basically common to the case $\beta\agt-4.35$. Figure 1 shows $\tilde{\rho}_c=\mbox{const}$ curves in the $(\varphi_0,N)$ plane, where the equilibrium solutions have been determined in the manner described in Sec. II. At a point on an envelope of the family of the curves seen in Fig. 1, the stability of the sequence of equilibrium solutions changes. Figure 2 shows the curves $(\varphi_0,\omega)$ with $N$ constant. In Fig. 2, the solid lines denote stable branches while the dotted lines denote unstable branches, where stability criteria (i) and (ii) are applied. Therefore, in region (A) in Fig. 1, only one stable equilibrium solution exists. For $\varphi_0=0$, this stable solution is identical to that of general relativity. In region (B) in Fig. 1, however, no stable solution exists. This is classified as the fold catastrophe in which the control space is two-dimensional. This catastrophe is elementary and structurally stable. Hence it is expected that this catastrophe structure is not changed by adding small higher-order terms to the exponent of the coupling function (\[eq:quadratic\]). The potential function $m$ is written locally around point $p(\varphi_{0p},N_{p})$ on the envelope (see Fig. 1) as, for $\varphi_{0p}>0$, $$\label{eq:fold}
  m = \frac{A}{3}(\omega-\omega_{p})^3
  +[B(\varphi_{0p}-\varphi_0)+B^{\prime}(N-N_{p})]
  (\omega-\omega_{p})+m_p,$$ where $A$, $B$, and $B^{\prime}$ are some 