$a$ because of the shape of the bifurcation set seen in Fig. 3. That is why Eq. (\[eq:cusp\]) does not contain the cubic term. We also note that, for the case of two-dimensional control space, the structurally stable catastrophe is classified as either the fold or cusp type by Thom’s theorem. Therefore, at point $a$, a second-order phase transition occurs. If we fix $N$ to $N_a$, $$\omega= C^{-1/3}\varphi_0^{1/3}.$$ This is stable. From Eqs. (\[eq:cube\])-(\[eq:critical\]), with $\varphi_0=0$ near point $a$, it is derived that the scalar susceptibility $\chi_{\varphi}$ is given by $$\chi_{\varphi}=\cases{
    D^{-1}(N_a-N)^{-1} & for $N<N_a$, \cr
    \frac{1}{2}D^{-1}(N-N_a)^{-1} & for $N>N_a$. \cr}$$ Near point $a$ in region (A) in Fig. 3, only one real root of the cubic equation (\[eq:cube\]) corresponds to the stable equilibrium solution, while, in region (B), the smallest and largest roots of three real roots correspond to the stable equilibrium solutions and the intermediate root corresponds to the unstable one. If $\varphi_0>0$, the largest root corresponds to the globally stable one. If $\varphi_0<0$, the smallest root corresponds to the globally stable one. If $\varphi_0=0$, the two stable equilibrium solutions have identical ADM energies. The bifurcation set $B_m$, which is the envelope $b^{\prime}ab$, is given by $$\begin{aligned}
  \left(\frac{\partial 