I'm currently studying the Rossler attractor, which is the following system: \begin{align*} \frac{\mathrm{d} x}{\mathrm{d}t}&=-(y+z)\\ \frac{\mathrm{d}y}{\mathrm{d}t}&=x+\alpha y \\ \frac{\mathrm{d}z}{\mathrm{d}t}&=\beta + z(x-\gamma). \end{align*} This system has a Jacobian of \begin{equation*} J^*=\begin{pmatrix} 0 & -1 & -1 \\ 1 & \alpha & 0 \\ z & 0 & x-\gamma \end{pmatrix}. \end{equation*}
I have found the equilibrium points of the system, which are \begin{align*} (x^*,y^*,z^*)&=\Bigg(\frac{\gamma + \sqrt{\gamma^2 -4\alpha \beta}}{2}, \frac{-\gamma - \sqrt{\gamma^2 -4\alpha \beta}}{2\alpha} , \frac{\gamma + \sqrt{\gamma^2 -4\alpha \beta}}{2\alpha}\Bigg) \text{, and }\\ (x^*,y^*,z^*)&=\Bigg(\frac{\gamma - \sqrt{\gamma^2 -4\alpha \beta}}{2}, \frac{-\gamma + \sqrt{\gamma^2 -4\alpha \beta}}{2\alpha} , \frac{\gamma - \sqrt{\gamma^2 -4\alpha \beta}}{2\alpha}\Bigg). \end{align*}
I now need to analyse the stability of the system at these equilibrium points, but when I attempt to do so I get the characteristic equations that are tough to solve.
I also need to find the bifurcation points, and plot a bifurcation diagram. I think that bifurcation will occur when $\gamma^2 =4\alpha \beta$, but I'm not sure that I can prove it. And I have no idea how to plot a bifurcation diagram.
Can anyone offer me some help on how to proceed?
Thanks in advance.