The asymptotic value of the scalar field $\varphi_0$ and baryon number $N$ form a two-dimensional control space. The equilibrium solutions are uniquely parametrized by two parameters, i.e., the central value of the scalar field, $\varphi_c$, and the central total baryonic density, $\tilde{\rho}_c$.

We adopt the following stability criteria [@sorkin1982]:

\(i) The stability of $X(\varphi_c,\tilde{\rho}_c)$ can change typically only at a “turning point.” Here the “turning point” $(\varphi_c^0,\tilde{\rho}_c^0)$ is a point where there exists a nontrivial vector $(\delta\varphi_c,\delta\tilde{\rho}_c)$ such that $$\begin{aligned}
  \label{eq:tp1}
  \delta \varphi_0 &=& \left(\frac{\partial \varphi_0}{\partial
  \varphi_c}\right)_{\tilde{\rho}_c}
  \delta \varphi_c + \left(\frac{\partial \varphi_0}{\partial
  \tilde{\rho_c}}\right)_{\varphi_c}
  \delta \tilde{\rho}_c=0,\\
  \label{eq:tp2}
  \delta N &=& \left(\frac{\partial N}{\partial \varphi_c}
  \right)_{\tilde{\rho}_c}  
  \delta \varphi_c +  \left(\frac{\partial N}{\partial \tilde{\rho_c}}
  \right)_{\varphi_c}
  \delta \tilde{\rho}_c=0.\end{aligned}$$ From Eqs. (\[eq:tp1\]) and (\[eq:tp2\]), $$\frac{\partial(\varphi_0,N)}{\partial(\varphi_c,\tilde{\rho}_c)}=0$$ at the turning point. Therefore the change of stability can be detected as envelopes of a family of curves $\tilde{\rho}_c=\mbox{const}$ in the $(\varphi_0, N)$ plane. Of course, this is also true for a family of curves $\varphi_c=\mbox{const}$.

\(ii) In order to specify an unstable branch at the turning point, we draw the sequence of equilibrium solutions in the $(\varphi_0,\omega)$ plane, maintaining $N$ constant. Then, as one proceeds along the curve in a counterclockwise direction, a branch beyond the turning point is unstable. 