Thought Toys · Cycles & change · Exhibit 73
A system with two stable states and one drive, r, that you nudge back and forth. Push r up slowly enough and the state clings to its branch — right up until the branch vanishes out from under it and the state snaps to the other one. Push r back down by the same distance, and it does not retrace its steps: the return door is somewhere else entirely.
The equilibrium curve, and where you've been stable unstable you are here
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The chart plots a state x against a drive r you control. The Z-shaped curve running through it is every point where the system could sit still — but the middle stretch, dashed, is a rest point that immediately falls away from the slightest nudge: it separates two solid, genuinely stable branches, one low and one high. For a wide middle range of r, both stable branches exist at once. Which one you're actually on depends entirely on history — on which direction you arrived from.
Drag r slowly to the right from the lower branch and the live dot rides the lower curve upward, obediently — right up until r crosses the amber landmark near +0.385. At that exact point the lower branch runs out: there is no nearby rest point left to sit on. The dot doesn't creep to the upper branch, it jumps there, nearly straight up on the chart. Now drag r back down by the same distance. It does not jump back at the same place. The upper branch survives all the way down to a different landmark, near −0.385, before it, too, runs out and the dot snaps back down. Sweep up and back down and the trail behind the dot draws an unmistakable loop — not a line retraced, a loop.
Press sweep the full loop to watch this happen slowly and automatically, start to finish, or drag the slider yourself. Either way, notice what the loop is not: it isn't friction blurring a single answer, and it isn't the system being slow to respond. Ask it to sit anywhere on either solid branch and it will, indefinitely. The loop is that the same r, mid-loop, has two honest answers, and the one you get is whichever branch you haven't fallen off of yet.
The folk version — it snaps back the same way it snapped forward — is exactly what doesn't happen here, and that's the whole point. The two jump points are set by two completely different conditions (the lower branch running out going up; the upper branch running out coming down), and nothing forces them to coincide. For this system they sit at precisely r = ±2⁄(3√3) — the same distance from zero, but on opposite sides, with a wide stretch of memory in between.
improve/verify/73-hysteresis.js, 35 checks): that fold value is
confirmed three independent ways — calculus, the cubic discriminant 4−27r² changing sign at exactly
that r, and an 800,000-point numerical search — and root count/stability are checked directly
against the discriminant at seven sample r. A full quasi-static sweep (RK4-simulating the actual
differential equation) shows an upward jump near +0.385, a downward jump near −0.385, clearly separated, and a
large enclosed loop area by the shoelace formula. Two negative controls fail as they must: a monostable
comparison system (x′ = r − x) run through the identical sweep shows a loop
over 15× smaller — ordinary lag, not hysteresis — and sweeping this same bistable system without
crossing either fold (r kept inside ±0.2) is nearly reversible, over 20× smaller still. Bistability
alone isn't the story; crossing the folds is.
Also in Cycles & change: Predator & prey →
Also in Cycles & change: Predator & prey →
← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 73.