What is the reasoning behind the names sub- and super-critical bifurcations that occur in the context of pitchfork and Hopf bifurcations? Textbooks seem to introduce this terminology without any explanation as to why.
In one place it is said that "supercritical" means "fixed points are created after the critical point." However, "before" and "after" depends on the meaning of the control parameter, and switching from say, $r$ to $-r$ changes this meaning. Hence this explanation seems to be weak at best.
The notion of sub- and supercriticality in these bifurcations is connected to the stability of the "branching" ($x \neq 0$) fixed points.
Formally, for a system which is known to have such a bifurcation at $x = 0$ and $r = r_0$, the sub- or supercriticality of the bifurcation is determined by the following parameter: $$\frac{\partial^3 f}{\partial x^3}(0,r_0) > 0 \to \text{subcritical}$$ $$\frac{\partial^3 f}{\partial x^3}(0,r_0) < 0 \to \text{supercritical}$$
In the subcritical case, the equilibrium at $x = 0$ is stable and the other generated fixed points for $x \neq 0 $ are unstable when $r < r_0$; this is inverted in the supercritical case.