Bistability and hysteresis

The landscape of Chapter 3 was fixed. In reality the landscape itself shifts slowly with outside conditions: routines, surroundings, relationships, health. This chapter asks what happens to a system with two stable states when an outside condition $h$ changes slowly.

The hysteresis loop and “giving up halfway”

# Bistability, hysteresis and tipping: dx/dt = h + x - x^3   (U = x^4/4 - x^2/2 - h x).
# For |h| < 2/(3*sqrt(3)) = 0.385 there are two stable states; at the fold (saddle-node) one vanishes.
# Simplified model; h stands for slowly changing conditions, x for the state of a routine.
import math
DT = 0.01
H_C = 2 / (3 * math.sqrt(3))

def settle(x, h, t=60.0):
    for _ in range(int(t / DT)):
        x += (h + x - x ** 3) * DT
    return x

# 1. Slow sweep of h up and back down: the state jumps at different places on the way up and down.
x, h, jumps = -1.0, -0.6, []
for direction in (+1, -1):
    while (h < 0.6) if direction > 0 else (h > -0.6):
        x_old = x
        x = settle(x, h, t=2.0)
        if (x_old < 0) != (x < 0):
            jumps.append((direction, h))
        h += direction * 0.005
for d, hj in jumps:
    print(f"sweeping {'up  ' if d > 0 else 'down'}: state jumps at h = {hj:+.3f}  (fold at {d * H_C:+.3f})")

# 2. "Giving up halfway": push with h_push for a time T, then let go (h = 0). Does it stay switched?
print("push h   shortest push that still ends in the new state")
for h_push in (0.40, 0.45, 0.60, 1.00):
    T = 0.0
    while True:
        T += 0.05
        x = settle(-1.0, h_push, t=T)
        if settle(x, 0.0) > 0:
            break
    print(f"{h_push:5.2f}    T = {T:4.2f}")
# sweeping up  : state jumps at h = +0.420  (fold at +0.385)
# sweeping down: state jumps at h = -0.420  (fold at -0.385)
# push h   shortest push that still ends in the new state
#  0.40    T = 17.35
#  0.45    T = 7.30
#  0.60    T = 3.20
#  1.00    T = 1.40

The first part shows hysteresis: on the way up the state jumps at $h \approx +0.42$, on the way down at $h \approx -0.42$. Both are a little beyond the theoretical fold at 0.385 because $h$ changes at a finite speed and the system lags behind; the slower the change, the closer the jump comes to 0.385. Hysteresis means that which state the system is in depends not only on present conditions but on where it came from. At the same $h = 0$, a system arriving from the left stays in the old state, and one arriving from the right stays in the new.

The second part corresponds to “giving up halfway”. If, when the force is removed, the state has not yet passed the ridge (here $x = 0$), the system slides back to the old state; if it has, the system slides into the new state by itself. The shortest time needed depends strongly on the size of the force: with a force of 1.00, 1.4 time units suffice; with 0.40, just above the fold, it takes more than 17. This is critical slowing down near the fold: a lingering “bottleneck” keeps the system dawdling near where the old state used to be for a long time before it finally gets through (Strogatz 2015).

Two points correct the original claim. First, sliding back is not “instant”; it takes a relaxation time. Second, the system “slides naturally” into the new state after the ridge only if the landscape stays the same; if outside conditions later change back far enough to remove the new state, the system still jumps back.

Hysteresis as a ratchet

Hysteresis has a useful implication for changing habits. Suppose $h$ stands for a condition slowly built up by practice (skills, arrangements at home, the people around you). To jump from the old state to the new one, the condition must build past $+h_c$; but once the jump has happened, the system stays in the new state even if the condition falls back to zero, or nearly to $-h_c$. Keeping a state takes weaker conditions than switching into it. This works like a ratchet: progress already made is not lost when conditions slip a little.

The same mechanism explains why old habits are hard to change: the conditions that built them are long gone, and they remain.

Warning signals before a tipping point

Limits of the analogy

The cusp model has one state variable and one control parameter. Human behaviour is shaped by many factors at once, and the control parameter ($h$) often responds to the state itself: once someone starts exercising, their fitness, social circle and sense of self change, and these in turn change $h$. Such feedback may make the new state more robust than in the model, or more fragile. A further limit: in the model the jump is deterministic, while in people it is often mixed with the random barrier crossing of Chapter 3, and comes earlier or later.

Summary

A bistable system has memory: under the same conditions, where it rests depends on where it came from. This explains two things: giving up halfway slides you back to the old state, while once past the ridge the new state maintains itself even when conditions slip somewhat. Critical slowing down near the fold shows why a push that is only just large enough takes a long time to work. Chapter 5 turns from one person to groups: can other people act as the force that pushes the system over the ridge and keeps it from sliding back?