\boldmath Late time attractors of some varying Chaplygin gas cosmological models
A. Uthor
and A. Nother Author
Martiros_ms.tex
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\title{\boldmath Late time attractors of some varying Chaplygin gas cosmological models}


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%% \author{A. Uthor}
%% \author{and A. Nother Author}
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\author[a]{M. Khurshudyan,\note{Corresponding author.}}
\author[b]{R. Myrzakulov}

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\affiliation[a]{Armenian State Pedagogical University, 375010 Yerevan, Republic of Armenia}
\affiliation[b]{Eurasian International Center for Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan}


% e-mail addresses: one for each author, in the same order as the authors
\emailAdd{khurshudyan@yandex.ru}
\emailAdd{rmyrzakulov@gmail.com}




\abstract{Varying Chaplygin gas is one of the dark fluids actively studied in modern cosmology. It does belong to the group of the fluids which has an explicitly given EoS. From the other hand phase space does contain all possible states of the system. Therefore, phase space analysis of the cosmological models does allow to understand qualitative behavior and estimate required characteristics of the models. Phase space analysis is a convenient approach to study a cosmological model, because we do not need to solve a system of differential equations for a given initial conditions, instead, we need to deal with appropriate algebraic equations. The goal of this paper is to find late time attractors for the cosmological models, where a varying Chaplygin gas is one of the components of the large sale universe. We will pay our attention to some non linear interacting models.}



\begin{document}
\maketitle
\flushbottom

\section{Introduction}\label{sec:INT}
In modern cosmology there are several key open problems and various approaches to solve them~\cite{Liddle}~-~\cite{Overduin}. In this work we have a goal to consider various cosmological models and obtain appropriate conditions to have late time accelerated expansion of the universe~\cite{Riess}~-~\cite{Verde}. Phase space portrait describing the dynamics of the model is one of the mechanisms allowing us to have a qualitative understanding of the situation. This does allow us to have all states of the model and we do not need to solve a system of differential equations, instead, we need to work with appropriate algebraic equations~\cite{Xi-ming}~-~\cite{Rong}. In this section we will provide some aspects related to the cosmological models which we will study in this work. Dark fluids are actively used to explain an accelerated expansion of the large scale universe. One of the dark fluids it is Chaplygin gas~\cite{Kamenshchik0}~-~\cite{Havard}
\begin{equation}\label{eq:CH}
P_{c} = A\rho_{c} - \frac{B}{\rho_{c}^{\alpha}},
\end{equation}
where $A$, $B$ and $\alpha$ are positive constants and $\rho_{c}$ it is the energy density of the gas. In Physical Literature there are various modifications of this fluid~\cite{Naji}~-~\cite{Fu}. One of the modifications it is varying Chaplygin gas, where it is assumed that either $A$ or $B$ are not constants. It is possible that both of them are not constants. In this paper we will consider two models of varying Chaplygin gas given~(see for instance \cite{Naji} and \cite{Fu}) as
\begin{equation}\label{eq:M1}
P_{c} = A\rho_{c} - \frac{BH^{-n}}{\rho_{c}^{\alpha}}, 
\end{equation}
and
\begin{equation}\label{eq:M2}
P_{c} = A\rho_{c} - \frac{Ba^{-n}}{\rho_{c}^{\alpha}},
\end{equation}
where $H$ it is the Hubble parameter, $a$ it is the scale factor, while $n$ it is a constant. A phenomenological assumption about the content of the universe it is a usual procedure in modern cosmology. It is well known that to describe the dynamics of the large scale flat FRW universe~(supported from observational data) we need field equations of General Relativity
\begin{equation}\label{eq:Fridmmanvlambda}
H^{2}=\frac{\dot{a}^{2}}{a^{2}}=\frac{8\pi G\rho}{3},
\end{equation}
\begin{equation}\label{eq:fridman}
\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3P),
\end{equation}
where $\rho$ it is the energy density of the effective fluid, while $P$ it is the pressure. Furthermore, we will work with the models, where the cosmological constant $\Lambda=0$, the gravitational constant $G$ and $c$ are constants with $c=8\pi G=1$. It is easy to see from Eq~(\ref{eq:Fridmmanvlambda}) and Eq.~(\ref{eq:fridman}), that we should make an assumption, for example, about the structure of the effective fluid giving appropriate expanding behavior of the universe. From observational data it is known, that two kind of energy are involved into the dynamics of the large scale universe. This does mean that the energy density and pressure of the effective fluid  could be approximated as 
\begin{equation}\label{eq:rhoeff}
\rho = \rho_{DM} + \rho_{DE},
\end{equation}
and 
\begin{equation}\label{eq:Peff}
P  = P_{DM} + P_{DE},
\end{equation}
where $\rho_{DE}$ and $P_{DE}$ are the energy density and pressure of dark energy, while $\rho_{DM}$ and $P_{DM}$ are the energy density and pressure of dark matter. There are various assumptions about the dark energy and dark matter, but all of them are still assumptions. There is another interesting fact, that should be mentioned about Chaplygin gas Eq.~(\ref{eq:CH}). It is a joint model of dark energy and dark matter. Some critics on this issue is also does exist in Physical Literature~(see for instance~\cite{Havard}). There is a huge number of viscous dark fluids~\cite{Arturo}~(and references therein). Dark fluids could be either linear or non linear. Dark fluids could be even solutions of different algebraic or differential equations~\cite{Nojiri}. Explanation of the accelerated expansion of the large scale universe could involve additional ideas despite to dark energy. In Physical Literature, there are various ideas on this topic. From the other hand it is well known, that modern cosmology has several other problems, and we believe that the solutions some of them are hidden into the other ones. An interesting situation has been noticed recently. Considered various cosmological models have proved that an interaction between dark energy and dark matter is a phenomenologically good idea. Practically there is nothing against to this idea. An interaction between dark components of the universe, can improve some unpleasant aspects of the models. This is a right motivation to continue a research on interacting cosmological models. Particularly, when there is an increase of the number of the interacting viable models. There are different parameterizations of the interaction term. Some linear and non linear interactions are used intensively. Recently, sign changeable interactions started to be used very intensively~\cite{Yang}~-~\cite{Arevalo}. Interaction in this context should be understood as either dark energy transfers into dark matter or dark matter transfers into dark energy. If we consider interacting models, we do mean that the following should take a place
\begin{equation}\label{eq:firstfluid}
\dot{\rho}_{DM}+3H(\rho_{DM}+P_{DM})=Q,
\end{equation}
and
\begin{equation}\label{eq:secondfluid}
\dot{\rho}_{DE}+3H(\rho_{DE}+P_{DE})=-Q.
\end{equation}
To construct cosmological models of this paper, we assume that dark matter is a pressureless fluid. The other aspect allowing us to study our models it is an assumption of the form of the interaction term $Q$. Particularly, we are interested in non linear interactions, which will be presented in section~\ref{sec:IntModels}. \\\\
The paper is organised as follows: In section~\ref{sec:IntModels} we will present basics on the phase space analysis to find late time attractor solutions of the field equations of General Relativity. In section~\ref{sec:PSA} phase space analysis is performed, late time attractors are found and classified according to their cosmological applicability. Finally, discussion on obtained results are summarised in section~\ref{sec:Discussion}.

\section{Interacting models and autonomous system}\label{sec:IntModels}
In Physical Literature there is a huge number of works devoted to phase space analysis of various cosmological models. In order to start phase space study of a model, an appropriate autonomous system should be found first. The critical points are solutions of the autonomous system. They are stable if appropriate Jacobian matrix has negative trace and positive determinant. This is in case of linear stability. Following Ref.~\cite{Huang} for our models we set
\begin{equation}\label{eq:x}
x = \frac{\rho_{c}}{3H^{2}},
\end{equation}
\begin{equation}\label{eq:y}
y = \frac{P_{c}}{3H^{2}},
\end{equation}
\begin{equation}\label{eq:z}
z = \frac{\rho_{m}}{3H^{2}},
\end{equation}
and 
\begin{equation}\label{eq:N}
N=\ln{a},
\end{equation}
where $a$ it is the scale factor. It is not hard to see, that for interacting models the autonomous system does read as
\begin{equation}\label{eq:xPrime}
x^{\prime} = 3x (1+y) - \frac{Q+3H\rho_{c} (1+\omega_{c})}{3H^{3}},
\end{equation}
and
\begin{equation}\label{eq:yPrime}
y^{\prime} = 3y(1+y) + \frac{\dot{P}_{c}}{3H^{3}},
\end{equation}
where $\prime$ it is the derivative with respect to $N$, dot it is the derivative with respect to cosmic time, $Q$ it is interaction term. Explicit forms of Eq.~(\ref{eq:xPrime}) and Eq.~(\ref{eq:yPrime}) are obtained when the form of $Q$ is known. To have physically reasonable solutions we should have the following constraints $0 \leq x \leq 1$ and $0 \leq z \leq 1$. Therefore, we need to find the range of the model parameters such, that to have a physically reasonable stable critical points i.e.  $0 \leq x_{c} \leq 1$ and $0 \leq z_{c} \leq 1$. From the other hand a stable critical point it is an attractor, which we are looking for.  At the same time we should remember that $x$ and $z$ according to Eq.~(\ref{eq:Fridmmanvlambda}), Eq.~(\ref{eq:x}) and Eq.~(\ref{eq:z}) should satisfy to the following constraint
\begin{equation}\label{eq:FConst}
x+z =1.
\end{equation}
In terms of $x$ and $y$ EoS parameter of the Chaplygin gas does read as
\begin{equation}\label{eq:EoSC}
\omega_{c} = \frac{y}{x}.
\end{equation}
EoS parameter of the effective fluid does read as
\begin{equation}\label{eq:EoSeff}
\omega_{eff} = \frac{P_{c}}{\rho_{c} + \rho_{m}} = y,
\end{equation}
because dark matter is considered as a pressureless fluid, while it is not hard to show that the deceleration parameter $q$ does read as
\begin{equation}\label{eq:q}
q = -1 - \frac{\dot{H}}{H^{2}} = \frac{1}{2}(1 + 3 y).
\end{equation}
In this work we will consider various cosmological models involving a non linear interaction into the darkness of the universe. Particularly, we are interested to consider an interaction $Q$ of the following form
\begin{equation}\label{eq:Int}
Q = 3H b \left( \rho + \frac{\hat{\rho}}{\rho}\right ),
\end{equation}
where $\rho$ could be either the energy density of the effective fluid, or the energy density one of the components of the effective fluid. $\hat{\rho}$ it is product of the energy densities of the components of the effective fluid i.e. three possibilities could be $\rho_{c}^{2}$, $\rho_{m}^{2}$ and $\rho_{c}\rho_{m}$. From the next section we will start our analysis.

\section{Phase space analysis}\label{sec:PSA}
In this section we will find late time attractors of the field equations for various interacting varying Chaplygin gas cosmological scenarios. For that we need to determine interaction term $Q$ to have Eq.~(\ref{eq:xPrime}) and Eq.~(\ref{eq:yPrime}). In this work we will pay attention to some fixed sign non linear interactions. Non linear interactions having similar structure as here, were considered in Ref.~\cite{Martiros}. In Ref.~\cite{Martiros} various cosmological models with Chaplygin gas Eq.~(\ref{eq:CH}) have been studied, including models with some non linear sign changeable interactions. To simplify study, we will combine an experience known from Physical Literature to impose some constraints on the model parameters. Namely, for varying Chaplygin gas we will impose that
\begin{equation}\label{eq:alpha}
0 < \alpha \leq 1,
\end{equation}
\begin{equation}\label{eq:n}
0 \leq n \leq 5,
\end{equation}
\begin{equation}\label{eq:A}
A\geq 0~~~~and~~~~B>0.
\end{equation}
For parameter $b$ describing interaction $Q$ we take
\begin{equation}\label{eq:b}
0\leq b < 1,
\end{equation}
which is the best range supported from observational data. $b=0$ does correspond to non interacting models. From the other hand, constraints on the deceleration parameter $q$
\begin{equation}
-1 \leq q < 0,
\end{equation}
and on the EoS parameter of varying Chaplygin gas
\begin{equation}\label{eq:omegaconst}
-2 \leq \omega_{c} < 0,
\end{equation}
will give us an option to find late time attractors for the large scale universe with an accelerated expansion. It is not excluded that the phantom divide about $z\approx 0.2$ redshift took place in our universe, therefore constraint Eq.~(\ref{eq:omegaconst}) is considered by us~\cite{Nesseris}~-~\cite{Mohseni}. Conditional attractors are interesting possibilities, because depends on the constraints on the model parameters, different late time states of the universe will be obtained. The goal of this section is to find and discus stability of the critical points for appropriate cosmological scenarios for $6$ different cosmological models. 

\subsection{Varying Chaplygin gas $P_{c} = A\rho_{c} - \frac{BH^{-n}}{\rho_{c}^{\alpha}}$}
We did various assumptions allowing us to study cosmological models for the large scale universe. We start our analysis of two different cosmological models involving different forms of non linear interactions Eq.~(\ref{eq:Int}) for each of them. We will start from the models where varying Chaplygin gas Eq.~(\ref{eq:M1}) interacts with dark matter.
 
\subsubsection{Interaction $Q = 3 H b \left( \rho_{c} + \frac{\rho_{m}^{2}}{ \rho_{c} + \rho_{m}} \right )$}\label{ssec:Q1}
The first cosmological model that we would like to study is a model where interaction between varying Chaplygin gas Eq.~(\ref{eq:M1}) and dark matter is given as
\begin{equation}\label{eq:Int1}
Q = 3 H b \left( \rho_{c} + \frac{\rho_{m}^{2}}{ \rho_{c} + \rho_{m}} \right ).
\end{equation}
Such consideration allowed us to find all real critical points analytically. Obtained solutions are presented in Table~\ref{tab:Table1}. In this case $3$ different physically reasonable critical points are exist and only two of them are stable. However only one of them does correspond to the state of the large scale universe. Stable critical points are $E.1.1$ and $E.1.3$. Critical point $E.1.1$ is late time scaling attractor, because 
\begin{equation}
r = \frac{\Omega_{m}}{\Omega_{c}} = \frac{1-b - \sqrt{1+ (2-3b) b}}{2(b-1)},
\end{equation} 
tends to a constant. For this universe the decelerated parameter is $q=-1$ and $\omega_{eff} = -1$, while the EoS parameter of the varying Chaplygin gas Eq.~(\ref{eq:M1}) is
\begin{equation}
\omega_{c} = - \frac {2b} { b - 1 +  \sqrt {1+ (2-3b) b} }
\end{equation}
It is not hard to see that in this case varying Chaplygin gas Eq.~(\ref{eq:M1}) is a phantom dark energy. Discussed behavior is obtained when the parameters satisfy to Eq.-s~(\ref{eq:alpha})~-~(\ref{eq:A}) and $0 < b < 2/3$. Critical point $E.1.3$ to be stable when $0 < \alpha \leq 1$, $0 \leq n \leq 5$, $b=0$ and $A > 1+ \alpha + n/2$ should be. In this case a matter will dominate in our universe, since $x=0$. In such universe accelerated expansion is not possible. Therefore, we left only $E.1.1$ as a physically reasonable solution for the large scale universe. 

\begin{table}[tbp]
\centering
\begin{tabular}{|lcc|c|}
\hline
 $S. P.$ & $x$ & $y$ \\
\hline
$E.1.1$  & $\frac{b-1+ \sqrt{1+(2-3b)b}}{2b}$ & $-1$ \\

$E.1.2$  &  $\frac{A-b + \sqrt{(A-b)(A+3b)} }{2 (A-b) }$ & $ \frac{A(A-b + \sqrt{(A-b)(A+3b)}) }{2(A-b)}$ \\

$E.1.3$  &  $\frac{A-b - \sqrt{(A-b)(A+3b)} }{2 (A-b) }$ & $ \frac{A(A-b - \sqrt{(A-b)(A+3b)}) }{2(A-b)}$ \\
\hline
\end{tabular}
\caption{\label{tab:Table1} Critical points corresponding to interacting varying Chaplygin gas Eq.~(\ref{eq:M1}) for interaction $Q$ determined via Eq.~(\ref{eq:Int1}).}
\end{table}


\subsubsection{Interaction  $Q = 3 H b \left( \rho_{c} + \frac{\rho_{m} \rho_{c}}{ \rho_{c} + \rho_{m}} \right )$}\label{sseq:Q2}
In the second model the interaction between varying Chaplygin gas Eq.~(\ref{eq:M1}) and dark matter is taken to be
\begin{equation}\label{eq:Int2}
Q = 3 H b \left( \rho_{c} + \frac{\rho_{m} \rho_{c}}{ \rho_{c} + \rho_{m}} \right ).
\end{equation}
In this case physically reasonable two critical points do exist, which are presented in Table~\ref{tab:Table2}. $E.2.1$ is a physically reasonable solution, moreover it is late time scaling attractor when $0 \leq n \leq 5$, $0 < \alpha \leq 1$, $A \geq 0$ and $0 < b < 2/3$. This solution represents a state of the universe where varying Chaplygin gas Eq.~(\ref{eq:M1}) is a phantom dark energy with
\begin{equation}
\omega_{c} = -\frac{1+2b + \sqrt{1+4b^{2}} }{2}
\end{equation} 
This model is free from the cosmological coincidence problem due to
\begin{equation}
r = \frac{\Omega_{m}}{\Omega_{c}} = \frac{2b -1 + \sqrt{1+ 4b^{2}}}{2}.
\end{equation}
Critical point $E.2.2$ is physically reasonable solution when $b=0$, this does mean that $x =1$ and $y=A$. Therefore, this is the state of the universe where an accelerated expansion is not possible, since $q>0$. This is the state of the universe where varying Chaplygin gas Eq.~(\ref{eq:M1}) completely dominates in the dynamics of the universe as a usual fluid. As was expected $E.2.2$ is not a stable critical point. Fig.~(\ref{fig:Fig1}) does represent phase space portraits for $E.1.2$ and $E.2.1$ critical points. For a symmetry in plots, we considered $-1 \leq x \leq 1$ interval, however, we should remember that physically reasonable range for $x$ it is $(0,1]$.

\begin{table}[tbp]
\centering
\begin{tabular}{|lcc|c|}
\hline
 $S. P.$ & $x$ & $y$ \\
\hline
$E.2.1$  &  $\frac{1+2b - \sqrt{1+ 4b^{2}} }{2b}$ & $-1$ \\
 $E.2.2$  & $\frac{A+2b}{A+b}$ & $\frac{A(a+2b)}{A+b}$ \\
\hline
\end{tabular}
\caption{\label{tab:Table2} Critical points corresponding to interacting varying Chaplygin gas Eq.~(\ref{eq:M1}) for interaction $Q$ determined via Eq.~(\ref{eq:Int2}).}
\end{table}

\begin{figure}[tbp]
\centering % \begin{center}/\end{center} takes some additional vertical space
\includegraphics[width=.45\textwidth]{fig1a.jpg}
\hfill
\includegraphics[width=.45\textwidth]{fig1b.jpg}
% "\includegraphics" is very powerful; the graphicx package is already loaded
\caption{\label{fig:Fig1} Phase space portraits for the interacting varying Chaplygin gas Eq.~(\ref{eq:M1}) models. Left plot does represent the model where interaction $Q$ is determined via Eq.~(\ref{eq:Int1}). Right plot does represent the model where interaction $Q$ is given via Eq.~(\ref{eq:Int2}).}
\end{figure}

\subsubsection{Interaction $Q = 3 H b \left( \rho_{c} +\rho_{m} + \frac{\rho_{m}^{2}}{ \rho_{c} + \rho_{m}} \right )$} \label{ssec:Q3}
The other possibility that we are going to consider in this subsection, it is a cosmological model where interaction term $Q$ does read as
\begin{equation}\label{eq:Int3}
Q = 3 H b \left( \rho_{c} +\rho_{m} + \frac{\rho_{m}^{2}}{ \rho_{c} + \rho_{m}} \right ).
\end{equation}
In this case $3$ critical points are physically acceptable~(among $4$ critical points). They are collected into Table.~\ref{tab:Table3}. Critical point $E.3.1$ is a late time scaling solution since
\begin{equation}
r = \frac{\Omega_{m}}{\Omega_{c}} = \frac{\sqrt{1- 4 b^{2}}}{2(1-2b)} - \frac{1}{2},
\end{equation}
and does represent the state of the universe, where varying Chaplygin gas Eq.~(\ref{eq:M1}) is a phantom dark energy with
\begin{equation}
\omega_{c} = -\frac{2b}{2b - 1 + \sqrt{1-4b^{2}}}.
\end{equation}
This is possible late time attractor when $0 \leq n \leq 5$, $0 < b \leq 2/5$, $0 < \alpha \leq 1$ and $A \geq 0$. Critical point $E.3.2$ is physically reasonable when $b = 0$, and it is unstable, therefore, we will not discuss cosmological consequences related to this solution. From the other hand we found that $E.3.3$ is a stable critical point, when $b=0$ and $A > 1+ \alpha + n/2$ for $0 \leq n \leq 5$ and $0 < \alpha \leq 1$. It is not hard to see that in this case our universe will be in a matter dominating state. Accelerated expansion in this case is not possible. 

\begin{table}[tbp]
\centering
\begin{tabular}{|lcc|c|}
\hline
 $S. P.$ & $x$ & $y$ \\
\hline
 $E.3.1$  & $\frac{2b - 1 + \sqrt{1-4b^{2}}}{2b}$ & $-1$ \\
$E.3.2$  &  $\frac{A-2b + \sqrt{A^{2} + 4Ab - 4b^{2}} }{2(A-b)}$ & $ \frac{A(A-2b + \sqrt{A^{2} + 4Ab - 4b^{2}}) }{2(A-b)}$ \\
$E.3.3$  &  $\frac{A-2b - \sqrt{A^{2} + 4Ab - 4b^{2}} }{2(A-b)}$ & $\frac{A(A-2b - \sqrt{A^{2} + 4Ab - 4b^{2}} )}{2(A-b)}$ \\
\hline
\end{tabular}
\caption{\label{tab:Table3} Critical points corresponding to interacting varying Chaplygin gas Eq.~(\ref{eq:M1}) for Q determined via Eq.~(\ref{eq:Int3}).}
\end{table}

\subsection{Varying Chaplygin gas $P_{c} = A\rho_{c} - \frac{Ba^{-n}}{\rho_{c}^{\alpha}}$}
In subsections~\ref{ssec:Q1}~-~\ref{ssec:Q3} we considered cosmological models where varying Chaplygin gas was given via Eq.~(\ref{eq:M1}). In next three subsections we will consider cosmological models, where another model of varying Chaplygin gas is associated to a part of energy of our universe. Namely, we will consider models, where varying Chaplygin gas is given via Eq.~(\ref{eq:M2})

\subsubsection{Interaction $Q = 3 H b \left( \rho_{c} + \frac{\rho_{m}^{2}}{ \rho_{c} + \rho_{m}} \right )$} \label{ssec:Q4}
As in subsection~\ref{ssec:Q1}, first of all, we will consider interaction given via Eq.~({\ref{eq:Int1}}). In this case $4$ critical points are obtained~(Table~\ref{tab:Table4}). To simplify future analysis we can take some particular values for the parameters of the model. We will consider a particular example with $0 \leq b \leq 1/10$, when $n=1$. In case of $n=1$, when $A\geq b$, only critical point $E.4.4$ is a late time scaling attractor with
\begin{equation}
r = \frac{\Omega_{m}}{\Omega_{c}} = \frac{1}{2} \left ( -1 + \frac{2 + 3 \alpha + 9 b (1+\alpha)}{ \sqrt{ (2+3\alpha - 3b (1+\alpha)) (2+3\alpha + 9 b(1+\alpha))} } \right ).
\end{equation}
Obtained constraints on the parameters provide the deceleration parameter $q$
\begin{equation}
q = -1 + \frac{1}{2(1+\alpha)},
\end{equation}
to be $0 < q \leq -1$. In this case varying Chaplygin gas Eq.~(\ref{eq:M2}) is a quintessence dark energy
\begin{equation}
\omega_{c} = - \frac{2b (2+3\alpha)}{3b (1+\alpha) - 2 - 3\alpha + \sqrt{(2 - 3b - 3\alpha (b-1)) (2 + 3\alpha + 9b (1+\alpha))} }.
\end{equation}
Our study of the general case showed that $E.4.4$~(only one stable critical point for this model) it is a conditional late time scaling attractor. This does mean, that for some constraints on the model parameters we can obtain the state of the universe, where varying Chaplygin gas Eq.~(\ref{eq:M2}) will be a quintessence dark energy. From the other hand, $E.4.4$ could describe the state of the universe where varying Chaplygin gas Eq.~(\ref{eq:M2}) will be a phantom dark energy. This is very interesting aspect missing in previously considered models.

\begin{table}[tbp]
\centering
\begin{tabular}{|lcc|c|}
\hline
 $S. P.$ & $x$ & $y$ \\
\hline
 $E.4.1$  & $\frac{A-b + \sqrt{(A-b)(A+3b)}}{2(A-b)}$ & $\frac{A(A-b + \sqrt{(A-b)(A+3b)})}{2(A-b)}$ \\
 $E.4.2$  &  $\frac{A-b - \sqrt{(A-b)(A+3b)}}{2(A-b)}$ &  $\frac{A(A-b - \sqrt{(A-b)(A+3b)})}{2(A-b)}$ \\
 $E.4.3$  &  $\frac{n+3(b-1)(1+\alpha)-\hat{r}}{6b(1+\alpha)}$ & $-1+\frac{n}{3(1+\alpha)}$ \\
 $E.4.4$  &  $\frac{n+3(b-1)(1+\alpha)+\hat{r}}{6b(1+\alpha)}$ & $-1+\frac{n}{3(1+\alpha)}$ \\
\hline
\end{tabular}
\caption{\label{tab:Table4} Critical points corresponding to interacting varying Chaplygin gas Eq.~(\ref{eq:M2}) for interaction $Q$ determined via Eq.~(\ref{eq:Int1}). $\hat{r}=\sqrt{(n- 3(1+\alpha)(1+3b)) (n - 3(1+\alpha)(1-b))}$.}
\end{table}

\subsubsection{Interaction  $Q = 3 H b \left( \rho_{c} + \frac{\rho_{m} \rho_{c}}{ \rho_{c} + \rho_{m}} \right )$}\label{sseq:Q5}
A cosmological model where varying Chaplygin gas Eq.~(\ref{eq:M2}) interacts with dark matter via interaction $Q$ Eq.~(\ref{eq:Int2}) has three critical points and one of them is a late time scaling attractor. Critical points for this model are collected in Table~\ref{tab:Table5}. Late time scaling solution $E.5.2$ for some values of parameters could represent a state of the universe, where varying Chaplygin gas Eq.~(\ref{eq:M2}) is a quintessence dark energy~(Table~\ref{tab:Table6}), while for some other values of the parameters we will have a state of the universe, where varying Chaplygin gas Eq.~(\ref{eq:M2}) is a phantom dark energy~(Table~\ref{tab:Table7}). To finish this subsection we provide expressions for the deceleration parameter $q$, $\omega_{eff}$, $\omega_{c}$ and $r$
\begin{equation}
q = -1 + \frac{n}{2(1+\alpha)},
\end{equation}
\begin{equation}
\omega_{eff} = -1 + \frac{n}{3(1+\alpha)},
\end{equation}
\begin{equation}
\omega_{c} = - \frac{3(1+2b)(1+\alpha) - n + \hat{r}}{6(1+\alpha)},
\end{equation}
and 
\begin{equation}
r = \frac{3(1-2b)(1+\alpha) - n - \hat{r}}{2(n - 3(1+\alpha))},
\end{equation}
where  $\hat{r} = \sqrt{3b b^{2}(1+\alpha)^{2} + (n-3(1+\alpha))^{2}}$, for $E.5.2$ attractor.

\begin{table}[tbp]
\centering
\begin{tabular}{|lcc|c|}
\hline
 $S. P.$ & $x$ & $y$ \\
\hline
$E.5.1$  & $\frac{A+2b}{A+b}$ & $\frac{A(A+2b)}{A+b}$ \\
$E.5.2$  &  $\frac{3 (1+2b) (1+\alpha)- n - \hat{r}}{6b(1+\alpha)}$ &  $- 1 + \frac{n}{3(1+\alpha)}$ \\
$E.5.3$  &  $\frac{3 (1+2b) (1+\alpha)- n + \hat{r}}{6b(1+\alpha)}$ &  $- 1 + \frac{n}{3(1+\alpha)}$ \\
\hline
\end{tabular}
\caption{\label{tab:Table5} Critical points corresponding to interacting varying Chaplygin gas Eq.~(\ref{eq:M2}) for interaction $Q$ determined via Eq.~(\ref{eq:Int2}). $\hat{r} = \sqrt{3b b^{2}(1+\alpha)^{2} + (n-3(1+\alpha))^{2}}$.}
\end{table}

\begin{table}[tbp]
\centering
\begin{tabular}{|lcc|c|}
\hline
 $n$ & $\alpha$ & $b$\\
 \hline
 $0<n\leq \frac{3}{2}$  & $0\leq \alpha \leq 1$ & $0<b\leq \frac{n}{3 \alpha +n+3}$ \\

$\frac{3}{2}<n<2$  &  $0\leq \alpha \leq \frac{1}{3} (2 n-3)$ &  $ 0<b\leq \frac{6 \alpha -2 n+6}{9 \alpha +9}$ \\

$\frac{3}{2}<n<2$  &  $\frac{1}{3} (2 n-3)<\alpha \leq 1$ &  $ 0<b\leq \frac{n}{3 \alpha +n+3}$ \\

$2\leq n\leq 3$  &  $\frac{n-2}{2}<\alpha \leq \frac{1}{3} (2 n-3)$ &  $0<b\leq \frac{6 \alpha -2 n+6}{9 \alpha +9}$ \\

$2\leq n\leq 3$   &  $\frac{1}{3} (2 n-3)<\alpha \leq 1$ &  $0<b\leq \frac{n}{3 \alpha +n+3}$ \\

$3<n<4$  &  $\frac{n-2}{2}<\alpha \leq 1$ &  $0<b\leq \frac{6 \alpha -2 n+6}{9 \alpha +9}$  \\ 
\hline 
\end{tabular}
\caption{\label{tab:Table6} Constraints on the model parameters for the interacting varying Chaplygin gas Eq.~(\ref{eq:M2}) with interaction term $Q$ described. Eq.~(\ref{eq:Int2}). $E.5.2$ does describe the state of the universe where varying Chaplygin gas Eq.~(\ref{eq:M2}) is a quintessence dark energy.}
\end{table}
 

\begin{table}[tbp]
\centering
\begin{tabular}{|lcc|c|}
\hline
$n$ & $\alpha$ & $b$\\
      \hline
 $n=0$  & $0\leq \alpha \leq 1$ & $0<b<\frac{6 \alpha +6}{9 \alpha +9}$ \\

$0<n<\frac{3}{2}$  &  $0\leq \alpha \leq 1$ &  $\frac{n}{3 \alpha +n+3}<b\leq \frac{6 \alpha -2 n+6}{9 \alpha +9}$ \\

$\frac{3}{2}\leq n<3$  &  $ \frac{1}{3} (2 n-3)<\alpha \leq 1$ &  $\frac{n}{3 \alpha +n+3}<b\leq \frac{6 \alpha -2 n+6}{9 \alpha +9}$ \\
\hline
\end{tabular}
\caption{\label{tab:Table7} Constraints on the model parameters for the interacting varying Chaplygin gas Eq.~(\ref{eq:M2}) with interaction term $Q$ described. Eq.~(\ref{eq:Int2}). $E.5.2$ does describe the state of the universe where varying Chaplygin gas Eq.~(\ref{eq:M2}) is a phantom dark energy.}
\end{table}
 
\subsubsection{Interaction $Q = 3 H b \left( \rho_{c} +\rho_{m} + \frac{\rho_{m}^{2}}{ \rho_{c} + \rho_{m}} \right )$}\label{sseq:Q6}
 
 \begin{table}[tbp]
\centering
\begin{tabular}{|lcc|c|}
\hline
 $S. P.$ & $x$ & $y$ \\
\hline
$E.6.1$  & $\frac{A-2b + \sqrt{A^{2} + 4Ab - 4b^{2}}}{2(A-b)}$ & $\frac{A(A-2b + \sqrt{A^{2} + 4Ab - 4b^{2}})}{2(A-b)}$ \\
$E.6.2$  &  $\frac{A-2b - \sqrt{A^{2} + 4Ab - 4b^{2}}}{2(A-b)}$ & $\frac{A(A-2b + \sqrt{A^{2} + 4Ab - 4b^{2}})}{2(A-b)}$ \\
$E.6.3$  &  $\frac{n + 3 (- 1+2b) (1+\alpha)  - \hat{r}}{6b(1+\alpha)}$ &  $- 1 + \frac{n}{3(1+\alpha)}$ \\
$E.6.4$  &  $\frac{n + 3 (- 1+2b) (1+\alpha) + \hat{r}}{6b(1+\alpha)}$ &  $- 1 + \frac{n}{3(1+\alpha)}$ \\
\hline
\end{tabular}
\caption{\label{tab:Table8} Critical points corresponding to interacting varying Chaplygin gas Eq.~(\ref{eq:M2}) for interaction $Q$ determined via Eq.~(\ref{eq:Int3}). $\hat{r} = \sqrt{-3b b^{2}(1+\alpha)^{2} + (n-3(1+\alpha))^{2}}$.}
\end{table}

The last model that we will consider in this work it is a model where the interaction between varying Chaplygin gas Eq.~(\ref{eq:M2}) and dark matter is given via Eq.~(\ref{eq:Int3}). For this model $4$ real critical points do exist, which are presented in Table~\ref{tab:Table8}. However, only $E.6.4$ is a late time scaling attractor. Moreover, this attractor it is a conditional attractor. It is easy to obtain the following expressions 
\begin{equation}
q=-1 + \frac{n}{2(1+\alpha)},
\end{equation}
\begin{equation}
\omega_{eff} = -1 + \frac{n}{3(1+\alpha)},
\end{equation}
\begin{equation}
\omega_{c} = \frac{2b(n-3(1+\alpha))}{n + 3(-1 + 2b) (1+\alpha) + \hat{r}}
\end{equation}
and
\begin{equation}
r = -\frac{1}{2} - \frac{\hat{r}}{2 (n - 3(-1 + 2b) (1+\alpha))}, 
\end{equation}
where $\hat{r} = \sqrt{-3b b^{2}(1+\alpha)^{2} + (n-3(1+\alpha))^{2}}$, for $E.6.4$ solution. In Table~\ref{tab:Table9} constraints on the model parameters are collected, when attractor $E.6.4$ does describe a universe where varying Chaplygin gas Eq.~(\ref{eq:M2}) is a phantom dark energy. It is also possible, that for some values of the model parameters varying Chaplygin gas Eq.~(\ref{eq:M2}) is a quintessence dark energy. It is mathematically very easy problem and we omitted appropriate discussion on this issue to save a place.

\begin{table}[tbp]
\centering
\begin{tabular}{|lcc|c|}
\hline
$n$ & $\alpha$ & $b$ \\
      \hline

 $n=0$  & $0\leq \alpha \leq 1$ & $0<b<\frac{18 \alpha ^2+36 \alpha +18}{45 \alpha ^2+90 \alpha +45}$ \\

$0<n<\frac{3}{2}$  &  $0\leq \alpha \leq 1$ &  $\frac{-n^2+3 \alpha  n+3 n}{9 \alpha ^2+18 \alpha +n^2+9}<b\leq \frac{6 \alpha -2 n+6}{15 \alpha +15}$\\

$\frac{3}{2}\leq n<3$  &  $\frac{1}{3} (2 n-3)<\alpha \leq 1$ &  $ \frac{-n^2+3 \alpha  n+3 n}{9 \alpha ^2+18 \alpha +n^2+9}<b\leq \frac{6 \alpha -2 n+6}{15 \alpha +15}$ \\
\hline
\end{tabular}
\caption{\label{tab:Table9} Constraints on the model parameters for the interacting varying Chaplygin gas Eq.~(\ref{eq:M2}) with interaction term $Q$ described. Eq.~(\ref{eq:Int3}). $E.6.4$ does describe the state of the universe where varying Chaplygin gas Eq.~(\ref{eq:M2}) is a phantom dark energy.}
\end{table}
 
\section{Discussion}\label{sec:Discussion}
In this work we have considered 6 different cosmological scenarios. Three different structures for the interaction term $Q$ are used. Chaplyging gas~(and various modifications of it) is still very active subject for study in cosmology, because it is a joint model of dark energy and dark matter. One of the modifications of it, it is varying Chaplygin gas where a phenomenological assumption about $B$~(or $A$) parameter is used. We considered two models where two parametrizations of $B$ are taken into account. Assuming the structure of the darkness of the universe and the form of the interaction term $Q$ we performed phase space analysis. For all cases late time scaling attractors are found and appropriate constraints on the model parameters are obtained. For the first type cosmological models late time scaling attractors describe the state of the universe where varying Chaplygin gas it is a phantom dark energy. In the second models attractors are conditional late time scaling attractors. This does mean that for some values of the model parameters varying Chaplygin gas will be a phantom dark energy, while for some cases it will be a quintessence dark energy. An appropriate discrimination of discussed issues are presented in this article to give a right motivation for other works. We hope that very soon we will report about our new results on this topic.


\acknowledgments
M. Khurshudyan is grateful to Prof E. Kokanyan for warm hospitality, comprehensive support and stimulating discussion.

% The bibliography will probably be heavily edited during typesetting.
% We'll parse it and, using the arxiv number or the journal data, will
% query inspire, trying to verify the data (this will probalby spot
% eventual typos) and retrive the document DOI and eventual errata.
% We however suggest to always provide author, title and journal data:
% in short all the informations that clearly identify a document.

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% Please avoid comments such as "For a review'', "For some examples",
% "and references therein" or move them in the text. In general,
% please leave only references in the bibliography and move all
% accessory text in footnotes.

% Also, please have only one work for each \bibitem.


\end{thebibliography}
\end{document}

1509.07357.extracted_bib
{Liddle Andrew Liddle, An Introduction to Modern Cosmology, Willey (2015).
1509.07357.extracted_bib
{Jaewon Jaewon Yoo, Yuki Watanabe, Theoretical Models of Dark Energy, Int. J. Mod. Phys. D {21 (2012) 1230002.
1509.07357.extracted_bib
{Timothy Timothy Clifton et al., Modified Gravity and Cosmology, Physics Reports {513 (2012) 1.
1509.07357.extracted_bib
{Bousso Raphael Bousso, The Cosmological Constant Problem, Dark Energy, and the Landscape of String Theory, arXiv:1203.0307.
1509.07357.extracted_bib
{Villata M. Villata, On the nature of dark energy: the lattice Universe, Astrophys. Space Sci. {345 (2013) 1.
1509.07357.extracted_bib
{Velten H.E.S. Velten et al, Aspects of the cosmological "coincidence problem", Eur. Phys. J. C {74 (2014) 3160.
1509.07357.extracted_bib
{Bamba Kazuharu Bamba et al., Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests, Astrophys. and Space Sci. {342 (2012) 155.
1509.07357.extracted_bib
{Overduin James Overduin, Wolfgang Priester, Problems of modern cosmology: how dominant is the vacuum?, Naturwissenschaften {88 (2001) 229.
1509.07357.extracted_bib
{Riess A. G. Riess et al., Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, Astron. J. {116 (1998) 1009.
1509.07357.extracted_bib
{Perlmutter S. Perlmutter et al., Measurements of $\Omega$ and $\Lambda$ from 42 High-Redshift Supernovae, Astrophys. J. {517 (1999) 565.
1509.07357.extracted_bib
{Spergel D. N. Spergel et al., First-Year Wilkinson Microwave Anisotropy Probe (WMAP)* Observations: Determination of Cosmological Parameters, Astrophys. J. Suppl. {148 (2003) 175.
1509.07357.extracted_bib
{Tegmark M/ Tegmark et al., Cosmological parameters from SDSS and WMAP, Phys. Rev. D {69 (2004) 103501.
1509.07357.extracted_bib
{Abazajian K. Abazajian et al., The Third Data Release of the Sloan Digital Sky Survey, Astron. J. {129 (2005) 1755.
1509.07357.extracted_bib
{Abazajian1 K. Abazajian et al. , The Second Data Release of the Sloan Digital Sky Survey, Astron. J. {128 (2004) 502.
1509.07357.extracted_bib
{Hawkins E. Hawkins et al., The 2dF Galaxy Redshift Survey: Correlation functions, peculiar velocities and the matter density of the universe, Mon. Not. Roy. Astron. Soc {346 (2003) 78.
1509.07357.extracted_bib
{Verde L. Verde et al. The 2dF Galaxy Redshift Survey: The Bias of galaxies and the density of the Universe, Mon. Not. Roy. Astron. Soc. {335 (2002), 432.
1509.07357.extracted_bib
{Xi-ming Xi-ming Chen et al., Phase-space analysis of interacting phantom cosmology, JCAP {04 (2009) 001.
1509.07357.extracted_bib
{Mubasher Mubasher Jamil et al., Phase space analysis of interacting dark energy in $f(T)$ cosmology, Cent. Eur. J. Phys. {10 (2012) 1065.
1509.07357.extracted_bib
{Leona0 Genly Leona and Emmanuel N. Saridakisc, Dynamical analysis of generalized Galileon cosmology, JCAP {03 (2013) 025.
1509.07357.extracted_bib
{Hamid Hamid Shabani, Mehrdad Farhoudi, $f(R,T)$ Cosmological Models in Phase Space, Phys. Rev. D {88 (2013) 044048.
1509.07357.extracted_bib
{Dagoberto Dagoberto Escobar et al., Phase space analysis of quintessence fields trapped in a Randall-Sundrum braneworld: a refined study, Class. Quantum Grav. {29 175005 (2012).
1509.07357.extracted_bib
{Laur Laur Jarv et al., Phase Space Analysis of Quintessence Cosmologies with a Double Exponential Potential, JCAP {08 (2004) 016.
1509.07357.extracted_bib
{ChenXu Chen Xu et al, Phase-space analysis of teleparallel dark energy, JCAP {07 (2012) 005.
1509.07357.extracted_bib
{Genly Genly Leon et al., Quintom phase-space: Beyond the exponential potential, Phys. Lett. B {732 (2014) 285.
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