The most simple kind of bifurcation studied in texts is generally a fold, characteristic of equations such as:
$$\dot{y}=y^2+\lambda$$
There are two equilibria if $\lambda<0$, and no equilibria if $\lambda>0$. But what about a linear equation such as:
$$\dot{y}=\lambda y $$
In this case, there is a unstable equilibrium point if $\lambda>0$ and a stable equilibrium point if $\lambda<0$. While equilibria have not disappeared like in the fold above, it has "changed its sign". What's the name for this kind of bifurcation?
EDIT: The earlier equation did not show the stability behavior I wrote, this one should.
$$\dot y=λy$$ is not in general position. As such this is not a named bifurcation. To test this, add some small "random" term $$ \dot y=λy+ϵq(y), ~~~ q(y)=y(y+1), $$ note that $q$ is smooth and keeps the equilibrium point $y=0$. Then the perturbed right side has further roots at $$ y=-1-\fracλϵ. $$ You can change the type of the resulting bifurcation by changing the perturbation. So with $q(y)=-y^3$ you get the additional equilibria at $$ y^2=\fracλϵ. $$ This is however a special construction, the general type will remain the crossing like in the first example.