Early warning
Could you see it coming?
A ball sits in a well. Something outside the picture leans on the landscape, and the well gets shallower every year. The ball keeps returning to the bottom — until one day it doesn’t. The question this page exists to answer is whether the wobble of the ball, watched carefully, tells you the well is going before it goes.
Tune the analysis
What you are looking at
The left panel is a potential — a landscape whose valleys are the states the system can rest in. Its shape is U(x) = ¼x⁴ − ½x² − m x, and the ball rolls downhill on it while noise kicks it about. The pressure m tilts the whole landscape to the right. As it grows, the left valley shallows, slides toward the ridge, and at the exact value μ = 2√3∕9 ≈ 0.3849 the valley and the ridge merge and annihilate. That is a fold, and past it there is no left-hand state left to return to.
The shading in the well is filled to the exact height of the ridge — the level at which the basin would spill. What sits below that waterline is the kick the system can still absorb and recover from. It is resilience, drawn to scale, and it drains.
Watch how fast it goes. The buffer does not fall in step with the pressure; it collapses far ahead of it. By the time m is only halfway to critical, two thirds of the depth is already gone. At 84% of critical — with the landscape still visibly holding a valley — 94% of it has gone. For most of the run the reservoir is a sliver you can barely see, which is the honest picture: long before a system tips, almost all of its capacity to absorb a shock has quietly already been spent.
The right panel is the same instant, plotted as a time series — what an observer with a thermometer would actually have. The ball’s wobble is all they see.
Why the wobble gives it away
Near the bottom of a valley the restoring force is set by the curvature, λ = U″(x*). Flatten the valley and λ falls; the system takes longer and longer to shrug off a kick. For this kind of noise-driven system that has two measurable consequences: Var = σ²∕2λ and AR(1) = e−λΔt. As λ → 0 the variance blows up and the lag-1 autocorrelation climbs toward 1. That is critical slowing down, and the two indicator strips are measuring it live, in a trailing window, on detrended data. The Kendall τ beside each is simply how relentlessly it has been rising.
Two things worth noticing
The tip usually arrives early. Watch where the ball actually leaves against the dashed line marking the fold. Noise almost always kicks it over the shrinking ridge before the ridge disappears — so the crossing is a probability, not a date. Press New noise a few times: same physics, same pressure, different year.
AR(1) is the trustworthy one. Across a few hundred runs the autocorrelation trend is positive in about 99% of them; variance manages roughly 83%, and often only turns up at the very end. If you take one thing from the indicator strips, take the difference between them.
Model and method after Chris A. Boulton (University of Exeter), Early Warning Signals workshop for AMAZALERT, March 2013; generic indicators after Dakos et al. and the earlywarnings toolbox; critical slowing down after Scheffer et al. (2009), “Early-warning signals for critical transitions,” Nature 461, 53–59. Boulton’s original animation is CC BY 4.0.
Related: Tipping Interactive