their counterparts in general relativity. For $\varphi_0=0$, the unstable solution agrees with the stable solution in general relativity. One of the two stable equilibrium solutions disappears on the envelope $dcbab^{\prime}c^{\prime}d$. In region (C), no stable equilibrium solution exists. Point $a$ is a bifurcation point for $\varphi_0=0$. This is seen in Fig. 5 which displays the curves $(\tilde{\rho_c},m)$ and $(\tilde{\rho_c},N)$ for $\varphi_0=0$, where the solid lines denote stable branches and the dotted lines denote unstable branches. In this figure, two stable branches are degenerate because, for $\varphi_0=0$, two stable equilibrium solutions are identical except for the sign of the scalar field. The equilibrium solution of the bifurcated stable branches is more compact for smaller mass but less compact for larger mass than the general relativistic sequence. Figure 6 shows the equilibrium space $M_m$ near point $a$. This type of the catastrophe at point $a$ is classified as the cusp catastrophe in which the control space is two-dimensional. The map $\chi_m$ is a cusp catastrophe map. This catastrophe is elementary and structurally stable, which suggests that this structure is stable against adding small higher order terms to the exponent of the coupling function (\[eq:quadratic\]). Point $a$ is called a cusp point.

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