Module III · Exploration 10
Near the Edge
Extremes, Tipping Points, and Abrupt Change
A large peak can be an extreme excursion, a nonlinearly amplified response, or a transition to another regime. The peak alone does not identify the mechanism.
Central questionWhat happens after the disturbance ends?
Apply the same disturbance ↓
Watch first · short video
Can We Predict Tipping Points Before They Happen?
Begin with what tipping means, what warning signals may reveal, and why prediction near an edge remains conditional.
Watch video ↗Applied Mathematics in Geosciences · Episode 17
Related: Extreme Events Aren’t Just Bad Luck: Here’s Why ↗Then explore it yourself ↓
Excursion, amplification, or tipping?
Apply one finite pulse F(t) to all three systems. The disturbance is identical; only the dynamics differ.
Linear stableNonlinear one-regimeTwo-basin
Right · Diagnosis
Recovery time is the time after the disturbance ends until the state returns within 5% of its original equilibrium and stays there.
The potential is a one-dimensional teaching picture of stability and barriers—not a literal Earth landscape.
dx/dt = −λx + F(t), λ > 0
Restoring tendency grows proportionally with displacement. A large disturbance can create a large event without changing regime.
dx/dt = −x/(1 + gx²) − εx³ + F(t)
Restoration weakens over an intermediate range, amplifying the same push, but x(t) → 0 after forcing ends. This is a teaching comparator, not a specific Earth-system model.
dx/dt = r + x − x³ + F(t)
A transition is identified by the state approached after the disturbance—not simply by the peak.
Which model produced the largest peak?
Which model changed regime?
How much disturbance is needed to cause a transition?
Focus on the two-basin model. As the old stable state weakens, test the same standard pulse and then search for the smallest pulse that changes the final regime.
U(x) = x⁴/4 − x²/2 − rx, dx/dt = −dU/dx
Barrier highRecovery time 0.92Did this pulse tip? NO
The potential is a one-dimensional teaching picture of stability and barriers—not a literal Earth landscape.
As resilience weakens, a smaller disturbance can cause a lasting transition. The stable state may still exist while its restoring pull—and recovery speed—declines.
If we reverse the forcing, do we retrace the same path?
Slowly increase r from the left stable branch, then reverse it. No random forcing is used in this experiment.
The potential is a one-dimensional teaching picture of stability and barriers—not a literal Earth landscape.
The forward and return paths are different. History affects which stable state is occupied. Reversing the forcing may require going much farther before the earlier state is occupied again.
Hysteresis is a property of this bistable fold model, not a universal feature of every abrupt change.
What if random forcing acts while the barrier is weakening?
Every member starts in the same basin and experiences the same slowly changing r(t), but each receives a different random realization.
dX = [r(t) + X − X³]dt + σ dWt
Fraction tipping early 2%Median tipping value 0.387
Tipping can occur before the deterministic fold. A weakening barrier changes the probability of a noise-induced crossing while the old stable state still exists.
The deterministic fold remains a property of the drift dynamics. Random forcing changes when individual trajectories may cross the existing barrier.
Can you diagnose the event from the time series alone?
This synthetic record comes from a slowly weakening bistable system with random forcing and occasional pulses.
A single irregular record can contain an extreme excursion, a noise-induced switch, and evidence of weakening stability. Appearance alone does not separate them.
What does “rare” mean?
pu = P(X > u), Tu = 1/pu
Under a stationary annual-maximum law, rarity is defined relative to a threshold and probability law. Tipping is defined by a change in the state to which the system tends to return. A 100-year return period is an average probability statement, not a schedule.
Five equations, one diagnostic story
Each equation isolates one part of the experiment.
Restoring pull grows proportionally; after forcing, x(t) → 0.
The same push can be amplified without creating another stable state.
Equilibria satisfy r + x − x³ = 0; stability requires f′(x*) = 1 − 3(x*)² < 0.
Valleys are stable states; the hill separates their basins.
Random disturbances can cross an existing barrier; smaller barriers are easier to cross.
08 · Final synthesis1 · Large does not mean tipped 2 · Nonlinearity changes the response 3 · Resilience changes disturbance size 4 · Randomness changes transition timing
Near the edge, the ending is evidence
An extreme excursion and a tipping transition can both be large. Only the transition leaves the system in another regime.
A moderate disturbance can produce a disproportionately large event even when the original regime remains stable.
As the barrier weakens, a smaller disturbance can create a lasting transition and recovery becomes slower.
With random forcing, tipping becomes a distribution of possible transition points rather than one exact crossing.
disturbance → response → recovery → final regime
Diagnose the mechanism from the whole response—not from the peak alone.
Useful distinctions, not a universal detector
One state variable
Real transitions can begin locally, propagate through networks, or involve several interacting components.
Potential landscapes
Some non-equilibrium systems contain persistent circulation and fluxes that a one-dimensional potential cannot represent.
Mechanism dependence
Noise-induced crossing, fold loss, and hysteresis are distinct ideas; none is a universal description of every abrupt change.
These experiments isolate mechanisms and diagnostic questions. They are not calibrated forecasts of a specific Earth subsystem.
Sources, methods, and synthetic-data note
- Marten Scheffer and coauthors on critical slowing down and stability loss.
- Peter Ashwin and coauthors on bifurcation-, noise-, and rate-induced tipping.
- Standard dynamical-systems references on fold bifurcations, bistability, potentials, and hysteresis.
- Extreme-value references on exceedance probability and return periods.
All models and records are synthetic teaching examples. The fold threshold is a property of the deterministic drift; noisy transition points are generated with controlled pseudo-random realizations. Critical pulse amplitudes are found by numerical search.