hysteresis
/ hiss-ter-EE-sis /
Turn a dimmer switch up until a flickering bulb finally catches and glows steadily, then turn it back down — and notice it stays lit well below the level where it first switched on. The system 'remembers' which way you came. Where it ends up depends not just on the current setting but on the path you took to get there. This path-dependence, this lag between switching on and switching off, is hysteresis.
Hysteresis arises when a bifurcation diagram has an S-shaped or folded curve of equilibria, so that over a band of parameter values there are two coexisting stable states with an unstable one between them. As you slowly increase the parameter, the system clings to the lower stable branch until that branch ends at a fold (a saddle-node) and abruptly jumps up to the upper branch. Now decrease the parameter: the system clings to the upper branch and does not drop back at the same place — it holds on until the upper branch hits its own fold, lower down, and only then jumps back. Because the up-jump and the down-jump happen at two different parameter values, sweeping up and sweeping down trace different paths, enclosing a hysteresis loop. Inside that band, which state you are in is decided by your history.
Hysteresis is why many real switches are reliable rather than jittery: a thermostat that turned the heater on and off at exactly one temperature would chatter; a deliberate gap (a hysteresis band) makes it commit. The same mechanism explains sudden, hard-to-reverse transitions in nature and engineering — a buckled structure that does not unbuckle when the load is eased back, a lake that flips to a turbid state and resists cleaning, magnetization that persists after the field is removed. The lesson is sharp: a tipping point reached by pushing a parameter forward is generally not undone by pulling it the same distance back.
A system with x' = r + x - x^3 has an S-shaped equilibrium curve. Increase r from low: the state sits on the lower branch, then at the upper fold jumps high. Decrease r: it sits on the upper branch and only drops at the lower fold, a smaller r. The two jumps happen at different r, so the path up differs from the path down — a hysteresis loop.
Sweeping a parameter up and then down traces different paths around an S-shaped fold — the present state remembers the past.
Hysteresis needs two coexisting stable states (bistability), which in turn needs nonlinearity and the folded branch from a pair of saddle-nodes. A single-valued equilibrium curve has no memory — reverse the parameter and you retrace exactly the same states.