Difference between revisions of "Phase space structure of models with interaction"

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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
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\begin{equation}
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\label{IDE_s6_1}
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\begin{array}{l}
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\dot x=f(x,y,t),\\
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\dot y=g(x,y,t).
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\end{array}
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\end{equation}
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We will be interested in the so-called autonomous systems, for which the functions $f$ and $g$ do not contain explicit time-dependent terms.
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A point $(x_c,y_c)$ is said to be a fixed (a.k.a. critical) point of the autonomous system if
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\[f(x_c,y_c)=g(x_c,y_c)=0.\]
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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 $t\to\infty$.
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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
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\[x=x_c+\delta x,\quad y=y_c+\delta y.\]
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Substituting it into (\ref{IDE_s6_1}) leads to the first-order differential equations:
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\[\frac{d}{dN}\left(\begin{array}{c}\delta x\\ \delta y\end{array}\right) = \hat M \left(\begin{array}{c}\delta x\\ \delta y\end{array}\right).\]
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Taking into account the specifics of the problem that we are solving, we made the change \[\frac{d}{dt}\to\frac{d}{dN},\]
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where $N=\ln a$. The  matrix $\hat M$ is given by
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\[\hat M =
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\left(
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\begin{array}{lr}
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\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\
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{} & {}\\
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\frac{\partial g}{\partial x} & \frac{\partial g}{\partial y}
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\end{array}
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\right)
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\]
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The general solution for the linear perturbations reads
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\[\delta x=C_1e^{\lambda_1 N} + C_2e^{\lambda_2 N},\]
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\[\delta y=C_3e^{\lambda_1 N} + C_4e^{\lambda_2 N},\]
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The stability around the fixed points depends on the nature of the eigenvalues.
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Let us treat the interacting dark components as a dynamical system described by the equations
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\[\rho'_{de}+3(1+w_{de})\rho_{de}=-Q\]
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\[\rho'_{dm}+3(1+w_{dm})\rho_{dm}=Q\]
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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:
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\[Q_1=3\gamma_{dm}\rho_{dm},\quad Q_1=3\gamma_{de}\rho_{de},\quad Q_1=3\gamma_{tot}\rho_{tot}\]
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  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;"></p>
 +
  </div>
 +
</div></div>
 +
 +
 +
 +
<div id=""></div>
 +
<div style="border: 1px solid #AAA; padding:5px;">
 +
=== Problem 1 ===
 +
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
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Revision as of 11:22, 8 November 2013




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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