Press on a straight column and past a critical load it BUCKLES — snapping left or right, with no stable position in the middle. The centre isn’t a resting place; it’s a peak you roll off. Which way it goes depends on a whisker of history, and pushing the load back doesn’t return it the same way. That path-dependence is hysteresis. Slide the load past the critical point.
A buckling / snap-through system has potential V(x) = x⁴ − a·x². Below the critical load (a≤0) the single minimum at x=0 is stable. Above it (a>0) x=0 becomes a MAXIMUM (V″(0) = −2a < 0, unstable) and two stable minima appear at x = ±√(a/2) — the pitchfork bifurcation. There is no stable middle: the system must fall to one side, chosen by initial perturbation, and reversing the load traces a different path (hysteresis). A fail-loud self-check throws unless, past critical, the middle is unstable and the two outer states are stable. ◆ real bifurcation theory, node-verified.
The quartic potential is the normal form; the instability of the middle and the two-state, path-dependent outcome above critical are exact. ‘No stable rest between’ is literal — the same non-convergence the whole domain is about.