v_{1},v_{3}$), ($v_{1},v_{4}$), ($v_{2},v_{3}$),
($v_{2},v_{4}$), ($v_{3},v_{4}$), ($u_{1},v_{1}$) and ($u_{2},v_{2}$). The red
line in fig 11 (resp. fig 12) of the bottom array denotes the invariant set
$v_{1}=0$ (resp. $v_{2}=0$) in the projections $u_{1}$ vs $v_{1}$ (resp.
$u_{2}$ vs $v_{2}$). Notice in the projection $u_{1}$ vs $u_{2}$ (which
contains the center manifold) the origin is stable (but not asymptotically
stable) since any $\epsilon$-neighborhood of the origin will contain a
$\delta$-neighborhood of origin with other points apart of the origin with
$(u_{1}^{\prime},u_{2}^{\prime})|_{u_{1}=u_{1}^{*},u_{2}=u_{2}^{*}}=(0,0)$.
Therefore, they remain in $\delta$-neighborhood of origin. Figure 9: Some
solutions of (a) the system (123)-(125) and (b) the system (127) for the
potential $V_{III}(\phi,\psi)$ when $\omega_{m}=\frac{1}{3}$. From left to
right, from top to bottom: Projections in the planes $(\phi_{1},\Phi_{1})$,
$(\phi_{2},\Phi_{2})$, $(\phi_{3},\Phi_{3})$, $(u_{1},u_{2})$,
($v_{1},v_{2}$), ($v_{1},v_{3}$), ($v_{1},v_{4}$), ($v_{2},v_{3}$),
($v_{2},v_{4}$), ($v_{3},v_{4}$), ($u_{1},v_{1}$) and ($u_{2},v_{2}$). The red
line in fig 11 (resp. fig 12) of the bottom array denotes the invariant set
$v_{1}=0$ (resp. $v_{2}=0$) in the projections $u_{1}$ vs $v_{1}$ (resp.
$u_{2}$ vs $v_{2}$). This plot shows that the equilibrium point is a saddle
for $w_{m}>0$.

Using the previous variables we obtain the decoupled equations:

$\displaystyle u_{1}^{\prime}=0,$ (127a) $\displaystyle u_{2}^{\prime}=0,$
(127b) $\displaystyle v_{1}^{\prime}=-v_{1},$ (127c) $\displaystyle
v_{2}^{\prime}=-v_{2},$ (127d) $\displaystyle
v_{3}^{\prime}=\frac{\frac{(5w_{m}-1)(v_{3}+v_{4})^{2}}{-v_{3}-\frac{v_{4}}{w_{m}+1}+v_{4}+1}-4(w_{m}+1)(v_{3}+v_{4})+\frac{4(w_{m}(v_{3}-v_{4}-1)+v_{3}-1)(v_{3}w_{m}+v_{3}-v_{4}w_{m})}{w_{m}+1}+4(v_{3}+v_{4})}{4(2w_{m}+1)},$
(127e) $\displaystyle
v_{4}^{\prime}=\frac{(w_{m}+1)\left(\frac{(5w_{m}-1)(v_{3}+v_{4})^{2}}{4w_{m}\left(-v_{3}-\frac{v_{4}}{w_{m}+1}+v_{4}+1\right)}+\frac{v_{3}+v_{4}}{w_{m}}+\frac{(w_{m}(v_{3}-v_{4}-1)+v_{3}-1)(v_{3}w_{m}+v_{3}-v_{4}w_{m})}{w_{m}(w_{m}+1)}+v_{3}+v_{4}\right)}{2w_{m}+1}.$
(127f)

In figures 7-9 some solutions of the systems (123)-(125) and (127) for the
potential $V_{III}(\phi,\psi)$ when $\omega_{m}=-1$, $-\frac{1}{3}$ and
$\frac{1}{3}$ are represented. More specific, projections in the spaces
($\phi_{1}$, $\Phi_{1}$), ($\phi_{2}$, $\Phi_{2}$), ($\phi_{3}$, $\Phi_{3}$),
($u_{1},u_{2}$), ($v_{1},v_{2}$), ($v_{1},v_{3}$), ($v_{1},v_{4}$),
($v_{2},v_{3}$), ($v_{2},v_{4}$), ($v_{3},v_{4}$), ($u_{1},v_{1}$) and
($u_{2},v_{2}$) are represented. In these figures we have depicted a red line
in the projections ($u_{1},v_{1}$) and ($u_{2},v_{2}$) that denotes the
invariant set $v_{1}=0$ and $v_{2}=0$, respectively. Both lines are stable in
these projections. Notice that in figures 7 and 8 in the projection $u_{1}$ vs
$u_{2}$ (which contains the center manifold) the origin is stable (but not
asymptotically stable) since any $\epsilon$-neighborhood of the origin will
contain a $\delta$-neighborhood of origin with other points apart of the
origin with
$(u_{1}^{\prime},u_{2}^{\prime})|_{u_{1}=u_{1}^{*},u_{2}=u_{2}^{*}}=(0,0)$.
Therefore, they remain in $\delta$-neighborhood of origin if $-1<w_{m}<0$. On
the other hand, in figure 9 the origin behaves as a saddle point since
$w_{m}>0$.

###  VI.4 Scalar field potential $V_{IV}\left(\phi,\psi\right)$

In this section we study the stability of the solution (71), (72), (73) of
(67), (68), (69). We set for simplicity the integration constants
$t_{1},t_{2},t_{3}$ to zero because they are not relevant as
$t\rightarrow\infty$.

With the time variable $\tau=\ln(t)$, and defining the new variables
$X(\tau)=x(\tau)-x_{0}e^{\tau}$, $Y(\tau)=y(\tau)-y_{0}e^{\tau}$ and
$\mathcal{Z}(\tau)=6\bar{V}_{0}e^{2\tau}Z^{r}$ with
$r=-\frac{3w_{m}+1}{w_{m}-1}$ the equations (67), (68), (69) become

$\displaystyle
X^{\prime\prime}(\tau)-X^{\prime}(\tau)-\frac{w_{m}+1}{w_{m}-1}\beta\mathcal{Z}(\tau)=0,\quad
Y^{\prime\prime}(\tau)-Y^{\prime}(\tau)+\frac{w_{m}+1}{w_{m}-1}\mathcal{Z}(\tau)=0,$
$\displaystyle\mathcal{Z}^{\prime\prime}(\tau)=\frac{-(w_{m}-5)\mathcal{Z}(\tau)\mathcal{Z}^{\prime}(\tau)+4w_{m}\mathcal{Z}^{\prime}(\tau)^{2}-2(w_{m}+3)\mathcal{Z}(\tau)^{2}}{(3w_{m}+1)\mathcal{Z}(\tau)}-\frac{(w_{m}+1)(3w_{m}+1)Z_{1}\varepsilon
e^{-\frac{2\tau(w_{m}-1)}{3w_{m}+1}}\mathcal{Z}(\tau)^{\frac{7w_{m}+1}{3w_{m}+1}}}{(w_{m}-1)^{2}},$

where
$Z_{1}=\beta^{2}2^{\frac{1-w_{m}}{3w_{m}+1}}3^{\frac{2(w_{m}+1)}{3w_{m}+1}}{\bar{V}_{0}}^{\frac{1-w_{m}}{3w_{m}+1}}$.
The solution (71), (72), (73) becomes

$\displaystyle
X_{c}\left(\tau\right)=\frac{Z_{0}e^{\frac{\tau(w_{m}-1)}{2w_{m}}}}{3\beta},\;Y_{c}\left(\tau\right)=-\frac{Z_{0}e^{\frac{\tau(w_{m}-1)}{2w_{m}}}}{3\beta^{2}}\varepsilon,\;\mathcal{Z}_{c}\left(\tau\right)=-\frac{(w_{m}-1)^{2}Z_{0}e^{\frac{\tau(w_{m}-1)}{2w_{m}}}}{12\beta^{2}w_{m}^{2}}\varepsilon.$

Defining the dimensionless variables

$\phi_{1}=\frac{X}{X_{c}},\quad\phi_{2}=\frac{Y}{Y_{c}},\quad\phi_{3}=\frac{\mathcal{Z}}{\mathcal{Z}_{c}},$
(128)

we obtain for $w_{m}\neq 0,-1/3$ the dynamical system

$\displaystyle\phi_{1}^{\prime}=\Phi_{1},\;\Phi_{1}^{\prime}=\frac{\Phi_{1}}{w_{m}}+\frac{\left(w_{m}^{2}-1\right)\phi_{1}}{4w_{m}^{2}}+\frac{1}{4}\left(\frac{1}{w_{m}^{2}}-1\right)\phi_{3},$
(129)
$\displaystyle\phi_{2}^{\prime}=\Phi_{2},\;\Phi_{2}^{\prime}=\frac{\Phi_{2}}{w_{m}}+\frac{\left(w_{m}^{2}-1\right)\
\phi_{2}}{4w_{m}^{2}}+\frac{1}{4}\left(\frac{1}{w_{m}^{2}}-1\right)\phi_{3},$
(130)
$\displaystyle\phi_{3}^{\prime}=\Phi_{3},\;\Phi_{3}^{\prime}=\frac{\Phi_{3}}{w_{m}}+\frac{4\Phi_{3}^{2}w_{m}}{(3w_{m}+1)\phi_{3}}+\frac{(w_{m}+1)(3w_{m}+1)\phi_{3}^{\frac{7w_{m}+1}{3w_{m}+1}}}{4w_{m}^{2}}-\frac{(w_{m}+1)(3w_{m}+1)\phi_{3}}{4w_{m}^{2}}.$
(131)

Now we analyze the stability of the fixed point
$P:=(\phi_{1},\Phi_{1},\phi_{2},\Phi_{2},\phi_{3},\Phi_{3})=(1,0,1,0,1,0)$.
The Jacobian matrix of the full system is

$J:=\left(\begin{array}[]{cccccc}0&1&0&0&0&0\\\
\frac{1}{4}-\frac{1}{4w_{m}^{2}}&\frac{1}{w_{m}}&0&0&\frac{1}{4}\left(\frac{1}{w_{m}^{2}}-1\right)&0\\\
0&0&0&1&0&0\\\
0&0&\frac{1}{4}-\frac{1}{4w_{m}^{2}}&\frac{1}{w_{m}}&\frac{1}{4}\left(\frac{1}{w_{m}^{2}}-1\right)&0\\\
0&0&0&0&0&1\\\
0&0&0&0&\frac{(w_{m}+1)(7w_{m}+1)\phi_{3}^{\frac{4w_{m}}{3w_{m}+1}}-(w_{m}+1)(3w_{m}+1)-\frac{16\Phi_{3}^{2}w_{m}^{3}}{(3w_{m}+1)\phi_{3}^{2}}}{4w_{m}^{2}}&\frac{8\Phi_{3}w_{m}}{3w_{m}\phi_{3}+\phi_{3}}+\frac{1}{w_{m}}\\\
\end{array}\right).$ (132)

Evaluating $J$ at the fixed point $P$ the eigenvalues
$\left\\{-1,-\frac{w_{m}-1}{2w_{m}},-\frac{w_{m}-1}{2w_{m}},\frac{w_{m}+1}{2w_{m}},\frac{w_{m}+1}{2w_{m}},\frac{w_{m}+1}{w_{m}}\right\\}$
are obtained. For $-1<w_{m}<0$ $P$ is a sink, or a saddle for $w_{m}>0$.

Defining the variables

$\displaystyle
v_{1}=\frac{(w_{m}+1)\left(w_{m}^{2}(6\Phi_{2}+2\Phi_{3}+3\phi_{2}+\phi_{3}-4)-2w_{m}(-\Phi_{2}+\Phi_{3}+\phi_{2}+\phi_{3}-2)-\phi_{2}+\phi_{3}\right)}{4w_{m}^{2}(3w_{m}+1)},$
$\displaystyle
v_{2}=\frac{(w_{m}-1)\left(2w_{m}(w_{m}(3\Phi_{2}+\Phi_{3}+2)+\Phi_{2}-\Phi_{3}+2)-(w_{m}+1)(3w_{m}+1)\phi_{2}+w_{m}^{2}(-\phi_{3})+\phi_{3}\right)}{4w_{m}^{2}(3w_{m}+1)},$
$\displaystyle
v_{3}=\frac{\Phi_{3}w_{m}+w_{m}-(w_{m}+1)\phi_{3}+1}{2w_{m}+1},$
$\displaystyle
v_{4}=\frac{(w_{m}+1)\left(w_{m}^{2}(6\Phi_{1}+2\Phi_{3}+3\phi_{1}+\phi_{3}-4)-2w_{m}(-\Phi_{1}+\Phi_{3}+\phi_{1}+\phi_{3}-2)-\phi_{1}+\phi_{3}\right)}{4w_{m}^{2}(3w_{m}+1)},$
$\displaystyle
v_{5}=\frac{(w_{m}-1)\left(2w_{m}(w_{m}(3\Phi_{1}+\Phi_{3}+2)+\Phi_{1}-\Phi_{3}+2)-(w_{m}+1)(3w_{m}+1)\phi_{1}+w_{m}^{2}(-\phi_{3})+\phi_{3}\right)}{4w_{m}^{2}(3w_{m}+1)},$
$\displaystyle v_{6}=\frac{(w_{m}+1)(\Phi_{3}+\phi_{3}-1)}{2w_{m}+1},$

(a) Projections in the space ($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) (left) and
($\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$) (right). We have represented the spheres
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=r^{2}$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=r^{2}$, $r\in\\{1,\sqrt{2}\\}$. (b)
From left to right, from top to bottom: Projections in the planes
($v_{1},v_{2}$), ($v_{1},v_{4}$), ($v_{1},v_{5}$), ($v_{2},v_{3}$),
($v_{2},v_{4}$), ($v_{2},v_{5}$), ($v_{3},v_{6}$), ($v_{4},v_{5}$) and
($v_{5},v_{6}$).

Figure 10: Some solutions of (a) the system (129)-(131) and (b) the system
(133) for the potential $V_{IV}(\phi,\psi)$ when $\omega_{m}=-\frac{1}{4}$.
The point $P$ is asymptotically stable.

(a) Projections in the space ($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) (left) and
($\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$) (right). We have represented the spheres
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=r^{2}$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=r^{2}$, $r\in\\{1,\sqrt{2}\\}$. (b)
From left to right, from top to bottom: Projections in the planes
($v_{1},v_{2}$), ($v_{1},v_{4}$), ($v_{1},v_{5}$), ($v_{2},v_{5}$),
($v_{2},v_{6}$), ($v_{3},v_{6}$), ($v_{4},v_{5}$), ($v_{4},v_{6}$) and
($v_{5},v_{6}$).

Figure 11: Some solutions of (a) the system (129)-(131) and (b) the system
(133) for the potential $V_{IV}(\phi,\psi)$ when $\omega_{m}=-\frac{1}{7}$.
The point $P$ is asymptotically stable.

(a) Projections in the space ($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) (left) and
($\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$) (right). We have represented the spheres
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=r^{2}$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=r^{2}$, $r\in\\{1,\sqrt{2}\\}$. (b)
From left to right, from top to bottom: Projections in the planes
($v_{1},v_{2}$), ($v_{1},v_{3}$), ($v_{1},v_{4}$), ($v_{1},v_{5}$),
($v_{1},v_{6}$), ($v_{3},v_{4}$), ($v_{3},v_{6}$), ($v_{4},v_{6}$) and
($v_{5},v_{6}$).

Figure 12: Some solutions of (a) the system (129)-(131) and (b) the system
(133) for the potential $V_{IV}(\phi,\psi)$ when $\omega_{m}=1$. The point $P$
is a saddle.

we obtain the system

$\displaystyle
v_{1}^{\prime}=\frac{(w_{m}-1)(w_{m}+1)^{2}\left(-v_{3}+v_{6}-\frac{v_{6}}{w_{m}+1}+1\right)^{\frac{7w_{m}+1}{3w_{m}+1}}}{8w_{m}^{3}}$
$\displaystyle+\frac{1}{8w_{m}^{3}(3w_{m}+1)^{2}(v_{3}+(v_{3}-v_{6}-1)w_{m}-1)}\Bigg{(}v_{3}^{2}(w_{m}-1)(5w_{m}+1)(w_{m}(w_{m}+6)+1)(w_{m}+1)^{2}$
$\displaystyle-2v_{3}\left(w_{m}\left(w_{m}\left(-2v_{1}(3w_{m}+1)^{2}+w_{m}(15w_{m}+8)-14\right)+v_{6}(w_{m}-1)(w_{m}(37w_{m}+10)+1)-8\right)-1\right)(w_{m}+1)^{2}$
$\displaystyle+w_{m}\Big{(}v_{6}^{2}w_{m}(w_{m}+1)(5w_{m}-1)(w_{m}-1)^{2}-w_{m}\left(4v_{1}(w_{m}+1)^{2}(3w_{m}+1)^{2}-w_{m}(w_{m}(3w_{m}(3w_{m}+8)+13)-16)+21\right)$
$\displaystyle-2v_{6}(w_{m}+1)(3w_{m}+1)(w_{m}(w_{m}(-5w_{m}+v_{1}(6w_{m}+2)-1)+5)+1)-8\Big{)}-1\Bigg{)},$
(133a) $\displaystyle
v_{2}^{\prime}=\frac{(w_{m}-1)^{2}(w_{m}+1)\left(-v_{3}+v_{6}-\frac{v_{6}}{w_{m}+1}+1\right)^{\frac{7w_{m}+1}{3w_{m}+1}}}{8w_{m}^{3}}$
$\displaystyle+\frac{1}{8w_{m}^{3}(3w_{m}+1)^{2}(v_{3}+(v_{3}-v_{6}-1)w_{m}-1)}\Bigg{(}v_{3}^{2}(w_{m}+1)(5w_{m}+1)(w_{m}(w_{m}+6)+1)(w_{m}-1)^{2}$
$\displaystyle+w_{m}\Big{(}v_{6}^{2}w_{m}(5w_{m}-1)(w_{m}-1)^{3}+2v_{6}(3w_{m}+1)\left(w_{m}\left((6v_{2}+5)w_{m}^{2}+2v_{2}w_{m}+w_{m}-5\right)-1\right)(w_{m}-1)$
$\displaystyle+w_{m}\left(4v_{2}\left(w_{m}^{2}-1\right)(3w_{m}+1)^{2}+w_{m}(3w_{m}+4)(w_{m}(3w_{m}-2)-3)+7\right)+6\Big{)}$
$\displaystyle-2v_{3}\left(w_{m}^{2}-1\right)\left(w_{m}\left(w_{m}\left(2v_{2}(3w_{m}+1)^{2}+w_{m}(15w_{m}+8)-14\right)+v_{6}(w_{m}-1)(w_{m}(37w_{m}+10)+1)-8\right)-1\right)+1\Bigg{)},$
(133b) $\displaystyle
v_{3}^{\prime}=\frac{\left(3w_{m}^{2}+4w_{m}+1\right)\left(-v_{3}+v_{6}-\frac{v_{6}}{w_{m}+1}+1\right)^{\frac{7w_{m}+1}{3w_{m}+1}}}{8w_{m}^{2}+4w_{m}}$
$\displaystyle+\frac{1}{4w_{m}(2w_{m}+1)(3w_{m}+1)(v_{3}+(v_{3}-v_{6}-1)w_{m}-1)}\Bigg{(}-(w_{m}-1)(w_{m}+1)(w_{m}(19w_{m}+8)+1)v_{3}^{2}$
$\displaystyle-2(w_{m}(w_{m}(w_{m}+2)(3w_{m}+10)+v_{6}(w_{m}(w_{m}(25w_{m}+37)+9)+1)+8)+1)v_{3}$
$\displaystyle+w_{m}\left((5v_{6}(v_{6}+6)+9)w_{m}^{3}+2((23-3v_{6})v_{6}+12)w_{m}^{2}+v_{6}(v_{6}+18)w_{m}+22w_{m}+2v_{6}+8\right)+1\Bigg{)},$
(133c) $\displaystyle
v_{4}^{\prime}=\frac{(w_{m}-1)(w_{m}+1)^{2}\left(-v_{3}+v_{6}-\frac{v_{6}}{w_{m}+1}+1\right)^{\frac{7w_{m}+1}{3w_{m}+1}}}{8w_{m}^{3}}$
$\displaystyle+\frac{1}{8w_{m}^{3}(3w_{m}+1)^{2}(v_{3}+(v_{3}-v_{6}-1)w_{m}-1)}\Bigg{(}v_{3}^{2}(w_{m}-1)(5w_{m}+1)(w_{m}(w_{m}+6)+1)(w_{m}+1)^{2}$
$\displaystyle-2v_{3}\left(w_{m}\left(w_{m}\left(-2v_{1}(3w_{m}+1)^{2}+w_{m}(15w_{m}+8)-14\right)+v_{6}(w_{m}-1)(w_{m}(37w_{m}+10)+1)-8\right)-1\right)(w_{m}+1)^{2}$
$\displaystyle+w_{m}\Big{(}v_{6}^{2}w_{m}(w_{m}+1)(5w_{m}-1)(w_{m}-1)^{2}-w_{m}\left(4v_{1}(w_{m}+1)^{2}(3w_{m}+1)^{2}-w_{m}(w_{m}(3w_{m}(3w_{m}+8)+13)-16)+21\right)$
$\displaystyle-2v_{6}(w_{m}+1)(3w_{m}+1)(w_{m}(w_{m}(-5w_{m}+v_{1}(6w_{m}+2)-1)+5)+1)-8\Big{)}-1\Bigg{)},$
(133d) $\displaystyle
v_{5}^{\prime}=\frac{(w_{m}-1)^{2}(w_{m}+1)\left(-v_{3}+v_{6}-\frac{v_{6}}{w_{m}+1}+1\right)^{\frac{7w_{m}+1}{3w_{m}+1}}}{8w_{m}^{3}}$
$\displaystyle+\frac{1}{8w_{m}^{3}(3w_{m}+1)^{2}(v_{3}+(v_{3}-v_{6}-1)w_{m}-1)}\Bigg{(}v_{3}^{2}(w_{m}+1)(5w_{m}+1)(w_{m}(w_{m}+6)+1)(w_{m}-1)^{2}$
$\displaystyle+w_{m}\Big{(}v_{6}^{2}w_{m}(5w_{m}-1)(w_{m}-1)^{3}+2v_{6}(3w_{m}+1)\left(w_{m}\left((6v_{5}+5)w_{m}^{2}+2v_{5}w_{m}+w_{m}-5\right)-1\right)(w_{m}-1)$
$\displaystyle+w_{m}\left(4v_{5}\left(w_{m}^{2}-1\right)(3w_{m}+1)^{2}+w_{m}(3w_{m}+4)(w_{m}(3w_{m}-2)-3)+7\right)+6\Big{)}$
$\displaystyle-2v_{3}\left(w_{m}^{2}-1\right)\left(w_{m}\left(w_{m}\left(2v_{5}(3w_{m}+1)^{2}+w_{m}(15w_{m}+8)-14\right)+v_{6}(w_{m}-1)(w_{m}(37w_{m}+10)+1)-8\right)-1\right)+1\Bigg{)},$
(133e) $\displaystyle
v_{6}^{\prime}=\frac{(w_{m}+1)^{2}(3w_{m}+1)\left(-v_{3}+v_{6}-\frac{v_{6}}{w_{m}+1}+1\right)^{\frac{7w_{m}+1}{3w_{m}+1}}}{4w_{m}^{2}(2w_{m}+1)}$
$\displaystyle+\frac{(w_{m}+1)}{4w_{m}^{2}(2w_{m}+1)(3w_{m}+1)(v_{3}+(v_{3}-v_{6}-1)w_{m}-1)}\Bigg{(}(w_{m}+1)(5w_{m}+1)(w_{m}(w_{m}+6)+1)v_{3}^{2}$
$\displaystyle-2(w_{m}+1)(5w_{m}+1)(w_{m}(3w_{m}+v_{6}(5w_{m}-1)+4)+1)v_{3}$
$\displaystyle+w_{m}\left(-w_{m}(w_{m}(19w_{m}+26)+3)v_{6}^{2}+2(w_{m}-1)(w_{m}+1)(3w_{m}+1)v_{6}+(3w_{m}+4)(w_{m}(3w_{m}+4)+2)\right)+1\Bigg{)}.$
(133f)

In figures 10-12 some solutions of the system (129)-(131) and of the system
(133) for the potential $V_{IV}(\phi,\psi)$ when $\omega_{m}=-\frac{1}{4}$,
$-\frac{1}{7}$ and $1$ are represented. More specific, in then upper panel the
solutions are projected in the space ($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) and
($\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$), where we have represented the spheres
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=r^{2}$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=r^{2}$ with $r\in\\{1,\sqrt{2}\\}$. In
the lower panel projections in the spaces ($v_{1},v_{2}$), ($v_{1},v_{4}$),
($v_{1},v_{5}$), ($v_{2},v_{3}$), ($v_{2},v_{4}$), ($v_{2},v_{5}$),
($v_{3},v_{6}$), ($v_{4},v_{5}$) and ($v_{5},v_{6}$) are represented. From
these figures, it is confirmed that the point $(1,0,1,0,1,0)$ is a saddle when
$\omega_{m}>0$ while for $-1<\omega_{m}<0$ is an attractor of the system, i.
e., it is asymptotically stable.

###  VI.5 Scalar field potential $V_{V}\left(\phi,\psi\right)$

In this section we study the stability of the solution (82), (83), (84) of
system (76), (77), (78). We set for simplicity the integration constants
$t_{1},t_{2},t_{3}$ to zero because they are not relevant as
$t\rightarrow\infty$.

With the time variable $\tau=\ln(t)$, and defining the new variables
$X(\tau)=x(\tau)-x_{0}e^{\tau}$, $\mathcal{Y}(\tau)=6\varepsilon
V_{0}e^{2\tau}Y^{r}$ with $r=-\frac{3w_{m}+1}{w_{m}-1}$ and
$Z(\tau)=z(\tau)-z_{0}e^{\tau}$ the equations (76), (77), (78) become

$\displaystyle\mathcal{Y}^{\prime\prime}(\tau)=\frac{4w_{m}\mathcal{Y}^{\prime}(\tau)^{2}}{3w_{m}\mathcal{Y}(\tau)+\mathcal{Y}(\tau)}+\frac{(5-w_{m})\mathcal{Y}^{\prime}(\tau)}{3w_{m}+1}-\frac{2(w_{m}+3)\mathcal{Y}(\tau)}{3w_{m}+1}$
$\displaystyle-
Y_{1}(w_{m}-1)^{2}(w_{m}+1)(3w_{m}+1)e^{-\frac{2\tau(w_{m}-1)}{3w_{m}+1}}\mathcal{Y}(\tau)^{\frac{7w_{m}+1}{3w_{m}+1}},$
$\displaystyle
X^{\prime\prime}(\tau)-X^{\prime}(\tau)-\frac{1}{3}\left(w_{m}^{2}-1\right)\beta\varepsilon\mathcal{Y}(\tau)=0,\quad
Z^{\prime\prime}(\tau)-Z^{\prime}(\tau)+\frac{\left(w_{m}+1\right)}{w_{m}-1}\varepsilon\mathcal{Y}(\tau)=0,$

where
$Y_{1}=\beta^{2}2^{\frac{1-w_{m}}{3w_{m}+1}}81^{-\frac{w_{m}}{3w_{m}+1}}V_{0}^{\frac{1-w_{m}}{3w_{m}+1}}\varepsilon^{\frac{1-w_{m}}{3w_{m}+1}}$.
The solution (82), (83), (84) becomes

$\displaystyle X_{c}(\tau)=\frac{\varepsilon
Y_{0}e^{\frac{w_{m}-1}{2w_{m}}\tau}}{\beta(w_{m}-1)^{2}},\quad\mathcal{Y}_{c}(\tau)=-\frac{3Y_{0}e^{\frac{\tau(w_{m}-1)}{2w_{m}}}}{4\beta^{2}(w_{m}-1)^{2}w_{m}^{2}},\quad
Z_{c}(\tau)=-\frac{3\varepsilon
Y_{0}e^{\frac{w_{m}-1}{2w_{m}}\tau}}{\beta^{2}(w_{m}-1)^{4}}.$ (134)

Defining the dimensionless variables

$\phi_{1}=\frac{X}{X_{c}},\quad\phi_{2}=\frac{\mathcal{Y}}{\mathcal{Y}_{c}},\quad\phi_{3}=\frac{Z}{Z_{c}},$
(135)

we obtain the dynamical system

$\displaystyle\phi_{1}^{\prime}=\Phi_{1},\;\Phi_{1}^{\prime}=\frac{\Phi_{1}}{w_{m}}+\left(\frac{1}{4}-\frac{1}{4w_{m}^{2}}\right)\phi_{1}+\frac{1}{4}\left(\frac{1}{w_{m}^{2}}-1\right)\phi_{2},$
(136a)
$\displaystyle\phi_{2}^{\prime}=\Phi_{2},\;\Phi_{2}^{\prime}=\frac{\Phi_{2}}{w_{m}}+\frac{4\Phi_{2}^{2}w_{m}}{3w_{m}\phi_{2}+\phi_{2}}+\frac{(w_{m}+1)(3w_{m}+1)\phi_{2}^{\frac{7w_{m}+1}{3w_{m}+1}}}{4w_{m}^{2}}-\frac{(w_{m}+1)(3w_{m}+1)\phi_{2}}{4w_{m}^{2}},$
(136b)
$\displaystyle\phi_{3}^{\prime}=\Phi_{3},\;\Phi_{3}^{\prime}=\frac{\Phi_{3}}{w_{m}}+\frac{1}{4}\left(\frac{1}{w_{m}^{2}}-1\right)\phi_{2}+\left(\frac{1}{4}-\frac{1}{4w_{m}^{2}}\right)\phi_{3}.$
(136c)

Now we analyze the stability of the fixed point
$P:=(\phi_{1},\Phi_{1},\phi_{2},\Phi_{2},\phi_{3},\Phi_{3})=(1,0,1,0,1,0)$.
The Jacobian matrix of the full system is

$J:=\left(\begin{array}[]{cccccc}0&1&0&0&0&0\\\
\frac{1}{4}-\frac{1}{4w_{m}^{2}}&\frac{1}{w_{m}}&\frac{1}{4}\left(\frac{1}{w_{m}^{2}}-1\right)&0&0&0\\\
0&0&0&1&0&0\\\
0&0&\frac{(w_{m}+1)(3w_{m}+1)(7w_{m}+1)\phi_{2}^{\frac{10w_{m}+2}{3w_{m}+1}}-16\Phi_{2}^{2}w_{m}^{3}-(w_{m}+1)(3w_{m}\phi_{2}+\phi_{2})^{2}}{4\phi_{2}^{2}w_{m}^{2}(3w_{m}+1)}&\frac{8\Phi_{2}w_{m}}{3w_{m}\phi_{2}+\phi_{2}}+\frac{1}{w_{m}}&0&0\\\
0&0&0&0&0&1\\\
0&0&\frac{1}{4}\left(\frac{1}{w_{m}^{2}}-1\right)&0&\frac{1}{4}-\frac{1}{4w_{m}^{2}}&\frac{1}{w_{m}}\\\
\end{array}\right).$ (137)

Evaluating $J$ at the fixed point $P$ the eigenvalues
$\left\\{-1,-\frac{w_{m}-1}{2w_{m}},-\frac{w_{m}-1}{2w_{m}},\frac{w_{m}+1}{2w_{m}},\frac{w_{m}+1}{2w_{m}},\frac{w_{m}+1}{w_{m}}\right\\}$
are obtained. For $-1<w_{m}<0$ $P$ is a sink, or a saddle for $w_{m}>0$.

(a) Projections in the space ($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) (left) and
($\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$) (right) for $\omega_{m}=-\frac{1}{4}$.
The point $P$ is asymptotically stable. (b) Projections in the space
($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) (left) and ($\Phi_{1}$, $\Phi_{2}$,
$\Phi_{3}$) (right) for $\omega_{m}=-\frac{1}{7}$. The point $P$ is
asymptotically stable. (c) Projections in the space ($\phi_{1}$, $\phi_{2}$,
$\phi_{3}$) (left) and ($\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$) (right) for
$\omega_{m}=\frac{1}{3}$. The point $P$ is a saddle.

Figure 13: Some solutions of the system (136a)-(136c) for the potential
$V_{V}(\phi,\psi)$ when $\omega_{m}=-\frac{1}{4}$, $-\frac{1}{7}$ and
$\frac{1}{3}$. In these figures we have represented the spheres
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=r^{2}$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=r^{2}$, $r\in\\{1,\sqrt{2}\\}$.

In figures 13(a)-13(c) some solutions of the system (136a)-(136c) are
represented for the potential $V_{V}(\phi,\psi)$ when
$\omega_{m}=-\frac{1}{4}$, $-\frac{1}{7}$ and $\frac{1}{3}$, respectively.
More specific, in each sub-figure the solutions are projected in the space
($\phi_{1}$, $\phi_{2}$, $\phi_{3}$) (left) and ($\Phi_{1}$, $\Phi_{2}$,
$\Phi_{3}$) (right), where additionally we have represented the spheres
$(\phi_{1}-1)^{2}+(\phi_{2}-1)^{2}+(\phi_{3}-1)^{2}=r^{2}$ and
$\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=r^{2}$ with $r\in\\{1,\sqrt{2}\\}$.
From these figures, it is confirmed that the point $(1,0,1,0,1,0)$ is a saddle
when $\omega_{m}>0$ while for $-1<\omega_{m}<0$ is an attractor of the system,
i. e., it is asymptotically stable.

###  VI.6 Scalar field potential $V_{VI}\left(\phi,\psi\right)$

In this section we study the stability of the powerlaw solution (90), (91),
(92) of the system (85), (86), (87). We set for simplicity the integration
constants $t_{1},t_{2},t_{3}$ to zero because they are not relevant as
$t\rightarrow\infty$.

With the time variable $\tau=\ln(t)$, and defining the new variables
$\mathcal{U}(\tau)=12\beta V_{0}e^{2\tau}U^{r}$ with
$r=-\frac{3w_{m}+1}{w_{m}-1}$, $X(\tau)=x(\tau)-x_{0}e^{\tau}$, and
$Z(\tau)=z(\tau)-z_{0}e^{\tau}$ the equations (85), (86), (87) become

$\displaystyle\mathcal{U}^{\prime\prime}(\tau)=\frac{4w_{m}\mathcal{U}^{\prime}(\tau)^{2}}{(1+3w_{m})\mathcal{U}(\tau)}+\frac{(5-w_{m})\mathcal{U}^{\prime}(\tau)}{3w_{m}+1}+(w_{m}+1)(3w_{m}+1)e^{-\frac{2\tau(w_{m}-1)}{3w_{m}+1}}\mathcal{U}(\tau)^{\frac{7w_{m}+1}{3w_{m}+1}}-\frac{2(w_{m}+3)\mathcal{U}(\tau)}{3w_{m}+1},$
$\displaystyle
X^{\prime\prime}(\tau)-X^{\prime}(\tau)-\gamma\left(w_{m}^{2}-1\right)\mathcal{U}(\tau)=0,\quad
Z^{\prime\prime}(\tau)-Z^{\prime}(\tau)+\frac{\left(w_{m}+1\right)}{w_{m}-1}\mathcal{U}(\tau)=0,$

where
$v_{1}=2^{\frac{2(3w_{m}-1)}{3w_{m}+1}}w_{m}^{\frac{2(w_{m}-1)}{3w_{m}+1}}U_{0}^{-\frac{4w_{m}}{3w_{m}+1}}\gamma^{\frac{4w_{m}}{3w_{m}+1}}\left(8-3\beta\gamma(w_{m}-1)^{2}\epsilon\right)^{\frac{4w_{m}}{3w_{m}+1}}$.

The powerlaw solution (90), (91), (92) transforms to

$\displaystyle\mathcal{U}(\tau)=\frac{U_{0}e^{\frac{\tau(w_{m}-1)}{2w_{m}}}}{8\gamma
w_{m}^{2}\left(8-3\beta\gamma(w_{m}-1)^{2}\varepsilon\right)},\quad
X(\tau)=-\frac{U_{0}e^{\frac{\tau(w_{m}-1)}{2w_{m}}}}{2(8-3\beta\gamma(w_{m}-1)^{2}\varepsilon)},$
$\displaystyle
Z(\tau)=\frac{U_{0}e^{\frac{\tau(w_{m}-1)}{2w_{m}}}}{2\gamma(w_{m}-1)^{2}\left(8-3\beta\gamma(w_{m}-1)^{2}\varepsilon\right)}.$
(138)

Defining the dimensionless variables

$\phi_{1}=\frac{X}{X_{c}},\quad\phi_{2}=\frac{\mathcal{U}}{\mathcal{U}_{c}},\quad\phi_{3}=\frac{Z}{Z_{c}}.$
(139)

we obtain the dynamical system (136a), (136b), (136c). Hence, as before, the
fixed point
$P:=(\phi_{1},\Phi_{1},\phi_{2},\Phi_{2},\phi_{3},\Phi_{3})=(1,0,1,0,1,0)$ is
a sink for $-1<w_{m}<0$ or a saddle for $w_{m}>0$ as shown in figure 13.

##  VII Conclusions

We have considered a cosmological model consisting of two-scalar fields
minimally coupled to gravity and an ideal gas. The two scalar fields interact
in the kinetic and in the potential term. In particular the kinematics of the
two scalar fields lie on a space of constant curvature. This kind of scalar
field models are known also as Chiral models. The Chiral model is the main
mechanism for the description of the hyperbolic inflation.

In the case of a spatially flat FLRW universe, the field equations form an
autonomous dynamical system which is described by a point-like Lagrangian. The
Lagrangian function depends on an unknown potential function which drives the
dynamics of the two scalar fields. In this work we focused on the
integrability properties for the field equations and we applied geometrical
selection rules in order to constraint the functional forms of the potential.

We applied the theory of symmetries of differential equations to constraint
the scalar field potential such that field equations admit conservation laws.
Specifically, we investigate the existence of variational symmetries where the
corresponding conservation laws are constructed with the use of Noether’s
theorem. Because the point-like dynamical system is defined on a three
dimensional space with dependent variables the scale factor and two scalar
fields $\left\\{a,\phi,\psi\right\\}$, and because the system is autonomous,
the Hamiltonian function is one conservation law. Thus, two additional
conservation laws should be found such that the system is Liouville
integrable.

Therefore, we performed a classification of the variational symmetries for the
point-like Lagrangian which describes the field equations and we found six
potential functions $V\left(\phi,\psi\right)$ where the field equations are
Liouville integrable. For these six potential functions we define the normal
coordinates, which is used to construct the analytic solution for the field
equations. The free parameters of the cosmological model are constrained in
order to describe analytic and exact solutions for the scale factor which
describe the hyperbolic inflation era.

Finally, the stability properties of exact closed-form solutions were
investigated using a dynamical systems formulation and numerical tools for
these potential functions. In particular, for potentials
$V_{I}\left(\phi,\psi\right)$ and $V_{II}\left(\phi,\psi\right)$ the scaling
solution is a saddle for $-1<w_{m}<1,w_{m}\neq 0$ and stable, but not
asymptotically stable for $w_{m}=0$; and for potential
$V_{III}\left(\phi,\psi\right)$ the scaling solution is a saddle when
$w_{m}>0$ while for $-1<w_{m}<0$ the scaling solution is stable, but not
asymptotically stable. For potentials $V_{IV}\left(\phi,\psi\right)$,
$V_{VI}\left(\phi,\psi\right)$ and $V_{VI}\left(\phi,\psi\right)$ the scaling
solutions are saddle when $w_{m}>0$ while for $-1<w_{m}<0$ are asymptotically
stable.

###### Acknowledgements.

This work is based on the research supported in part by the National Research
Foundation of South Africa (Grant Numbers 131604). The research of AG was
funded by Agencia Nacional de Investigación y Desarrollo - ANID through the
program FONDECYT Iniciación grant no. 1200293. The research of AP, GL and EG
was funded by Agencia Nacional de Investigación y Desarrollo - ANID through
the program FONDECYT Iniciación grant no. 11180126\. Additionally, GL was
funded by Vicerrectoría de Investigación y Desarrollo Tecnológico at
Universidad Catolica del Norte. Ellen de los M. Fernández Flores is
acknowledged for proofreading this manuscript and for improving the English.

## References

  * (1) M. Tegmark et al., Astrophys. J. 606, 702 (2004)
  * (2) D. N. Spergel et al., Astrophys. J. Supplt. 170, 377 (2007)
  * (3) T. M. Davis et al., Astrophys. J. 666, 716 (2007)
  * (4) M. Kowalski et al., Astrophys. J. 686, 749(2008)
  * (5) P.A.R. Ade, et al. (Planck 2015 Collaboration), A & A 594, A20 (2016)
  * (6) E. Di Valentino, A. Melchiorri, O. Mena, S. Pan and W. Yang, MNRAS 502, L23 (2021)
  * (7) W. Yang, S. Pan, R.C. Nunes and D.F. Mota, JCAP 04, 008 (2020)
  * (8) A.A. Starobinsky, Phys. Lett. B 91, 99 (1980)
  * (9) A. Guth, Phys. Rev. D 23, 347 (1981)
  * (10) J. Martin, Mod. Phys. Lett. A 23, 1252 (2008)
  * (11) S. Tsuikawa, Class. Quantum Grav. 30, 214003 (2013)
  * (12) T. Harko, F.S.N. Lobo and M.K. Mak, EPJC 74, 2784 (2014)
  * (13) N. Roy, N. Banarjee, Eur. Phys. J. Plus 129, 162 (2014)
  * (14) A. de la Macorra and G. Piccinelli, Phys. Rev. D 61, 123503 (2000)
  * (15) E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess and J. Silk, [arXiv:2103.01183 [astro-ph.CO]].
  * (16) L. Perivolaropoulos, [arXiv:0811.4684 [astro-ph]].
  * (17) D. Bertacca, S. Matarrese and M. Pietroni, Mod. Phys. Lett. A 22, 2893 (2007)
  * (18) D. Bertacca, N. Bartolo and S. Matarrese, JCAP 05, 005 (2008)
  * (19) L. Arturo Ureña-López, Front. Astron. Space Sci. 6, 47 (2019)
  * (20) J.S. Farnes, A & A 620, A92 (2018)
  * (21) S. Capozziello, J. Matsumoto, S. Nojiri and S.D. Odintsov, Phys. Lett. B 693, 198 (2010)
  * (22) P. Ratra and L. Peebles, Phys. Rev. D 37, 3406 (1988)
  * (23) R.R. Caldewell, M. Kamionkowski and N.N. Weinberg, Phys. Rev. Lett. 91, 071301 (2003)
  * (24) C.H. Brans and R.H Dicke, Phys. Rev. 124, 925 (1961)
  * (25) J. O’Hanlon, Phys. Rev. Lett. 29, 137 (1972)
  * (26) G.W. Hordenski, Int. J. Theor. Phys. 10, 363 (1974)
  * (27) V. Faraoni, Cosmology in Scalar-Tensor Gravity, Dordrecht, Kluwer Academic Press (2004)
  * (28) C. Deffayet, G. Esposito-Farese and A. Vikman, Phys. Rev. D 79, 084003 (2009)
  * (29) G. Kofinas and M. Tsoukalas, EPJC 76, 686 (2016)
  * (30) A.A. Coley and R.J. Van den Hoogen, Phys. Rev. D 62, 023517 (2000)
  * (31) Y.F. Cai, E.N. Saridakis, M.R. Setare and J.-Q. Xia, Phys. Rep. 493, 1 (2010)
  * (32) P.H.R.S. Moraes and J.R.L. Santos, Phys. Rev. D 89, 083516 (2014)
  * (33) S.V. Chervon, Quantum Matter 2, 71 (2013)
  * (34) P. Christodoulidis, D. Roest, E.I. Sfakianakis, JCAP 11, 012 (2019)
  * (35) A. Beesham, S.V. Chernov, S.D. Maharaj and A.S. Kubasov, Quantum Matter 2, 388 (2013)
  * (36) R.R. Abbyazov and S.V. Chernov, Grav. Cosmol. 18, 262 (2012)
  * (37) A.R. Browm, Phys. Rev. Lett. 121, 251601 (2018)
  * (38) S. Mizuno and S. Mukohyama, Phys. Rev. D 96, 103533 (2017)
  * (39) T. Bjorkmo and M.C. David Marsh, JHEP 04, 172 (2019)
  * (40) A. Paliathanasis and M. Tsamparlis, Phys. Rev. D 90, 043529 (2014)
  * (41) A. Paliathanasis, Class. Quantum Grav. 37, 195014 (2020)
  * (42) A.K. Halder, A. Paliathanasis and P.G.L. Leach, Symmety 10, 744 (2018)
  * (43) Noether, E. Invariante Variationsprobleme. Königlich Gesellschaft der Wissenschaften Göttingen NachrichtenMathematik-Physik Klasse 2, 235 (1918)
  * (44) M. Tsamparlis and A. Paliathanasis, Symmetry 10, 233 (2018)
  * (45) T. Pailas, N. Dimakis, A. Paliathanasis, P.A. Terzis and T. Christodoulakis, Phys. Rev. D 102, 063524 (2020)
  * (46) S. Dussault and V. Faraoni, EPJC 80, 1002 (2020)
  * (47) L. V. Ovsiannikov, Group analysis of differential equations, Academic Press, New York, (1982)
  * (48) S. Basilakos, M. Tsamparlis and A. Paliathanasis, Phys. Rev. D 83, 103512 (2011)
  * (49) T. Christodoulakis, N. Dimakis and P.A. Terzis, J. Phys. A: Math. Theor. 47, 095202 (2014)
  * (50) A. Paliathanasis and G. Leon, Class. Quantum Grav. (2021) 10.1088/1361-6382/abe2d7
  * (51) N. Dimakis and A. Paliathanasis, Class. Quantum Grav. (2021) 10.1088/1361-6382/abdaf6
  * (52) J. Socorro, S. Pérez-Payán, A. Espinoza-García and L. R. Díaz-Barrón, [arXiv:2101.05973 [gr-qc]].
  * (53) J. Socorro, S. Pérez-Payán, R. Hernández, A. Espinoza-García and L. R. Díaz-Barrón, [arXiv:2012.11108 [gr-qc]].
  * (54) P. Christodoulidis and A. Paliathanasis, [arXiv:2101.09582 [gr-qc]].
  * (55) J.R. Ray, J. Math. Phys. 13, 1451 (1972)
  * (56) O. Minazzoli and T. Harko, Phys. Rev. D 86, 087502 (2012)
  * (57) M. Tsamparlis and A. Paliathanasis, Gen. Rel. Gravit. 43, 1861 (2011)
  * (58) P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, (1993)
  * (59) V.I. Arnold, Mathematical Methods of Classical Mechanic, Springer-Verlag, New-York (1989)
  * (60) A. Paliathanasis and M. Tsamparlis, Phys. Rev. D 90, 043529 (2014)
  * (61) B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988)
  * (62) A. R. Liddle and R. J. Scherrer, Phys. Rev. D 59 (1999), 023509
  * (63) J. P. Uzan, Phys. Rev. D 59 (1999), 123510