restrict our attention to cusp point $a(0,N_a)$. ($m_b N_a \simeq 1.24 M_{\odot}$ for $\beta=-6$.) The potential function $m$ is written around cusp point $a$ as $$\label{eq:cusp}
  m=\frac{C}{4}\omega^4-\frac{D(N-N_a)}{2}\omega^2
  -\varphi_0\omega+m_a,$$ where $C$ and $D$ are some positive constants. The reason why the coefficient of $\varphi_0 \omega$ is determined is that Eq. (\[eq:dermderphi0\]) holds. This form of expansion agrees with the usual Landau ansatz for a second-order phase transition, which has been used to explain spontaneous scalarization by Damour and Esposito-Farèse [@de1996]. The scalar charge is given by the roots of the following cubic equation: $$\label{eq:cube}
  \left(\frac{\partial m}{\partial \omega}\right)
  _{\varphi_0,N}=
  C\omega^3-D(N-N_a)\omega-\varphi_0=0.$$ From Eq. (\[eq:cube\]), near cusp point $a$, the scalar charge is given by $$\omega=0,$$ for $N<N_a$ with $\varphi_0=0$. This is a stable branch. For $N>N_a$ with $\varphi_0=0$, $$\label{eq:critical}
  \omega=\cases{
    \pm\left(\frac{D}{C}\right)^{1/2}(N-N_a)^{1/2} 
    & for the stable branches, \cr
    0 & for the unstable branch. \cr }$$ At point $a$ the stable equilibrium solution changes [*continuously*]{}, but its derivative with respect to $N$ is [*discontinuous*]{}. If Eq. (\[eq:cusp\]) involved a cubic term in $\omega$, the number of roots of the equation $\partial m/\partial \omega=0$ did change at point $a$. But this catastrophe was classified as the fold type and therefore not the case for point 