Dynamical system with three equations

How can I study bifurcations in a dynamical system consisting of $\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}$? The system depends on two parameters, and I want to find the bifurcation curves and, then, to plot these curves in a $(a,b)$-plane, the plane of the parameters $a$ and $b$.

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Here's how to find the places where the fixed points bifurcate. First find the equilibria of the system by finding for each $(a,b)$ all values of $(x,y,z)$ with $F(x,y,z) = 0$, where the ODE is $\frac {d}{dt} (x,y,z) = F(x,y,z)$. Then find the eigenvalues of the linearisation $DF$ at these equilibria. For each equilibrium these eigenvalues may depend on $a,b$ -- you can find the bifurcation curves by finding the values of $(a,b)$ at which the eigenvalues cross the imaginary axis, which corresponds to the fixed point changing stability from stable to unstable, or vice versa.