positive constants. Then the terms in the square brackets cancel out on the envelope, and are negative in region (A) and positive in region (B). For $\varphi_{0p}<0$, replace $\omega-\omega_p$ and $\varphi_{0p}-\varphi_0$ with $\omega_{p}-\omega$ and $\varphi_{0}-\varphi_{0p}$, respectively. For simplicity, we describe the behavior of the scalar charge for the case $\varphi_{0p}>0$. From Eq. (\[eq:fold\]), near point $p$, the scalar charge is given by the roots of the following quadratic equation: $$\left(\frac{\partial m}{\partial \omega}\right)_{\varphi_0,N}
  =A (\omega-\omega_{p})^2 +
  [B(\varphi_{0p}-\varphi_0)+B^{\prime}(N-N_{p})]=0.$$ The scalar charge is then given near point $p$ in region (A) by $$\omega = \omega_{p}\pm A^{-1/2}
  [B(\varphi_{0}-\varphi_{0p})+B^{\prime}(N_{p}-N)]^{1/2},$$ where the upper sign denotes the stable branch and the lower denotes the unstable one. If a quadratic term in $(\omega-\omega_p)$ was involved in Eq. (\[eq:fold\]), the number of the roots of the equation $\partial m/\partial\omega=0$ did not change at point $p$. That is why Eq. (\[eq:fold\]) does not contain the quadratic term. The “scalar susceptibility” $\chi_{\varphi}$ is given near point $p$ by $$\chi_{\varphi}\equiv\left(\frac{\partial \omega}{\partial \varphi_0}
    \right)_{N}
  =\pm\frac{1}{2}A^{-1/2}B
  [B(\varphi_{0}-\varphi_{0p})+B^{\prime}(N_{p}-N)]^{-1/2}.$$ The bifurcation set $B_m\subset {\bf R}^2$, which is the envelope seen in Fig. 1, is given by $$\begin{aligned}
  \left(\frac{\partial m}{\partial \omega}\right)_{\varphi_0,N}
  &=&A (\omega-\omega_{p})^2 +
  [B(\varphi_{0p}-\varphi_0)+B^{\prime}(N-N_{p})]=0, \\
  \left(\frac{\partial^2 m}{\partial \omega^2}\right)
  _{\varphi_0,N}
  &=& 2A(\omega-\omega_p)=0,\end{aligned}$$ i.e., $$\varphi_0=\frac{B^\prime}{B}(N-N_p)+\varphi_{0p},$$ near point $p$. 