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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.

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2 Answers 2

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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.

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  • $\begingroup$ I appreciate your answer but it seems to just repeat the definition of sub- and super-criticality. I want to know the reason behind the choice of the words "subcritical and supercritical" in the definition. $\endgroup$
    – timur
    Nov 3, 2022 at 4:40
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My understanding is that the origin of supercritical versus subcritical comes from the analogy with first order phase transitions in physics. In a typical temperature-pressure diagram, the line where vapor and liquid co-exist terminates at a critical point. Beyond this point we have a supercritical fluid, i.e. we can move continuously, without phase transition between the liquid and the gaseous phase.

In analogy, in a supercritical bifurcation, the stable state changes continuously. For example in a supercritical Hopf bifurcation, a stable limit cycle contracts continuously onto a stable fixed point, which can also be seen as a degenerate periodic orbit. Therefore, if we follow a stable object across a supercritical bifurcation, we evolve continuously.

In contrast, a subcritical bifurcation involves a jump. As we track a stable fixed point, it becomes unstable and no other stable object is created close-by. Usually we therefore fall onto some other stable object, which is far away. This naturally gives rise to the phenomenona of hysteresis and multistability through the subcritical bifurcation. This is in analogy to a first order phase transition in thermodynamics below the critical point, which involves the co-existence of the two phases.

The analogy is not perfect of course. However, it seems to have provided the intuition people needed to distinguish two mathematically different cases by appealing to a similar situation they were familiar with in physics.

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