Walking the Line of Jumps to Its End

Bracket the jump in two controls, follow it, and find where it vanishes.

This is the generic line-following procedure; in the Criticality Compass it is the Navigator’s job.

On the previous page one control, the catalytic drive a, was ramped and the system jumped at a foldA stable and an unstable state merge and vanish, so the state must jump.. A fold is a point in one control but a line in two: change the food supply b as well, and the jump moves. That line of jumps is what the procedure follows.

Why the end point is the test: along the line the jump shrinks, and at a cuspWhere two fold lines meet and the jump shrinks to zero. it vanishes as a power law, exponent ½ in mean-field theory.² Finding such an end point in an autocatalytic chemistry would be evidence for the hypothesis. This model, the Schlögl scheme,¹ is a cusp by construction, so the page proves the procedure, not the hypothesis.

1. At a = 2.6, bracket the jump in b from 0 to 0.30 and bisect 14 times. Each probe starts from food only (x = 0), runs to steady state, and asks one thing: did it end above a/3?
2. Measure the jump: start a second probe high, at the bracket’s lower end, and take the difference Δx.
3. Step a down by 0.1, re-bracket inside a declared window of ±0.05 around the last bracket, and repeat while Δx > 0.02.
4. Where the jump has gone, bisect on a 8 times: the end point is reported as an interval.
5. Fit log Δx against log(a − â*) and compare with ½.

“Show the other edge” brackets the jump down, starting each probe high. The shaded wedge between the two edges is the hysteresis region; the edges meet at the end point. For larger a the lower edge leaves the plane below b = 0, and is drawn only where found.