Given differential equation
$\frac{dy}{dt} = f(y, \alpha),$
we solve $f(y, \alpha) = 0$ to find equilibrium solutions.
By definition, $\overline y$ is a bifurcation point and $\overline \alpha$ is a bifurcation value if $f(\overline y, \overline \alpha) = 0 \text{ and } f_y(\overline y, \overline \alpha) = 0$.
Then my professor said that bifurcations can only happen at such points (but might not). I have difficulty trying to come up with an example where these conditions are satisfied, but bifurcation doesn't occur. Can you give me an example of such $f(y, \alpha)$? Providing some conceptual explanation would help as well.
Take $f(y,\alpha)=\alpha y^2$, and the values $\overline y=0$ and $\overline \alpha=1$. There is no bifurcation in this case.