density distribution. By a linearization of we rewrite the ODE in dependence of some parameter $t$ to $$\frac{\mathrm{d}}{\mathrm{d}t}  
    \begin{pmatrix}
        \xi  \\ u
    \end{pmatrix}=
    \begin{pmatrix}
        u^2-4u\xi+3\xi^2-1 & 2u\xi-2\xi^2\\
        -\kappa                   & \alpha
    \end{pmatrix}_{\mathcal{D}_{1,2}}
    \begin{pmatrix}
        \xi \\ u
    \end{pmatrix}
    =\mathbf{J}_{\mathcal{D}_{1,2}}^{\left(\alpha,\kappa\right)}
    \begin{pmatrix}
        \xi \\ u
    \end{pmatrix}\label{eq:spanalysis}$$ where $\mathbf{J}_{\mathcal{D}_{1,2}}^{\left(\alpha,\kappa\right)}$ is the Jacobian matrix at the positions $\mathcal{D}_{1,2}$. An analysis of the eigenvalues and eigenvectors [@Logan2008] of $\mathbf{J}_{\mathcal{D}_{1,2}}^{\left(\alpha,\kappa\right)}$ at the related positions enables us to characterize the critical points and hence the behavior of the solutions next to them. The eigenvectors are denoted with $\boldsymbol{x}_{1,2}$ for $\mathcal{D}_1$ and $\boldsymbol{z}_{1,2}$ for $\mathcal{D}_2$. The eigenvector $\boldsymbol{x}_{1}=\left(0,1\right)^{\mathrm{T}}$ is similar in all cases and describes the unstable diverging behavior at $\mathcal{D}_1$ yielding a saddle point. A solution of is given by as inner BC, so we can characterize the behavior of the velocity at the origin.

 

        $\kappa$   $\alpha$       $\delta\xi_0$       $\delta u_{\mathrm{u}}$   $\delta u_{\mathrm{d}}$   $\delta S_{\mathrm{u}}$   $\delta S_{\mathrm{d}}$
  ---- ---------- ---------- ----------------------- ------------------------- ------------------------- ------------------------- -------------------------
          1/3         0       $3.29 \cdot 10^ {-5}$    $2.01 \cdot 10^ {-6}$     $4.28 \cdot 10^ {-5}$     $3.59 \cdot 10^ {-6}$     $5.72 \cdot 10^ {-5}$
   IP     1/2                 $2.63 \cdot 10^ {-5}$    $3.44 \cdot 10^ {-6}$     $3.15 \cdot 10^ {-5}$     $4.23 \cdot 10^ {-6}$     $4.37 \cdot 10^ {-5}$
          2/3                 