Concept

Hysteresis — where it appears

A dependence of the current state on the path taken to it, so that turning a parameter up and back down does not retrace. A stem's branch shows it: which lattice it carries depends on where it started as well as on where it now is.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Two paths down the same tree. Both start at the same first fork. Keeping the larger family every time reaches 137.473°; one different choice reaches 99.549°. Neither angle is in the arithmetic — both are limits of a path.

The tree and the attractor

The dynamical model settles on the golden angle over a range of one parameter and on the Lucas angle outside it, and it cannot reach the low-growth end at all. The lattice tree reaches everywhere, has no dynamics in it, and produces the same two angles as limits of two paths. Two routes, one pair of numbers.

emergence · Branch tree

One seed, two rates, two ladders. Both stems begin as forty nodes of Lucas lattice at a rise of 0.12. At 65 nodes per rung the divergence stays at 99.5° and the counts walk 1/3 → 3/4 → 4/7 → 7/11. At 131 it leaves for 137.7° and walks 1/3 → 2/3 → 3/5 → 5/8 → 8/13 instead.

The rate decides the branch

Forty nodes of Lucas lattice, carried down to the same fine rise twice. Hurried, the pattern holds the Lucas ladder through three more forks at 99.5°. Given room, it abandons it at the first fork it reaches and walks to 8/13 at 137.5°. Same seed, same rise, two ladders — with a threshold between them at about ninety nodes per rung.

emergence · Branch choice

Named alongside it

The objects these essays reach for when they reach for this one.

Bifurcation Branch Fibonacci Ladder Repulsion Rise Attractor Branch selection Continuity Critical rate Initial condition Metastability

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