\right)_{\varphi_0,N}=\left(\frac{\partial^2 m}
        {\partial \omega^2}
        \right)_{\varphi_0,N}=0\right\},$$ and the bifurcation set $B_m \subset {\bf R}^2$ satisfies $$B_m = \left\{(\varphi_0,N)\Biggm|
    \left( \frac{\partial m}{\partial \omega}
        \right)_{\varphi_0,N}=\left(\frac{\partial^2 m}
        {\partial \omega^2}
        \right)_{\varphi_0,N}=0\right\}.$$ The envelopes of the family of the curves $\tilde{\rho}_c=\mbox{const} $ in the $(\varphi_0,N)$ plane form a bifurcation set $B_m$ of the catastrophe map $\chi_m$ because the Jacobian of $\chi_m $ vanishes at points on the envelopes. Criterion (i) says that a sequence of the equilibrium solutions can change its stability only at the points of the bifurcation set.

From criteria (i) and (ii), we examine the stability of the equilibrium solutions of neutron stars in scalar-tensor theory. From the turning point method alone, however, we cannot say that an equilibrium solution is [*stable*]{}. Therefore the stability of an equilibrium solution must be investigated by perturbation study [*once for all*]{}. For this purpose, we examine the case in which $\alpha(\varphi_0)=0$ and $m/r_s$ is sufficiently small ($\tilde{\rho}_c$ is sufficiently small). In this case, there is an equilibrium solution that is identical to that in general relativity. For this solution, the second-order variation of $m$ by regular, adiabatic, time-symmetric, and spherically symmetric perturbations with $\varphi_0$ and $N$ constant is $$\delta^2 m = \mbox{general relativistic terms} +
  \frac{1}{2}
  