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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.

There are other sorts of bifurcations that may occur, but these are the simplest.

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