m}{\partial \omega}
  \right)_{\varphi_0,N}&=&
  C\omega^3-D(N-N_a)\omega-\varphi_0=0, \\
  \left(\frac{\partial^2 m}{\partial \omega^2}
  \right)_{\varphi_0,N}&=&
  3C\omega^2-D(N-N_a)=0,\end{aligned}$$ i.e., $$\varphi_0=\pm\left(\frac{4D^3}{27C}\right)^{1/2}(N-N_a)^{3/2},$$ near point $a$. The cusp catastrophe has been named after this shape. On the envelope $dcbab^{\prime}c^{\prime}d$ except for points $a$ and $d$, one of the two distinct stable equilibrium solutions, the locally but not globally stable one, disappears, and hence a first-order phase transition occurs if the system obeys a perfect delay convention. On envelope $ede^{\prime}$, the stable equilibrium solution disappears. The catastrophic feature on the envelopes except for point $a$ is the fold catastrophe described in the last subsection. Point $d$ is not a cusp point but the intersection of two folds.

We should comment that, for the near critical case $-4.9\alt\beta\alt -4.35$, the behavior of the stable equilibrium solutions around the point of the maximum baryon number is somewhat complicated, although the structure of the cusp catastrophe at cusp point $a$ is not changed. Figures 7 and 8 show the curves $(\tilde{\rho}_c,N)$ with $\varphi_0=0$, for $\beta=-4.5$ and $-4.85$, respectively. For $-4.8\alt\beta\alt-4.35$, the number of stable equilibrium solutions changes as 1, 2 (degenerate in Fig. 7), 3, 1, 0 as the control parameter $N$ is increased continuously, as is seen in Fig. 7. For $-4.9\alt\beta\alt-4.8$, the number 