Phase space structure of models with interaction
The evolution of a Universe filled with interacting components can be effectively analyzed in terms of dynamical systems theory. Let us consider the following coupled differential equations for two variables
\begin{equation}
\label{IDE_s6_1}
\begin{array}{l}
\dot x=f(x,y,t),\\
\dot y=g(x,y,t).
\end{array}
\end{equation}
We will be interested in the so-called autonomous systems, for which the functions $f$ and $g$ do not contain explicit time-dependent terms.
A point $(x_c,y_c)$ is said to be a fixed (a.k.a. critical) point of the autonomous system if
\[f(x_c,y_c)=g(x_c,y_c)=0.\]
A critical point $(x_c,y_c)$ is called an attractor when it satisfies the condition \(\left(x(t),y(t)\right)\to(x_c,y_c)\) for $t\to\infty$.
Let's look at the behavior of the dynamical system (\ref{IDE_s6_1}) near the critical point. For this purpose, let us consider small perturbations around the critical point
\[x=x_c+\delta x,\quad y=y_c+\delta y.\]
Substituting it into (\ref{IDE_s6_1}) leads to the first-order differential equations:
\[\frac{d}{dN}\left(\begin{array}{c}\delta x\\ \delta y\end{array}\right) = \hat M \left(\begin{array}{c}\delta x\\ \delta y\end{array}\right).\]
Taking into account the specifics of the problem that we are solving, we made the change \[\frac{d}{dt}\to\frac{d}{dN},\]
where $N=\ln a$. The matrix $\hat M$ is given by
\[\hat M =
\left(
\begin{array}{lr}
\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\
{} & {}\\
\frac{\partial g}{\partial x} & \frac{\partial g}{\partial y}
\end{array}
\right)
\]
The general solution for the linear perturbations reads
\[\delta x=C_1e^{\lambda_1 N} + C_2e^{\lambda_2 N},\]
\[\delta y=C_3e^{\lambda_1 N} + C_4e^{\lambda_2 N},\]
The stability around the fixed points depends on the nature of the eigenvalues.
Let us treat the interacting dark components as a dynamical system described by the equations \[\rho'_{de}+3(1+w_{de})\rho_{de}=-Q\] \[\rho'_{dm}+3(1+w_{dm})\rho_{dm}=Q\] Here, the prime denotes the derivative with respect to $N=\ln a$. Note that although the interaction can significantly change the cosmological evolution, the system is still autonomous. We consider the following specific interaction forms, which were already analyzed above: \[Q_1=3\gamma_{dm}\rho_{dm},\quad Q_1=3\gamma_{de}\rho_{de},\quad Q_1=3\gamma_{tot}\rho_{tot}\]
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