denotes the fixed points located in the four dimensionsal phase space, which are earmarked by the coordinates $(x_c,y_c,z_c,\lambda_c)$. $\lambda_*$ means that $\lambda$ can be any value. $\lambda_a$ is the value that makes $f(\lambda)=0$. $\mathbf{M}^T$ means the inverted matrix of the eigenvalues of the fixed points. $f_1(\Lambda)=\left.\frac{df(\lambda)}{d\lambda}\right|_{\lambda=\Lambda}$ with $\Lambda=0, \lambda_a$. $A=\left[2f(\lambda_a)+\lambda_a\left(
  \left.\frac{df(\lambda)}{d\lambda}\right|_{\lambda_a}\right)\right]$. U stands for unstable, and S stands for stable.

The properties of each fixed points are determined by the eigenvalues of the Jacobi matrix $$\begin{aligned}
\mathcal{M}= \left. \begin{pmatrix}
\frac{\partial F_1}{\partial x}&\frac{\partial F_1}{\partial y}&\frac{\partial F_1}{\partial z}&\frac{\partial F_1}{\partial \lambda}\\
\frac{\partial F_2}{\partial x}&\frac{\partial F_2}{\partial y}&\frac{\partial F_2}{\partial z}&\frac{\partial F_2}{\partial \lambda}\\
\frac{\partial F_3}{\partial x}&\frac{\partial F_3}{\partial y}&\frac{\partial F_3}{\partial z}&\frac{\partial F_3}{\partial \lambda}\\
\frac{\partial F_4}{\partial x}&\frac{\partial F_4}{\partial
y}&\frac{\partial F_4}{\partial z}&\frac{\partial F_4}{\partial
\lambda}
\end{pmatrix}
\right|_{(x_c,y_c,z_c,\lambda_c)}.\end{aligned}$$ According to Lyapunov’s linearization method, the stability of a linearized system is determined by the eigenvalues of the matrix $\mathcal{M}$ (see Chapter 3 of [@Slotine-book]). If all of the eigenvalues are strictly in the left-half complex plane, then the autonomous system is stable. If at least one eigenvalue is strictly in the right-half complex plane, then the system is unstable. If all of the eigenvalues are in the left-half complex plane, but at least one of them is on the $i\omega$ axis, then 