of the fixed point if at least one of the eigenvalues of $\mathcal{M}$ is on the $i\omega$ axis with the rest being in the left-half complex plane. The fixed point $P_3$ is exactly such a point. In this appendix, we use the center manifold theorem (see Chapter 8 of [@Khalil-non] , or [@DynamicalReduction]) to get the condition for stability of $P_3$. The coordinates of $P_3$ are $(0,1,0,0)$ and the eigenvalues are $(-3,-3\gamma,0,0)$. First, we transfer $P_3$ to $P'_3$ $(x_c=0,\bar{y}_c=y-1=0,z_c=0, \lambda_c=0)$. In this case, Eqs. (\[x’\])-(\[z’\]) and (\[l’\]) become $$\begin{aligned}
\frac{dx}{dN}&=&-3\,x-\frac12\,\sqrt {6}\lambda\, \left( \bar{y}+1
\right) ^{2}+x \left[ 3\,{x}^
{2}+\frac32\gamma\, \left((1+z)\right.\right.\nonumber\\
&&\left.\left.-{x}^{2}-\left( \bar{y}+1 \right) ^{2} \right)  \right]\left( 1-2z\right)\label{X'},\\
\frac{d \bar{y}}{dN}&=&\frac12\,\sqrt {6}\lambda\,x \left( \bar{y}+1
\right) + \left( \bar{y}+1 \right)
\left[3\,{x}^{2}+\frac32\gamma\, \left((1+z)\right.\right.\nonumber\\
&&\left.\left.-{x}^{2}-\left( \bar{y}+1 \right) ^{2} \right)
\right]  \left( 1-2z
\right)\label{Y'}, \\
\frac{dz}{dN}&=&-3\gamma z-3z\left(1-z\right)\left[2x^2-\gamma
x^2-\gamma
(\bar{y}+1)^2\right],\label{Z'}\\
\frac{d\lambda}{dN}&=& \sqrt
{6}{\lambda}^{2}\left(f(0)+f_1(0)\lambda\right) x \label{L'},\end{aligned}$$ where we have considered that the related variables $(x,\bar{y},z,\lambda)$ are small around point $(x_c,\bar{y}_c,z_c,\lambda_c)=\left(0,0, 0,0\right)$. Therefore the following Taylor series $$\begin{aligned}
&&\frac{1}{1-z}=1+{z}+\cdots,\nonumber\\
&&f(\lambda)=f(0)+f_1(0)\lambda+\cdots,\nonumber\end{aligned}$$ can be used, where $f_1(0)=\frac{df(\lambda)}{d\lambda}\left.\right|_{\lambda=0}$.

We can get the Jacobi matrix $\mathcal{M'}$ of the dynamical system Eqs. (\[X’\])-(\[L’\]) as $$\begin{aligned}
\mathcal{M'}= \begin{pmatrix}
-3& 0& 0&-\frac{\sqrt{6}} 2\\
0&-3\gamma& \frac32\gamma & 0\\
0&0& 0&0\\
0&0&0&0
\end{pmatrix}.\end{aligned}$$ It is easy to get the 