If I have a system that takes the form $\dot x =by$ $\leftrightarrow$ $\dot y = y^2 + x$, I get a Jacobian matrix at the origina of the form
$$ \begin{matrix} 0 & b & \\ 1 & 0 & \\ \end{matrix} $$
We see that $\tau$ = 0 and $\Delta$=-$b$ This means that that eigenvalues $\lambda$ = $\pm$$\sqrt b$. As the value $b$ passes through 0 the behavior around the origin goes from a center to a saddle.
What kind of bifurcation is this called?