Saddle point in pitchfork bifurcation?

Question

$$ \dot{x} = \sigma(y-x) \\ \dot{y} = r \ x - y - xz \\ \dot{z} = -\beta z + xy $$ For a Lorentzian system, the node at (0,0,0) is stable for value of parameter $r<1$. I found that it turns in to a saddle node when $r$ is more than 1, i.e. i found that for $r>1$, some eigenvalues are more than 0 and some are less than zero. For example, one eigenvalue is always equal to $-\beta$, the other two are both negative for $r<1$ and then become of opposite signs for $r>1$. Many different sources state that biforcation at $r=1$ is a pitchfork biforcation. I though pitchfork biforcation has to have a stable node or an unstable node at the center. Is my result wrong or am I misunderstanding the pitchfork biforcation?

Answer 1

As an experiment consider the system $$\dot{x} = x(x^2-\lambda),\\ \dot{y} = \alpha y, \\ \dot{z} = \beta z.$$ When $\lambda = 0$ the system undergoes pitchfork bifurcation. Dynamics of each variable is independent from other variables, $Ox$ axis is a center manifold for equilibrium at $O$ when $\lambda = 0$. But you see that by choice of $\alpha$ and $\beta$ you can change what kind of equilibria you get after bifurcation. For a bifurcation it's important what happens on center manifold: this is what distinguishes one bifurcation from another.