Extended Phase Space Analysis of Interacting Dark Energy Models in Loop Quantum Cosmology
Hmar Zonunmawia Wompherdeiki Khyllep Nandan Roy Jibitesh Dutta Nicola Tamanini









^1Department of Mathematics,  North Eastern Hill
University, NEHU Campus, Shillong - 793022 ( INDIA )

^2 Department of Mathematics, St. Anthony's College, Shillong, Meghalaya 793001, India

^3 Departamento de Física, DCI Campus
 León  Universidad de Guanajuato, 37150 León, Guanajuato, México

^4 Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad-211019, India


^5 Mathematics Division, Department of Basic
Sciences and Social Sciences,  North Eastern Hill University, NEHU
Campus, Shillong - 793022 (INDIA)

^6 Inter University Centre for Astronomy and Astrophysics, Pune 411 007, India

^7Institut de Physique Théorique, CEA-Saclay,
CNRS UMR 3681, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France 







The present work deals with the dynamical system investigation of interacting dark energy models (quintessence and phantom) in the
framework of Loop Quantum Cosmology by taking into account a broad class of self-interacting scalar field potentials.
The main reason for studying potentials beyond the exponential type is to obtain additional critical points which can yield more interesting cosmological solutions.
The stability of critical points and the asymptotic behavior of the phase space are analyzed using dynamical system tools and numerical techniques.
We study two class of interacting dark energy models and consider two specific potentials as examples: the hyperbolic potential and the inverse power-law potential.
We found a rich and interesting phenomenology including the avoidance of big rip singularities due to loop quantum effects, smooth and non-linear transitions from matter domination to dark energy domination and finite periods of phantom domination with dynamical crossing of the phantom barrier.








 Extended Phase Space Analysis of Interacting Dark Energy Models in Loop Quantum Cosmology
      Hmar Zonunmawia^1[zonunmawiah@gmail.com], Wompherdeiki Khyllep^2[sjwomkhyllep@gmail.com], Nandan Roy^3,4[nandan@fisica.ugto.mx], Jibitesh Dutta^5,6[jdutta29@gmail.com, jibitesh@nehu.ac.in] and Nicola Tamanini^7[nicola.tamanini@cea.fr]
    December 30, 2023
================================================================================================================================================================================================================================================



Keywords : Loop quantum cosmology, big rip singularity, Dark energy dark matter interaction, self-interacting potentials.




§ INTRODUCTION


The accelerated expansion of our universe is by now confirmed by several observations, e.g. Cosmic Microwave Background (CMB) anisotropies <cit.>, large scale galaxy surveys <cit.> and type Ia supernovae <cit.>, but the nature of the entity that causes it, named dark energy (DE), is still obscure.
The cosmological constant represents the most simple and popular choice as a candidate for DE, but it is troubled by different theoretical issues: in particular the cosmological constant problem and the coincidence problem <cit.>.
In order to find alternative explanations for the observed acceleration of the universe, there are in general two different approaches one can follow: modified gravity models and dynamical DE models.
Within the modified gravity framework, the underlying gravitational theory determining the cosmological evolution is different from general relativity, while dynamical DE models do not modify the gravitational interaction, but rather introduce a new type of exotic matter component in the universe to describe the accelerated expansion.
In both approaches the cosmic dynamics can often be effectively described by the action of a single scalar field.
Several dynamical scalar field models have been proposed and studied: examples are quintessence <cit.>, phantom DE <cit.> and k-essence <cit.>.
Scalar field models with self-interacting potentials can provide a useful cosmological evolution which mimics the effect of a cosmological constant at the present epoch.
In a late-time cosmological context, a canonical scalar field is commonly known as quintessence and
can be motivated by the low energy limit of some well known high energy theories, e.g. string theory.
The quintessence equation of state w (EoS) can take values in the range -1≤ w ≤ 1, implying accelerated expansion for w<-1/3.
A canonical scalar field however cannot produce the so-called phantom regime (w<-1) which is slightly favoured by astronomical observations <cit.>.
For this reason, another well-known scalar field model of DE has been proposed: the phantom field with negative kinetic energy.
Phantom fields are plagued by instabilities at the quantum level <cit.>, but if considered from an effective phenomenological perspective they can be used as interesting cosmological solutions, which may better fit the observational data.

In standard Einstein cosmology (EC), phantom DE models usually lead to a cosmic end described by a future big rip singularity <cit.>.
This behaviour is however expected to be corrected by loop quantum effects.
Loop Quantum Gravity (LQG) <cit.> is one of the well-known approaches to a quantum theory of gravity <cit.>.
Its main aim consists in quantizing gravity with a non-perturbative and background independent method <cit.>.
The application of LQG in the context of cosmology is called Loop Quantum Cosmology (LQC) <cit.>.
LQC modifies the standard Friedmann equation by adding a term depending on a fix energy density imposed by quantum corrections
<cit.>, which essentially encodes the discrete quantum geometric nature of spacetime <cit.>.
This modifying contribution in the standard Friedmann equation can be used to avoid any past and future singularity <cit.>.
In fact loop corrections become important when the total energy density of the universe approaches the critical high energy value predicted by the theory (cf. Eq. (<ref>)), in which case a cosmic bounce might occur and the big rip and other singularities will never be reached <cit.>.
In both EC and LQC when DE is modelled as a scalar field, this is usually assumed to interact only with itself.
However there is no fundamental argument to ignore a possible coupling between DE and dark matter (DM), and DE models where a scalar field interacts non-gravitationally with the matter sector have been proposed as well; see e.g. <cit.>.
Interacting DE models are known to produce late time accelerated scaling attractors which can be used to alleviate the cosmic coincidence problem <cit.>.
However since there is no experimental evidence for a dark sector interaction and the fundamental nature of both DE and DM is still obscure, any coupling between the two dark components is phenomenologically constructed at the level of the equations of motion, though some recent attempts to built an effective interaction at the Lagrangian level have been advanced <cit.>.

In this work, using dynamical system tools, we investigate the dynamics of interacting scalar field (quintessence and phantom) in the framework of LQC for a broad class of self-interacting potentials.
Similar studies, restricted to the exponential potential case, have already been performed <cit.>, and the analysis of different scalar field potential will allow us to better understand the complete cosmological potential of these DE models, at least at the background level.
This type of generalization has been widely studied in different cosmological frameworks: e.g. standard quintessence models <cit.>, braneworld theories <cit.>, k-essence <cit.>, chameleon theories <cit.>, scalar-fluid theories <cit.> and (non-interacting) LQC <cit.>.
In these investigations the dimension of the resulting dynamical system increases by one if compared to that of the exponential potential case, making the analysis slightly more complicated.
The analysis of other scalar field potentials, beyond the exponential one, helps to better relate these phenomenological models with more fundamental high-energy theories.
Moreover, from a mathematical point of view, these generalizations usually yield additional non-hyperbolic points where linear stability theory fails and the stability properties can only be determined analytically using center manifold theory or Lyapunov functions <cit.>.
One can alternatively use numerical methods, for example analysing the behaviour of perturbed trajectories near the non-hyperbolic critical point <cit.>, while for the case of normally hyperbolic points, namely a non-isolated set of critical points with one vanishing eigenvalue, their stability is determined by the signature of the remaining non-vanishing eigenvalues <cit.>.
For a better understanding of the cosmological dynamics of models presenting non-hyperbolic critical points, in what follows we consider two concrete potentials as examples: the hyperbolic potential V=V_0 cosh^-μ(λϕ) and the inverse power-law potential V=M^4+n/ϕ^n.

In our analysis we consider two interacting models based on two different coupling functions for both the quintessence and phantom fields.
The first one arises in string theory and it has already been studied in the case of standard EC (for both quintessence and phantom case) <cit.>.
The same interaction has also been studied in the LQC framework for a phantom field with exponential potential <cit.>, showing that in such case the big rip singularity can be avoided.
The second kind of interaction was studied recently for quintessence in the EC context <cit.>, showing that the coincidence problem can be alleviated.
It was also investigated in braneworld theories with both quintessence and phantom fields as DE <cit.>.
In both interactions, depending on the choice of the scalar field potential, we find that there is a late time attractor with
contribution from loop quantum gravity.

The paper is organized as follows.
Section <ref> deals with the basic equations of interacting dark energy in LQC and shows how they can be recast into an autonomous system of equations.
In Sections <ref> and <ref> we consider two interacting dark energy models and we investigate the corresponding cosmological evolution using dynamical system tools.
In each of these section, we present two subsections: one for quintessence and the another for phantom DE, wherein we consider two scalar field potentials as examples.
The last two sections <ref> and <ref> are devoted to discussing the cosmological implications and drawing conclusions, respectively.









§ DYNAMICS OF INTERACTING SCALAR FIELD DARK ENERGY IN LOOP QUANTUM COSMOLOGY



In a flat universe the effective modified Friedmann equation in the framework of LQC is given by <cit.>

    3H^2= 8 π G/c^4ρ(1-ρ/ρ_c),

where H is the Hubble parameter, ρ=ρ_m+ρ_ϕ is the total energy density and ρ_ϕ, ρ_m are the energy densities of dark energy and dark matter respectively.
The constant

    ρ_c=√(3)/16 π^2γ^3 G^2ħ ,

is the critical loop quantum density, where γ is the dimensionless Barbero-Immirzi parameter <cit.>.
To simplify the notation in what follows we shall use units where 8 π G ≡ c ≡ 1.
Although the single energy components ρ_m and ρ_ϕ may not be conserved separately, the total energy density is conserved

    ρ̇ + 3 H ( ρ + p) = 0  ,

where p is the total pressure and an over-dot denotes differentiation with respect to the time t.
One can also write the modified Raychaudhuri equation of the system as

    Ḣ = - 1/2 (ρ + p) (1 - 2 ρ/ρ_c)  .

We assume that DE is described by either quintessence or a phantom scalar field.
The general Lagrangian for both these scalar fields can be generally written as

    ℒ = 1/2 ϵ ∂^μϕ∂_μϕ - V(ϕ)  ,

where ϵ = 1 corresponds to quintessence and ϵ = -1 corresponds to the phantom field.
Here V(ϕ) is the self-interacting potential for the scalar field ϕ.
The energy density and pressure of the scalar field are respectively given by

    ρ_ϕ= ϵ 1/2ϕ̇^2 +V(ϕ)
    and p_ϕ=ϵ 1/2ϕ̇^2  -V(ϕ).

In our investigation we assume that the scalar field interacts with dark matter and the energy conservation equations take the form

    ρ̇_̇ϕ̇+3H(1+w_ϕ)ρ_ϕ   = -Q  ,
    ρ̇_̇ṁ+3Hρ_m    = +Q  ,

where Q is the interaction term and w_ϕ = p_ϕ/ρ_ϕ is the EoS of DE.
If Q is positive, then the energy transfer takes place from quintessence/phantom DE to DM, whereas for a negative Q energy flows from DM to DE.
From Eqs. (<ref>) and (<ref>) the evolution equation of the scalar field can be expressed as
  
    ϕ̈=-3Hϕ̇-1/ϵdV/dϕ-Q/ϵ ϕ̇ ,

while from Eq. (<ref>) the effective modified Raychaudhuri equation can be rewritten as

    Ḣ=-1/2(ρ_m+ϵϕ̇^2)(1-2ρ/ρ_c)  .


In order to write our system of equations as an autonomous system of ordinary differential equations, we introduce the following set of dimensionless phase space variables
  
    x≡ϕ̇/√(6) H ,   y≡√(V(ϕ))/√(3) H,
         z≡ρ/ρ_cand s≡ -1/VdV/dϕ.

Using the new dimensionless variables (<ref>), the effective modified Friedmann equation (<ref>), can be rewritten in a dimensionless form as
   
    1 =(ρ_m/3H^2+ϵ x^2+y^2)(1-z)  .

This provides a constraint that can be used to eliminate ρ_m in favour of the dimensionless variables (<ref>) in all equations that follow.
Using equations Eqs. (<ref>)–(<ref>), we obtain the following autonomous system of differential equations,

    x'   =-3x+1/ϵ√(3/2) sy^2+x[3/2(1/1-z-ϵ x^2-y^2)+ϵ3x^2](1-2z)-Q/ϵ √(6) H^2ϕ̇,
    
    y'   =-√(3/2) sxy +y[3/2(1/1-z-ϵ x^2-y^2)+ϵ3x^2](1-2z),
    
    z'   =-3z-3z(1-z)(ϵ x^2-y^2),
    
    s'   =-√(6) xf(s),

where a prime denotes differentiation with respect to N=ln a (a being the usual scale factor) and we defined

    f(s)=s^2(Γ(s)-1)         Γ= Vd^2V/dϕ^2(dV/dϕ)^-2 .

We consider a class of potentials where Γ is a function of s, with Γ=1 being the case of an exponential potential.
In general Γ may not be function of s, in which case one has to take into account the higher derivatives of the scalar field potential <cit.> or choose new dimensionless variables <cit.>, resulting in the increase of dimension of the dynamical system for both cases.
In terms of the dimensionless variables (<ref>), the scalar field relative energy density, the relative energy density of dark matter, the relative energy contribution due to loop quantum corrections, the total cosmic energy EoS and the deceleration parameter are given by
  
    Ω_ϕ   ≡ρ_ϕ/3H^2=ϵ x^2+y^2,
    Ω_m   ≡ρ_m/3H^2=1/1-z-ϵ x^2-y^2,
    Ω_c    ≡ 1 - Ω_m - Ω_ϕ = z/z-1,
    
      w_ tot   ≡p_ϕ/ρ_ϕ+ρ_m=(1-z)(ϵ x^2-y^2),
    
     q   ≡ -1-Ḣ/H^2=-1+[3/2(1/1-z-ϵ x^2-y^2)+ϵ3x^2](1-2z).

The dynamical systems analysis yields the asymptotic behavior of the system, i.e., the beginning and the ultimate fate of the universe. For the numerical calculations that follows, we choose initial conditions in such a way that the final state of the universe is in agreement with the present observational data: Ω_m = 0.31, Ω_ϕ = 0.69 and q=-0.62 <cit.>.

In order to close the autonomous system (<ref>)–(<ref>), we need to consider a particular form of the interaction Q for which the last term in Eq. (<ref>) can be expressed as a function of the variables (<ref>).
In the next two sections, we choose the following two particular interaction terms


(I) Q=αρ_mϕ̇         (II) Q=βρ̇_ϕ  ,



with α and β dimensionless constants.
The first of these couplings arises naturally from scalar-tensor theories in the Einstein frame <cit.> and is well motivated from string theory. The second one is instead purely phenomenological <cit.> and it will be used to expose some interesting properties of a more complex interacting model.
In what follows, we tried to keep the analysis as general as possible but in order to study the stability of some interesting critical points in more detail, we consider two particular forms for the scalar field potential as examples: V=V_0 cosh^-μ(λϕ) and V=M^4+n/ϕ^n.
We note also that in non-interacting DE models, the energy density of DM is always taken to be non-negative, while in models of interacting DE the condition Ω_m<0 can be allowed (see e.g. <cit.>).
In particular this means that the resulting phase spaces are generally not compact.
This implies that critical points at infinity should be analyzed by compactifying the phase space using the Poincare compactification technique; see e.g. <cit.>.
Nevertheless in what follows we only determine the dynamics near the finite critical points, which is enough from a phenomenological point of view since our aim is to find physically viable solutions, namely trajectories connecting DM to DE domination.
Once we find such type of solutions, then given the right initial conditions the observed evolution of our universe can be described by the DE model under consideration (at least at the background level).








§ INTERACTING MODEL I: Q=ΑΡ_MΦ̇



This type of interaction term arises naturally in scalar-tensor theory <cit.>, where the energy terms are separately conserved in the Jordan frame but become coupled in the Einstein frame.
The dynamical system investigation of quintessence with this interaction and an exponential potential has been studied in both standard EC <cit.> and LQC <cit.>.
In this section, we provide the dynamical analysis of this DE model with a general scalar field potential in the LQC framework.
Here we only show the properties of the phase space and characterize the full cosmological dynamics of this model.
Discussions about the cosmological implications are postponed to section <ref>.

As noted above the dark sector interaction modifies only the x^' equation in the system (<ref>)-(<ref>), while the other equations remain unaffected.
For this particular interaction the system of equations (<ref>)-(<ref>) becomes

    x'   =-3x+1/ϵ √(3/2) sy^2 -√(6)/ϵ 2α(1/1-z-ϵ x^2-y^2)+x[3/2(1/1-z-ϵ x^2-y^2)+ϵ3x^2](1-2z),
    
    y'   =-√(3/2) sxy +y[3/2(1/1-z-ϵ x^2-y^2)+ϵ3x^2](1-2z),
    
    z'   =-3z-3z(1-z)(ϵ x^2-y^2),
    
    s'   =-√(6) xf(s) .

It is easy to check that the above dynamical system (<ref>)-(<ref>) is invariant under the transformation y→ -y, meaning that we need to analyze only the phase space for positive values of y.
The critical points of the system (<ref>)-(<ref>) and the relevant parameters are listed in Table <ref> and the eigenvalues of the corresponding Jacobian matrix are listed in Table <ref>.
In these tables, and throughout the rest of the paper, s_* is a solution of the equation f(s)=0 and df(s_*) denotes the value of the derivative of f at s=s_*.
From Tables <ref> and <ref>, we can note that all critical points depend on the specific form of potentials: C_1, C_2, C_3, C_4, C_5 all depend on s_* and df(s_*), whereas C_6 depends on the value of f(0) for its stability.
We now discuss the stability of the critical points listed in Table <ref> for quintessence and phantom dark energy separately.














 §.§ Quintessence dark energy (ϵ=1)


We briefly discuss the properties of the critical points listed in Table <ref> with ϵ=1.
The stability regions in the (s_*,α) parameter space for each critical point are shown in Fig. <ref>.




  * Point C_1: This point corresponds to a decelerated, stiff matter dominated universe (w_ tot=1). It is stable whenever α<-√(6)/2, s_*>√(6), df(s_*)>0, otherwise it is saddle.


  * Point C_2: This point corresponds to a decelerated, scaling solution. It is stable whenever α^2<3, α s_*<-3, α df(s_*)<0, otherwise it is saddle.



  * Point C_3: This point corresponds to an accelerated solution (q=-1) with loop quantum contribution and negative DM energy density (Ω_m<0). It corresponds to a late time attractor (stable spiral) when α s_*>0 and α df(s_*)>0.


  * Point C_4: Point C_4 exists when s_*^2<6. It corresponds to an accelerated solution for s_*^2<2 and describes a scalar field dominated universe. It is stable whenever s_*(α+s_*)<3 and s_*df(s_*)<0, otherwise it is saddle.



  * Point C_5: Point C_5 corresponds to a late time accelerated, scaling solution for some values of the parameters α and s_*. This is confirmed numerically from Fig. <ref> by plotting the region of stability and the region of acceleration for C_5 in the (s_*,α) parameter space.


  * Set of points C_6: A set of non-isolated critical points C_6 corresponds to the case where s=0 (i.e. the potential is effectively constant). It is a non-hyperbolic set with one vanishing eigenvalue if f(0) ≠ 0. This type of
non-isolated points form a normally hyperbolic set <cit.>.
The center manifold of a set of non-isolated critical points is
determined by the direction of the eigenvectors corresponding to
vanishing eigenvalues, while the signature of the non-vanishing
eigenvalues determine its stability. For points C_6 the real
components of the non-vanishing eigenvalues are negative only if
f(0)>0, implying that this set of points corresponds to a late
time stable attractor only if f(0)>0. It is a stable node if
0<f(0)<3/4(1-z), while it is a stable spiral if
f(0)>3/4(1-z). In any other case it is saddle. Further
investigation is required when f(0)=0, which will be studied for
the examples below once the scalar field potential is specified.
This additional set of points is interesting from a phenomenological perspective as it shows the effect of loop quantum corrections to explain late time acceleration of the universe.


The properties of points C_1, C_2, C_4, C_5 are the same as in standard EC <cit.>. Point C_3 shows the effect of loop quantum corrections but implies a negative DM energy density.
The additional set of points C_6 is interesting as it corresponds to a scalar field dominated solution where the effects of loop quantum gravity corrections can be used to explain the late time accelerated universe.
From the above analysis, we note that critical points C_1, C_2, C_4, C_5 cannot be late time attractors simultaneously.
However there is a possibility of multiple late time attractors: points C_3 and C_4 on one side, and C_3 and C_5 on the other side.
This kind of situation is interesting from both a phenomenological and mathematical point of view, and it usually leads to a more complex cosmological dynamics.
Depending on the choices of parameters and initial conditions, the interacting DE model considered here can successfully describe the late time evolution of the universe.
Since the existence and stability properties of all critical points heavily depend on s_*, df(s_*) and f(0), the full dynamics of the phase space can be derived only once a particular scalar field potential has been chosen.
For this reason in what follows we consider two particular potentials.












  §.§.§ Example 1: V=V_0 cosh^-μ(λϕ)
 

Here we consider the potential V=V_0 cosh^-μ(λϕ), (where V_0 and λ are two constants of suitable dimensions while μ is a dimensionless parameter).
This potential was proposed to explain the exit of the universe from a scaling regime to a de-Sitter like accelerated attractor through a mechanism of spontaneous symmetry breaking <cit.>.
The dynamical behaviour of standard EC with this scalar field potential has been studied in <cit.>.
For this potential we find f(s)=s^2/μ-μλ^2, so that

    s_*=±μλ df(s_*)=2s_*/μ = ± 2 λ .

Each of the critical points C_1, C_2, C_3, C_4, C_5
appears twice in the phase space according to the two solutions
s_*=±μλ.
We will denote with C_i^+ the critical point associated with s_*=μλ and with C_i^- the ones associated with s_*=-μλ (i=1, 2, 3, 4, 5).
The stability regions of points C_i^+ and C_i^- (i=1,2,3,4,5) for μ=1 in the (λ,α) parameter space are given in Figs. <ref> and <ref> respectively.
Note the symmetry between the stability regions of points C_i^+ and points C_i^-.
This is due to the invariance of the hyperbolic potential under the λ↦ -λ transformation.
The properties of the critical points for the hyperbolic potential are as follow:



  * Points C_1^+, C_1^-: These points correspond to decelerated, stiff matter dominated universes (w_ tot=1). Point C_1^+ is stable whenever α<-√(6)/2, μλ>√(6), λ>0, otherwise it is saddle. Point C_1^- is stable whenever α<-√(6)/2, μλ<-√(6), λ<0, otherwise it is saddle.


  * Points C_2^+, C_2^-: These points correspond to a decelerated, scaling solutions. Point C_2^+ is stable whenever α^2<3, α μλ<-3, α λ<0, otherwise it is saddle. Point C_2^- is stable whenever α^2<3, α μλ>3, α λ>0, otherwise it is saddle.


  * Points C_3^+, C_3^-: These points correspond to an accelerated solution (q=-1) with negative DM energy density. Point C_3^+ corresponds to a late time attractor (stable spiral) for μ >0 and α λ>0. Point C_3^- corresponds to a late time attractor  for μ >0 and α λ<0.


  * Points C_4^+, C_4^-: These points exist when μ^2λ^2<6. They correspond to accelerated, scalar field dominated universes. Point C_4^+ is stable whenever μλ(α+μλ)<3, μ<0, otherwise it is saddle. Point C_4^- is stable whenever μλ(α-μλ)>-3, μ<0, otherwise it is saddle.


  * Points C_5^+, C_5^-: These points are cosmologically interesting as they correspond to late time accelerated scaling solutions for some values of the parameters α, μ and λ (cf. Fig. <ref>).
For example, if we take α=-2.8, λ=-2, μ=1, we obtain λ_1=-1.25, λ_2=-1.18-3.38 i, λ_3=-1.18+3.38 i, λ_4=-2.5, with q=-0.37, Ω_m=0.28, Ω_ϕ=0.71 for point C_5^+, while point C_5^- is saddle in nature.
On the other hand for α=-2.8, λ=2, μ=1, point C_5^- is stable but point C_5^+ is saddle.
This implies that choosing the appropriate combination of parameters, critical points C_5^+ and C_5^-  describe a late time accelerated scaling attractor (stable spiral) with DM and DE energy density values in agreement with present observations.


  * Set of points C_6: This normally hyperbolic set of critical points C_6 corresponds to a late time attractor only if μ<0.



Fig. <ref> shows the contribution of loop quantum gravity corrections at early time together with the evolution of the relevant cosmological parameters (<ref>)–(<ref>).
One particular trajectory is considered which evolves from a matter phase dominated by interacting energy (point C_2) and settles in the DE dominated point C_4.
This model can thus describe the late time DE dominated observed phase of our universe; see Sec. <ref> for further discussions.







  §.§.§ Example 2: V=M^4+n/ϕ^n 


In this example we consider the inverse power-law potential V(ϕ)=M^4+n/ϕ^n (where M is a mass scale, while n is a dimensionless parameter), which can lead to tracking behavior in EC <cit.>.
For this potential, the function f(s) is given by
 
    f(s)=s^2/n ,

meaning that

    s_*=0   df(s_*)=2s_*/n = 0  .

This implies that in this case we have only one copy for each of the points C_1, C_2, C_3, C_4, C_5 listed in Table <ref>.
The properties of the critical points are as follow:







 
  * Point C_1: This describes a decelerated, stiff matter dominated point (w_ tot=1) and is always saddle.

 
  * Point C_2: This point corresponds to a decelerated, scaling solution. It is always saddle, in contrast with the hyperbolic potential for which it can be stable.

 
  * Point C_3: This point corresponds to an accelerated solution with negative DM energy density. It is a non-hyperbolic point, for which linear stability fails to determine its stability. In order to determine its stability we numerically stream plotted projections onto the (x,s) slice of the dynamical flow around its neighbourhood, as shown for example in Fig. <ref>.
 For several interesting combinations of the model parameters, we have checked numerically that this point cannot be stable as some trajectories are always repelled from it.

 
  * Point C_4: Point C_4 reduces to point (0,1,0,0) where the scalar field potential energy dominates.
 It is a non-hyperbolic point, so linear stability fails to determine its stability.
 Usually the stability of this type of critical point can be determined analytically by employing center manifold theory, otherwise one can also use numerical methods; see e.g. <cit.> for some applications in the recent literature.
 Numerically one can plot the projection of trajectories around the critical point separately on the x, y, z and s axes (see Fig. <ref> for example).
 We have checked numerically that nearby trajectories asymptotically approach the coordinates corresponding to point C_4 only for n>0, while the parameter α can be arbitrary.
 Hence depending on the choices of the model parameters this point can correspond to a late time attractor.
 
 
  * Point C_5: This point corresponds to an accelerated solution (q=-1) with negative DM energy density Ω_m<0. It is always a saddle since the eigenvalues λ_2 and λ_3 are always of opposite sign.

 
  * Set of points C_6: For the inverse power-law potential, this set of non-isolated critical points is non-hyperbolic but not normally hyperbolic since it presents two vanishing eigenvalues.
 We use numerical techniques to check stability by plotting the projection of nearby trajectories separately on the x, y, z and s axes.
 We find that these trajectories approach the set of points C_6 as N→∞ for n>0.
 In fact the effective EoS parameter is always -1  for any point belonging to this set.
 




 §.§ Phantom dark energy (ϵ=-1)


We now turn our attention to phantom DE with ϵ=-1. Region of stability in the (s_*,α) parameter space for the critical points listed in Table <ref> are given in Fig. <ref>.
Their properties are as follow:


 
  * Point C_1: Point C_1 does not exist for ϵ=-1.

 
  * Point C_2: This point corresponds to an accelerated solution for α^2>1/2, with negative DE energy density parameter. It is stable when α^2<3/2, 2α (α+s_*)>3 and α df(s_*)<0.
 
 
  * Point C_3 : This point corresponds to an accelerated solution with negative DE energy density parameter. It is a stable node if 3/2<α^2<2, α s_*>0, α df(s_*)>0, it is stable spiral if α^2>2, α s_*>0, α df(s_*)>0, otherwise it is saddle.
 
 
  * Point C_4: Point C_4 exists for any values s_*. It corresponds to an accelerated scalar field dominated universe and it is saddle. Although in LQC this point cannot be a late time attractor, it is found to be stable in EC where it corresponds to a future big rip singularity <cit.>.
 This behavior was first noticed in the case of an exponential potential <cit.> and constitutes an interesting example of how cosmological singularities can be avoided by LQG effects.
 
 
  * Point C_5: Point C_5 corresponds to an accelerated, scaling solution for some values of parameter α and s_*. However this point is always saddle as the eigenvalues λ_2 and λ_3 have opposite sign.

 
  * Set of points C_6: As in the case of the quintessence field, this non-isolated set of critical points C_6 is a normally hyperbolic set.
 It corresponds to a late time attractor only if f(0)<0.
 It is a stable node if -3/4(1-z)<f(0)<0, it is stable spiral if f(0)<-3/4(1-z), otherwise it is saddle.
 Further investigation is required when f(0)=0 and we shall postpone this once a particular scalar field potential has been chosen. Again this set of points is interesting as it shows the effects of loop quantum corrections in determining the late time accelerated phase of the universe.
 Interestingly this point is a late time attractor which is not present in the case of the exponential potential <cit.>.
 

From the above general analysis, we note that from Fig. <ref> points C_2, C_3 cannot be late time attractors simultaneously.
Depending on the choice of the scalar field potential and the initial conditions, we thus find that the universe evolves towards an accelerated, scalar field dominated set of critical points C_6 or towards a negative DE density critical points C_2, C_3.
Again since the existence and stability properties of the critical points depend heavily on s_*, df(s_*) and f(0), in what follows we analyze their properties selecting two specific potentials.




  §.§.§ Example 1: V=V_0 cosh^-μ(λϕ)






Here we report the properties of the critical points in Table <ref> for ϵ=-1 and the potential V=V_0 cosh^-μ(λϕ).
Again here we have two copies of each critical points C_2, C_3, C_4, C_5, one for each of the two solutions s_*=±μλ.
As in the case of quintessence each point C_i with s_*=μλ is denoted as C_i^+ while point with s_*=-μλ are denoted as C_i^- (i=2,3,4,5).
The stability regions of C_2^+, C_3^+ are given in Fig. <ref> and that of C_2^-, C_3^- are given in Fig. <ref>.
Again the symmetry of these two figures is due to the invariance of the hyperbolic potential under the λ↦ -λ transformation.



 
  * Points C_2^+, C_2^-: These points correspond to an accelerated solution for α^2>1/2, with negative DE energy density parameter. Point C_2^+ is stable when α^2<3/2, 2α (α+λμ)>3 and α λ<0. Point C_2^- is stable when α^2<3/2, 2α (α-λμ)>3 and α λ>0.

 
  * Points C_3^+, C_3^-: Point C_3^+ is a stable node if 3/2<α^2<2, μ>0, α λ>0, it is a stable spiral if α^2>2, μ>0, α λ>0, otherwise it is saddle. Point C_3^- is a stable node if 3/2<α^2<2, μ>0, α λ<0, it is a stable spiral if α^2>2, μ>0, α λ<0, otherwise it is saddle.

 
  * Points C_4^+, C_4^-: They correspond to an accelerated, scalar field dominated universe. However both of them are saddle in nature for any choice of model parameters.

 
  * Points C_5^+, C_5^-: These accelerated scaling solutions are saddle in nature for any choice of model parameters.

 
  * Set of points C_6: This non-isolated normally hyperbolic set of critical points C_6 corresponds to a late time attractor only if μ>0.



From the analysis above, we see that depending on the initial conditions the universe can evolve towards an accelerated, scalar field dominated set of critical points C_6.
This can be seen in Fig. <ref> which shows the evolution of cosmological parameters towards an accelerating phase q=-1 (point C_6).
Another interesting observation is that a transient phantom epoch described by points C_4^+, C_4^- (q=-1.5) might be possible.
Although the universe undergoes a super-accelerated expansion (Ḣ>0) at this point, the future big rip singularity is avoided by loop quantum effects which turn this point from an attractor to a saddle.





  §.§.§ Example 2: V=M^4+n/ϕ^n 


We recall that for this potential we have s_*=0 and df(s_*)=0, implying in particular that there is only one copy for each of the point C_2, C_3, C_4, C_5.
The properties of the critical points are as follow:








  * Point C_2: Point C_2 describes a solution with negative DE energy density. It is always saddle.


  * Point C_3: Point C_3 corresponds to an accelerated solution with negative DE density. We have numerically checked that it is never stable for any choice of parameters. An example of the flow around this point projected on the (x,s)-plane is given in Fig. <ref>.


  * Point C_4: Point C_4 exists for any values α, μ and λ. It is a non-hyperbolic critical point and it is not stable as confirmed numerically. An example of the dynamical flow around this point, projected on the (x,s)-plane, is given in Fig. <ref> for some values of the model parameters.


  * Point C_5: Point C_5 corresponds to an accelerated, scaling solution. This point is always saddle.


  * Set of points C_6: Since this set contains point C_4 and trajectories on the (x,s) sub-space do not approach the coordinates of point C_4, this set cannot be a late time attractor, as it is for the hyperbolic potential. This has been checked numerically for different set of model parameters of phenomenological interest.


From this analysis, we see that there is no finite late time attractor.
This interacting phantom model cannot thus explain the late time behavior of our universe, although future big rip singularities appearing in EC can be avoided.










§ INTERACTING MODEL II: Q=ΒΡ̇_Φ



The dynamical investigation of this interaction term for quintessence dark energy in the framework of standard EC has been recently
studied in <cit.> where it was found that this model can alleviate the coincidence problem.
A generalised form of this interaction has been used in <cit.> to build a coupled DE model where the sign of the dark interaction changes during the cosmological history.
The specific interaction Q=βρ̇_ϕ is also motivated from the dimensional point of view since Q has the dimension of a time rate of energy density <cit.>.
Again in this section we will only expose the dynamical system analysis of the interacting model under study, while leaving the discussion on the cosmological implications to section <ref>.

For this interaction the system (<ref>)-(<ref>) becomes

    x'   = -3x/(1+β)+1/ϵ √(3/2) sy^2  +x[3/2(1/1-z-ϵ x^2-y^2)+ϵ3x^2](1-2z),
    
    y'   =-√(3/2) sxy +y[3/2(1/1-z-ϵ x^2-y^2)+ϵ3x^2](1-2z),
    
    z'   =-3z-3z(1-z)(ϵ x^2-y^2),
    
    s'   =-√(6) xf(s) .

It can be seen that the system (<ref>)-(<ref>) is symmetric under the transformation y → -y.
It is also
symmetric under the transformation (x,s) → (-x,-s) for
potential with f(s)=f(-s). The critical points of the system
(<ref>)-(<ref>) and their properties are shown in
Table <ref> and the corresponding eigenvalues of the
Jacobian matrix are given in Table <ref>.
The dynamics of this model is simple if compared to that of the interacting model I analysed in section <ref>.
The system (<ref>)-(<ref>) contains only two critical points and
two non-isolated set of critical points.
The non-isolated set D_1
is completely independent from the scalar field for its existence
and stability. On the other hand, critical points D_2, D_3 and the
non-isolated set D_4 depend on the specific choice of the scalar
field potential for their existence and stability. As before, in
what follows, we analyze the cosmological dynamics separately
for quintessence and phantom DE.





 §.§ Quintessence dark energy (ϵ=1)








The properties of the critical points in table <ref> with ϵ = 1 are the following:



  * Set D_1:  This non-isolated set of critical points corresponds to a decelerated (q=1/2), matter dominated universe (Ω_m=1). It is always saddle.



  * Point D_2: This point corresponds to a decelerated,
scaling solution. It is always saddle since the eigenvalues
λ_1 and λ_2 are of opposite signs within the
region of existence.


  * Point D_3: The stability of this scaling solution is challenging to determine analytically due to the complicated expression of the eigenvalues of the Jacobian matrix of the system evaluated at this point.
Instead we analyzed its stability numerically by plotting the regions where λ_1<0 and λ_3<0 on the (s_*,β) parameter space as shown in Fig. <ref>.
Looking at Fig. <ref> we can conclude that this point is always saddle since the region of negativity of the eigenvalues λ_1 and λ_3 are disjoint.
This is different from the result obtained in standard EC, where this point is a late time accelerated attractor <cit.>.


  * Set of points D_4: The set of critical points D_4 exists only for s=0.
It is also a normally hyperbolic set with one vanishing eigenvalue if f(0) ≠ 0.
This set corresponds to a late time attractor only if β>-1 and f(0)>0.
It is a stable node if β>-1 and 0<f(0)<1-z/4(β+1)^2, it is stable spiral if β>-1 and f(0)>1-z/4(β+1)^2, otherwise it is saddle.
Further investigation is required when f(0)=0 for which we shall postpone the analysis once a particular potential has been chosen.
Again this set of points is interesting as it shows the effect of loop quantum corrections to explain the late time accelerated universe.


From this analysis, we see that the universe can evolve from a matter dominated phase (point D_1) towards an accelerated scalar field dominated attracting set (set D_4) through a scaling solution (point D_2 or D_3).
This scenario might be used to explain the late time transition of our universe from a matter dominated phase towards a dark energy dominated phase.
Since the existence and stability behavior of points D_2, D_3 and D_4 depends on s_*, df(s_*) and f(0), we shall better analyze their properties for particular scalar field potentials.
In what follows we investigate again the two potentials considered in the examples above.

 




 









  §.§.§ Example 1: V=V_0 cosh^-μ(λϕ)


For this potential we have again two copies of critical points D_2 and D_3, since there are two solutions s_*=±μλ. As before, we shall assign D_i^+ for s_*= μλ and D_i^- for s_*=-μλ (i=2,3). Point D_1 does not depend on the choice of the potential, while the properties of the other critical points are as follow:




  * Points D_2^+, D_2^-: As discussed in the general case, these points are saddle in nature as its eigenvalue λ_1 is negative while λ_2 is positive for any values of s_*.



  * Points D_3^+, D_3^-: Since point D_3 is always saddle for s_*>0 and s_*<0 as shown in Fig. <ref> for general potentials. This implies that points D_3^+, D_3^- are saddle in nature.



  * Set of points D_4: This set corresponds to a late time attractor only if β>-1 and μ<0, otherwise it is saddle.



Fig <ref> shows the evolution of the relevant cosmological parameters given suitable initial conditions.
From this figure, we see that the universe can evolve towards an accelerated phase with q=-1 (set D_4), with a long lasting matter phase dominated by the interacting energy.
From Fig. <ref> instead we see that the universe undergoes a transition to a DE dominated phase (set D_4 with z=0) from a long lasting matter phase dominated by DM (point D_1).
In both cases, we find a long lasting matter phase (w_ tot = 0) as required for the formation of the cosmic structure, before transition to DE domination prevails.




  §.§.§ Example 2: V=M^4+n/ϕ^n 


We recall that for this potential one finds s_*=0.
Again the properties of point D_1 are independent of the choice of the scalar field potential, while the other critical points behave as follows:








  * Point D_2: This point is non-hyperbolic but it behaves as a saddle since its eigenvalues λ_1 and λ_2 are of opposite sign.


  * Point D_3: This point does not exist for this potential since s_*=0.


  * Set of points D_4: This set of points is no longer
normally hyperbolic since now f(0)=0 and two eigenvalues are
zero. The stability can be determined numerically by plotting the
projection of trajectories near the set separately on the x, y,
z, s axes (see for example
Fig. <ref>). We find numerically that
trajectories approach points lying on this set only for n>0 and for any choice of β.
This set can thus behave as a late time attractor.






 §.§ Phantom dark energy (ϵ=-1)


We turn now our attention to phantom DE.
The properties of critical points in table <ref> with ϵ = -1 are as follow:



 
  * Point D_1: As in the case of quintessence, this point is always saddle.


  * Point D_2: This point corresponds to an accelerated universe (if β>2) with negative DE density. It is stable when s_*√(β-1/β+1)>6/β+1, otherwise it is saddle.


  * Point D_3: As in the case of quintessence, the complicated stability of this scaling solution can only be determined numerically by plotting the region of negativity of eigenvalues λ_1 and λ_3 on the (s_*,β) parameter space as shown in Fig. <ref>.
This point is again always saddle since the region where λ_1<0 and λ_3<0 are disjoint.


  * Set of points D_4: This set corresponds to a late time attractor only if β>-1 and f(0)<0, otherwise it is saddle. Further investigation is required when f(0)=0 and we shall postpone this till a particular potential is chosen.



The dynamics of this particular model is simple if compared to the interacting model I investigated in section <ref>.
From the above analysis, we see that the universe can evolve from a matter dominated phase (point D_1) towards an accelerated scalar field dominated phase (set D_4).
This might explain the late time transition of the universe from a matter dominated phase to a DE dominated phase.
However since the existence and stability behavior of points D_2, D_3 and D_4 depend on s_*, df(s_*) and f(0), we shall better analyze their properties for the two potentials considered in our examples.


 

  §.§.§ Example 1: V=V_0 cosh^-μ(λϕ)


We recall again that for this potential we obtain s_* = ±μλ, implying that two copies of points D_2 and D_3 are present in the phase space.
As before point D_1 does not depend on the particular form of the scalar field potential and its properties do not change.
For the other critical points we have:




  * Points D_2^+, D_2^-: Point D_2^+ is stable when √(β-1/β+1)>6/β+1 and point D_2^- is stable when √(β-1/β+1)<6/β+1, otherwise they are saddle.


  * Points D_3^+, D_3^-:  As we have seen in the general case, these scaling solution are always saddle for s_*>0 and s_*<0 (see Fig. <ref>). This implies that points D_3^+, D_3^- are saddle in nature.


  * Set of points D_4: This set corresponds to a late time attractor  only if β>-1 and μ>0, otherwise it is saddle.



The phantom model can yield an interesting and particular dynamics including finite stages of phantom domination.
An example is given in Fig. <ref> where the universe undergoes a transition from a matter phase dominated by the interacting energy, to a DE final era characterized by oscillations of w_ tot which bounces the universe between quintessence and phantom domination before eventually stabilising to an effective cosmological constant behavior (w_ tot = -1).
The same model can however leads to a dynamics very similar to standard ΛCDM, as reported in the example in Fig. <ref>.
A long lasting matter dominated phase is in fact followed by a smooth transition to DE domination mimicking the effects of a cosmological constant.


 

  §.§.§ Example 2: V=M^4+n/ϕ^n 





For this example we have again s_* = 0.
The properties of the critical points are then:

 


  * Point D_2:
This is now a non-hyperbolic point with two vanishing eigenvalues.
Its stability can be determined numerically by plotting the stream plot trajectories on the (x,s) plane.
In fact no matter the values of the model parameters, point D_2 is always a saddle in the (x,s) plane, implying that it can never be stable in general, as shown for example in Fig. <ref>.


  * Point D_3:  This point does not exists since for this potential s_*=0.


  * Set of points D_4:
Again the stability of this set of points is verified numerically.
Independently from the values of the model parameters this set of critical points never represents a stable attractor, as it can be seen in the example of Fig. <ref>.



We can conclude that the phantom field with power-law potential has no finite late time attractor, implying that the final state of the universe is characterized by some critical point at infinity.
This is in contrast with the example above for the hyperbolic potential, where both points D_2 and D_4 could be stable.






§ COSMOLOGICAL IMPLICATIONS
 


In this section we extract the cosmological features obtained from the interacting DE models analysed above.
The cosmological dynamics of both models is full of phenomenologically interesting solutions and it moreover shows the contribution of loop quantum gravity corrections in both the late and early time behaviors of our universe. 
In what follows we discuss each physically relevant solution separately:



	
  * Late-time DE dominated solutions: These critical points are late time attractors characterized by a cosmic phase dominated by DE (Ω_ϕ = 1) with an effective EoS mimicking a cosmological constant behavior (w_ tot≃ -1). Although this kind of solutions naturally arise for quintessence in EC, they do not appear explicitly in the two models investigated here. There are however other solutions which well describe this cosmological behavior. For example in model I the set of points C_6 well represent a late time attractor with w_ tot = -1, as also shown in the example with an hyperbolic potential (see Fig. <ref>). The same situation emerges in model II where the set of point D_4 acts as a late time attractor mimicking a cosmological constant behavior for both the quintessence and phantom fields (see Figs. <ref> and <ref>). In all these cases a smooth transition from a matter dominated phase to an effective DE era is attained in agreement with the observed dynamics of our universe.

	
  * Accelerating scaling solutions: These solutions are identified by a constant finite ratio between Ω_ϕ and Ω_m, implying that the DM energy density evolves at the same rate of the DE energy density. A scenario where a late time attractor is characterised by an accelerating scaling solution is commonly used to alleviate the cosmic coincidence problem, especially in models of interacting DE <cit.>. In the models analysed here several scaling solutions appear. For example critical points C_2 and C_3 in model I, or point D_2 in model II. Depending on the choice of the parameters, all these points can describe a stable accelerating scaling solution for either quintessence or phantom DE. Point C_3 is of particular interest since not only Ω_ϕ scales as Ω_m, but also the contribution due to loop correction Ω_c scales accordingly remaining a non negligible fraction of the energy content in the universe even at late times.

	
  * Phantom behavior: The phantom regime is associated with w_ tot< -1, which can indeed be attained by phantom DE in EC. In such cases however the final state of the universe is a big rip singularity at some finite time in the future. Loop quantum corrections are known for being able to avoid this singularity. An example of this situation is provided in Figs. <ref> and <ref> where phantom domination is turned into cosmological constant-like behavior at late times. Within this scenario the big rip is avoided and the universe eventually reach an expanding de Sitter solution. Nevertheless this leads to a finite period of phantom regime which characterizes the dynamics of the universe after a standard period of matter domination.

	
  * Loop quantum effects: Loop quantum corrections are important whenever the term z = ρ / ρ_c is not negligible. This always happens at early times when ρ≫ρ_c and Ω_c dominates over the energy budget of the universe (see Figs. <ref>, <ref> and <ref>). Note that although Ω_c dominates the effective EoS w_ tot is either zero or one, implying that the universe at early times is always well described by either a matter dominated phase or a stiff fluid dominated phase. On the other hand late time loop quantum effects do also appear. These are described by the sets of critical points C_6 and D_4, which can be used to describe the late time accelerated expansion of the universe, mimicking a cosmological constant behavior. For these solutions the DE energy density Ω_ϕ scales according to the contribution of quantum correction Ω_c, similar to a sort of scaling solution (see e.g. Fig. <ref>).

	
  * Possible observational signatures: The models investigated in this paper present also some other particular features which might provide distinguishing observational signatures with respect to the standard ΛCDM model. Clear examples are shown in Figs. <ref> and <ref>, where the effective EoS w_ tot instead of smoothly changing from zero to -1 presents particular oscillations with excursions in the phantom regime. This scenario could in fact be constrained by future data once astronomical observations will better determine the dynamics and EoS of DE at higher redshift. From Fig. <ref> we note also that early DE might appear in the dynamics of model I. These scenarios might provide distinguishing observational features, especially at the perturbations level, with respect to ΛCDM, and can thus in principle be constrained by present and future observations.








§ CONCLUSION



In this paper, we studied the dynamics of interacting quintessence and phantom DE in the LQC framework.
The scope was to perform a complete dynamical system investigation of interacting DE with an arbitrary scalar field potentials.
We have considered two specific interactions between DE and DM of the form α ρ_m ϕ̇ (model I; see Section <ref>) and βρ̇_̇ϕ̇ (model II; see section <ref>).
For each of these interactions, we have analysed two forms of dark energy: the quintessence and the phantom fields.
To better characterize the dynamical properties of these interacting models beyond the standard exponential potential, we followed the well known approach of considering the quantity Γ (cf. Eq. (<ref>)) as a general function of the parameter s (cf. Eq. (<ref>)).
Within this analysis the dimension of the resulting autonomous systems increases from three to four if compared with the corresponding dynamical systems obtained assuming an exponential potential, and the number of critical points multiply making these new systems more difficult to study in detail.
Furthermore in order to better understand the dynamics of these models, especially concerning non-hyperbolic critical points, in each case we considered two concrete non exponential potentials, namely the hyperbolic potential V=V_0 cosh^-μ(λϕ) and the inverse power-law potential V(ϕ)=M^4+n/ϕ^n.

We found some interesting and unique features arising from these interacting dark energy models beyond the exponential potential.
In model I with a quintessence field, we obtained one additional set of critical points C_6 (see Table <ref>), which does not appear in EC.
In fact this set implies the contribution of loop quantum gravity phenomena to a late time accelerated universe.
We also observed in the case of quintessence that there is the possibility of attaining a transient period of phantom acceleration (see Fig. <ref>), predicting in this way some specific observational signatures of this model.
Furthermore for some choices of the model parameters, we found multiple late time attractors.
This situations are interesting from the mathematical point of view since the dynamics strongly depends on the choice of initial conditions and can be handled using bifurcation theory.
In the case of phantom field, the set C_6 is a late time attracting set only for the hyperbolic potential and has no equivalent in the exponential potential case <cit.>.
This seems to imply that the scalar field potential plays an important role in determining the contribution of loop quantum effect for driving the late time accelerating universe.
As in the case of exponential potential, we also observe that future big rip singularities can be avoided with the universe passing from phantom domination to an accelerating de Sitter phase (q=-1).
On the other hand for power-law potential interacting model I cannot explain the observed late time behavior of the universe, as there are no late time accelerating attractors.

In the interacting model II with a quintessence field, apart from a standard matter dominated point D_1, we obtained one additional late time accelerated set of critical points (set D_4) with contributions from loop quantum effects.
We found that the scaling solution D_3 is a saddle, which is in contrast with standard EC, where it is a late time, accelerated
scaling solution <cit.>.
This model can also explain the late time transition from matter domination to a DE dominated phase (see Fig. <ref>).
In the case of phantom field, the set D_4 is a late time attracting set only for the hyperbolic potential but not for the power-law potential.
This again suggests the important role of the scalar field potential to obtain a late time accelerated universe in the framework of LQC.
As it happens for model I, for the power-law potential the interacting model II cannot explain the late time behavior of the universe as there are no late time accelerating attractors.

In conclusion we found that the results obtained with an exponential potential can not only be recovered for a wider class of potentials, but also new interesting phenomenology appears (see Sec. <ref>).
This is the case for both the quintessence and phantom DE fields, where several interesting solutions have been derived: for example late time DE domination, accelerating scaling solutions, phantom domination, DM to DE transition.
The rich background dynamics obtained from the models investigated in this work may lead to new interesting signatures to look for in present and future observations.
The next logical step to further analyse these models would be to study the dynamics of cosmological perturbations (linear or non-linear) and to compare the results against observational data in order to constrain the theoretical parameters.
Such investigation is beyond the scope of the present paper and will be left for future works.






J.D. is thankful to IUCAA for warm hospitality and facility of
doing research works.
N.T. acknowledge support from an Enhanced Eurotalents Fellowship and the Labex P2IO.







Bennett:2003bz
  C. L. Bennett et al. [WMAP Collaboration],
  
  Astrophys. J. Suppl.  148 (2003) 1
  doi:10.1086/377253
  [astro-ph/0302207].


Spergel:2003cb
  D. N. Spergel et al. [WMAP Collaboration],
  
  Astrophys. J. Suppl.  148 (2003) 175
  doi:10.1086/377226
  [astro-ph/0302209].


Masi:2002hp
  S. Masi et al.,
  
  Prog. Part. Nucl. Phys.  48 (2002) 243
  doi:10.1016/S0146-6410(02)00131-X
  [astro-ph/0201137].

Ade:2013zuv
  P. A. R. Ade et al. [Planck Collaboration],
  
  Astron. Astrophys.  571 (2014) A16
  doi:10.1051/0004-6361/201321591
  [arXiv:1303.5076 [astro-ph.CO]].

Ade:2015xua
  P. A. R. Ade et al. [Planck Collaboration],
  
  Astron. Astrophys.  594 (2016) A13
  doi:10.1051/0004-6361/201525830
  [arXiv:1502.01589 [astro-ph.CO]].


Scranton:2003in
  R. Scranton et al. [SDSS Collaboration],
  
  astro-ph/0307335.


Riess:1998cb
  A. G. Riess et al. [Supernova Search Team],
  
  Astron. J.  116 (1998) 1009
  doi:10.1086/300499
  [astro-ph/9805201].


Tonry:2003zg
  J. L. Tonry et al. [Supernova Search Team],
  
  Astrophys. J.  594 (2003) 1
  doi:10.1086/376865
  [astro-ph/0305008].


Perlmutter:1998np
  S. Perlmutter et al. [Supernova Cosmology Project Collaboration],
  
  Astrophys. J.  517 (1999) 565
  doi:10.1086/307221
  [astro-ph/9812133].

Weinberg:1988cp
  S. Weinberg,
  
  Rev. Mod. Phys.  61 (1989) 1.
  doi:10.1103/RevModPhys.61.1

Martin:2012bt
  J. Martin,
  
  Comptes Rendus Physique 13 (2012) 566
  doi:10.1016/j.crhy.2012.04.008
  [arXiv:1205.3365 [astro-ph.CO]].


Caldwell:1997ii
  R. R. Caldwell, R. Dave and P. J. Steinhardt,
  
  Phys. Rev. Lett.  80 (1998) 1582
  doi:10.1103/PhysRevLett.80.1582
  [astro-ph/9708069].


Sahni:2002kh
  V. Sahni,
  
  Class. Quant. Grav.  19 (2002) 3435
  doi:10.1088/0264-9381/19/13/304
  [astro-ph/0202076].


Carroll:1998zi
  S. M. Carroll,
  
  Phys. Rev. Lett.  81 (1998) 3067
  doi:10.1103/PhysRevLett.81.3067
  [astro-ph/9806099].



Caldwell:1999ew
  R. R. Caldwell,
  
  Phys. Lett. B 545 (2002) 23
  doi:10.1016/S0370-2693(02)02589-3
  [astro-ph/9908168].



Singh:2003vx
  P. Singh, M. Sami and N. Dadhich,
  
  Phys. Rev. D 68 (2003) 023522
  doi:10.1103/PhysRevD.68.023522
  [hep-th/0305110].



Elizalde:2004mq
  E. Elizalde, S. Nojiri and S. D. Odintsov,
  
  Phys. Rev. D 70 (2004) 043539
  doi:10.1103/PhysRevD.70.043539
  [hep-th/0405034].




ArmendarizPicon:2000ah
  C. Armendariz-Picon, V. F. Mukhanov and P. J. Steinhardt,
  
  Phys. Rev. D 63 (2001) 103510
  doi:10.1103/PhysRevD.63.103510
  [astro-ph/0006373].


Novosyadlyj:2013nya
  B. Novosyadlyj, O. Sergijenko, R. Durrer and V. Pelykh,
  
  JCAP 1405, 030 (2014)
  doi:10.1088/1475-7516/2014/05/030
  [arXiv:1312.6579 [astro-ph.CO]].


Carroll:2003st
  S. M. Carroll, M. Hoffman and M. Trodden,
  
  Phys. Rev. D 68 (2003) 023509
  doi:10.1103/PhysRevD.68.023509
  [astro-ph/0301273].

  
Caldwell:2003vq
  R. R. Caldwell, M. Kamionkowski and N. N. Weinberg,
  
  Phys. Rev. Lett.  91 (2003) 071301
  doi:10.1103/PhysRevLett.91.071301
  [astro-ph/0302506].



cr
 C. Rovelli, Quantum Gravity Cambridge University Press, Cambridge (2004).



Han:2005km
  M. Han, W. Huang and Y. Ma,
  
  Int. J. Mod. Phys. D 16 (2007) 1397
  doi:10.1142/S0218271807010894
  [gr-qc/0509064].

  
Thiemann:2002nj
  T. Thiemann,
  
  Lect. Notes Phys.  631 (2003) 41
  doi:10.1007/978-3-540-45230-0-3
  [gr-qc/0210094].


Ashtekar:2003hd
  A. Ashtekar, M. Bojowald and J. Lewandowski,
  
  Adv. Theor. Math. Phys.  7 (2003) no.2,  233
  doi:10.4310/ATMP.2003.v7.n2.a2
  [gr-qc/0304074].

   


  





Ashtekar:2006rx
  A. Ashtekar, T. Pawlowski and P. Singh,
  
  Phys. Rev. Lett.  96 (2006) 141301
  doi:10.1103/PhysRevLett.96.141301
  [gr-qc/0602086].


Bojowald:2001ep
  M. Bojowald,
  
  Class. Quant. Grav.  18 (2001) L109
  doi:10.1088/0264-9381/18/18/101
  [gr-qc/0105113].


Date:2004zd
  G. Date and G. M. Hossain,
  
  Class. Quant. Grav.  21 (2004) 4941
  doi:10.1088/0264-9381/21/21/012
  [gr-qc/0407073].


Ashtekar:2006uz
  A. Ashtekar, T. Pawlowski and P. Singh,
  
  Phys. Rev. D 73 (2006) 124038
  doi:10.1103/PhysRevD.73.124038
  [gr-qc/0604013].



 
  
  
  
  


Banerjee:2005ga
  K. Banerjee and G. Date,
  
  Class. Quant. Grav.  22 (2005) 2017
  doi:10.1088/0264-9381/22/11/007
  [gr-qc/0501102].


Ashtekar:2006wn
  A. Ashtekar, T. Pawlowski and P. Singh,
  
  Phys. Rev. D 74 (2006) 084003
  doi:10.1103/PhysRevD.74.084003
  [gr-qc/0607039].


Singh:2006sg
  P. Singh,
  
  Phys. Rev. D 73 (2006) 063508
  doi:10.1103/PhysRevD.73.063508
  [gr-qc/0603043].


Copeland:2005xs
  E. J. Copeland, J. E. Lidsey and S. Mizuno,
  
  Phys. Rev. D 73 (2006) 043503
  doi:10.1103/PhysRevD.73.043503
  [gr-qc/0510022].




Sami:2006wj
  M. Sami, P. Singh and S. Tsujikawa,
  
  Phys. Rev. D 74 (2006) 043514
  doi:10.1103/PhysRevD.74.043514
  [gr-qc/0605113].


Naskar:2007dn
  T. Naskar and J. Ward,
  
  Phys. Rev. D 76 (2007) 063514
  doi:10.1103/PhysRevD.76.063514
  [arXiv:0704.3606 [gr-qc]].


Samart:2007xz
  D. Samart and B. Gumjudpai,
  
  Phys. Rev. D 76 (2007) 043514
  doi:10.1103/PhysRevD.76.043514
  [arXiv:0704.3414 [gr-qc]].



  


  


Amendola:1999er
  L. Amendola,
  
  Phys. Rev. D 62 (2000) 043511
  doi:10.1103/PhysRevD.62.043511
  [astro-ph/9908023].

Billyard:2000bh
  A. P. Billyard and A. A. Coley,
  
  Phys. Rev. D 61 (2000) 083503
  doi:10.1103/PhysRevD.61.083503
  [astro-ph/9908224].


Chimento:2003iea
  L. P. Chimento, A. S. Jakubi, D. Pavon and W. Zimdahl,
  
  Phys. Rev. D 67 (2003) 083513
  doi:10.1103/PhysRevD.67.083513
  [astro-ph/0303145].

Cai:2004dk
  R. G. Cai and A. Wang,
  
  JCAP 0503 (2005) 002
  doi:10.1088/1475-7516/2005/03/002
  [hep-th/0411025].

Huey:2004qv
  G. Huey and B. D. Wandelt,
  
  Phys. Rev. D 74 (2006) 023519
  doi:10.1103/PhysRevD.74.023519
  [astro-ph/0407196].

Boehmer:2008av
  C. G. Boehmer, G. Caldera-Cabral, R. Lazkoz and R. Maartens,
  
  Phys. Rev. D 78 (2008) 023505
  doi:10.1103/PhysRevD.78.023505
  [arXiv:0801.1565 [gr-qc]].

Boehmer:2009tk
  C. G. Boehmer, G. Caldera-Cabral, N. Chan, R. Lazkoz and R. Maartens,
  
  Phys. Rev. D 81 (2010) 083003
  doi:10.1103/PhysRevD.81.083003
  [arXiv:0911.3089 [gr-qc]].

Pourtsidou:2013nha
  A. Pourtsidou, C. Skordis and E. J. Copeland,
  
  Phys. Rev. D 88 (2013) no.8,  083505
  doi:10.1103/PhysRevD.88.083505
  [arXiv:1307.0458 [astro-ph.CO]].

Boehmer:2015kta
  C. G. Boehmer, N. Tamanini and M. Wright,
  
  Phys. Rev. D 91 (2015) no.12,  123002
  doi:10.1103/PhysRevD.91.123002
  [arXiv:1501.06540 [gr-qc]].

Boehmer:2015sha
  C. G. Boehmer, N. Tamanini and M. Wright,
  
  Phys. Rev. D 91 (2015) no.12,  123003
  doi:10.1103/PhysRevD.91.123003
  [arXiv:1502.04030 [gr-qc]].

Koivisto:2015qua
  T. S. Koivisto, E. N. Saridakis and N. Tamanini,
  
  JCAP 1509 (2015) 047
  doi:10.1088/1475-7516/2015/09/047
  [arXiv:1505.07556 [astro-ph.CO]].

 
Fu:2008gh
  X. Fu, H. W. Yu and P. Wu,
  
  Phys. Rev. D 78 (2008) 063001
  doi:10.1103/PhysRevD.78.063001
  [arXiv:0808.1382 [gr-qc]].

  
Gumjudpai:2007fc
  B. Gumjudpai,
  
  arXiv:0706.3467 [gr-qc].



Roy:2014yta
  N. Roy and N. Banerjee,
  
  Eur. Phys. J. Plus 129, 162 (2014)
  doi:10.1140/epjp/i2014-14162-7
  [arXiv:1402.6821 [gr-qc]].

  Paliathanasis:2015gga
  A. Paliathanasis, M. Tsamparlis, S. Basilakos and J. D. Barrow,
  
  Phys. Rev. D 91, no. 12, 123535 (2015)
  doi:10.1103/PhysRevD.91.123535
  [arXiv:1503.05750 [gr-qc]].

   Dutta:2015jaq
  J. Dutta and H. Zonunmawia,
  
  Eur. Phys. J. Plus 130, no. 11 (2015)
  doi:10.1140/epjp/i2015-15221-3
  [arXiv:1601.00283 [gr-qc]].

  Dutta:2016dnt
  J. Dutta, W. Khyllep and E. Syiemlieh,
  
  Eur. Phys. J. Plus 131, no. 2, 33 (2016)
  doi:10.1140/epjp/i2016-16033-7
  [arXiv:1602.03329 [gr-qc]].

Escobar:2013js
  D. Escobar, C. R. Fadragas, G. Leon and Y. Leyva,
  
  Astrophys. Space Sci.  349, 575 (2014)
  doi:10.1007/s10509-013-1650-8
  [arXiv:1301.2570 [gr-qc]].


  Dutta:2016bbs
  J. Dutta, W. Khyllep and N. Tamanini,
  
  Phys. Rev. D 93, no. 6, 063004 (2016)
  doi:10.1103/PhysRevD.93.063004
  [arXiv:1602.06113 [gr-qc]].

   Roy:2014hsa
  N. Roy and N. Banerjee,
  
  Annals Phys.  356, 452 (2015)
  doi:10.1016/j.aop.2015.03.013
  [arXiv:1411.1164 [gr-qc]].



  
Dutta:2017kch
  J. Dutta, W. Khyllep and N. Tamanini,
  
  Phys. Rev. D 95, no. 2, 023515 (2017)
  [arXiv:1701.00744 [gr-qc]].



Xiao:2011nh
  K. Xiao and J. Y. Zhu,
  
  Phys. Rev. D 83 (2011) 083501
  doi:10.1103/PhysRevD.83.083501
  [arXiv:1102.2695 [gr-qc]].


Coley A. A. Coley, Dynamical systems and cosmology. (Kluwer Academic Publishers, Dordrecht Boston London, 2003).

swig  S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. (Springer, New York Heidelberg Berlin, 1990).

lper L. Perko, Differential Equations and Dynamical Systems. (Springer Verlag, 1991).




  






Shahalam:2015sja
  M. Shahalam, S. D. Pathak, M. M. Verma, M. Y. Khlopov and R. Myrzakulov,
  
  Eur. Phys. J. C 75 (2015) no.8,  395
  doi:10.1140/epjc/s10052-015-3608-1
  [arXiv:1503.08712 [gr-qc]].



Lazkoz:2007mx
  R. Lazkoz, G. Leon and I. Quiros,
  
  Phys. Lett. B 649 (2007) 103
  doi:10.1016/j.physletb.2007.03.060
  [astro-ph/0701353].


Nunes:2000yc
  A. Nunes and J. P. Mimoso,
  
  Phys. Lett. B 488 (2000) 423
  doi:10.1016/S0370-2693(00)00919-9
  [gr-qc/0008003].



  






Wetterich:1994bg
  C. Wetterich,
  
  Astron. Astrophys.  301 (1995) 321
  [hep-th/9408025].

Holden:1999hm
  D. J. Holden and D. Wands,
  
  Phys. Rev. D 61 (2000) 043506
  doi:10.1103/PhysRevD.61.043506
  [gr-qc/9908026].





  



 


  




   


  





  
Quartin:2008px
  M. Quartin, M. O. Calvao, S. E. Joras, R. R. R. Reis and I. Waga,
  
  JCAP 0805, 007 (2008)
  doi:10.1088/1475-7516/2008/05/007
  [arXiv:0802.0546 [astro-ph]].

  


  


Tamanini:2014nvd
  N. Tamanini,
  Dynamical systems in dark energy models,
  PhD thesis, University College London (2014).


Amendola:1999qq
  L. Amendola,
  
  Phys. Rev. D 60 (1999) 043501
  doi:10.1103/PhysRevD.60.043501
  [astro-ph/9904120].




Zhou:2007xp
  S. Y. Zhou,
  
  Phys. Lett. B 660 (2008) 7
  doi:10.1016/j.physletb.2007.12.020
  [arXiv:0705.1577 [astro-ph]].


Fang:2008fw
  W. Fang, Y. Li, K. Zhang and H. Q. Lu,
  
  Class. Quant. Grav.  26 (2009) 155005
  doi:10.1088/0264-9381/26/15/155005
  [arXiv:0810.4193 [hep-th]].




  
 
 
 


Steinhardt:1999nw
  P. J. Steinhardt, L. M. Wang and I. Zlatev,
  
  Phys. Rev. D 59 (1999) 123504
  doi:10.1103/PhysRevD.59.123504
  [astro-ph/9812313].


Guo:2004xx
  Z. K. Guo, R. G. Cai and Y. Z. Zhang,
  
  JCAP 0505 (2005) 002
  doi:10.1088/1475-7516/2005/05/002
  [astro-ph/0412624].



Wei:2010fz
  H. Wei,
  
  Nucl. Phys. B 845, 381 (2011)
  doi:10.1016/j.nuclphysb.2010.12.010
  [arXiv:1008.4968 [gr-qc]].

  
Wei:2010cs
  H. Wei,
  
  Commun. Theor. Phys.  56, 972 (2011)
  doi:10.1088/0253-6102/56/5/29
  [arXiv:1010.1074 [gr-qc]].









