Tipping points in the climate system

Builds on Dynamical system Equilibrium point Saddle-node bifurcation Bifurcation theory Stommel box model of thermohaline circulation Budyko–Sellers model and ice–albedo feedback

Definition

Model a component of the climate system by a dynamical system $\dot x = f(x, \mu)$ , possibly with noise. Here $x$ is the state and $\mu$ is a forcing parameter. A tipping point is a value or a rate of change of $\mu$ with the following property. A small further change of the forcing makes the state leave the neighbourhood of a stable equilibrium or attractor, and move to a distant one. Three mechanisms are distinguished:

bifurcation-induced tipping: as $\mu$ crosses a critical value $\mu_c$ , the stable state the system follows loses stability or disappears, typically in a saddle-node bifurcation;
noise-induced tipping: random fluctuations push the state over the boundary of its basin of attraction while $\mu$ stays fixed;
rate-induced tipping: $\mu$ changes faster than a critical rate, and the state cannot track its moving stable state, which exists for each fixed $\mu$ .

Remarks

Example: the scalar system $\dot x = \mu + x - x^3$ has two stable equilibria for $\abs{\mu} < 2 / (3 \sqrt 3)$ . If $\mu$ decreases slowly through $-2/(3\sqrt3)$ , the upper equilibrium merges with the middle one and disappears, and $x$ jumps to the lower branch; this is clause 1. To return, $\mu$ must rise above $+2/(3\sqrt3)$ , so the change is not reversed by undoing the forcing. The Stommel box model and the Budyko–Sellers model have tipping points of this kind.

Non-example: the linear system $\dot x = \mu - x$ has the single stable equilibrium $x = \mu$ , which moves continuously with $\mu$ and attracts every state; it has no tipping point.

Near a saddle-node bifurcation the linear rate of return to the stable state tends to $0$ (critical slowing down). In a noisy system this raises the variance and the lag-one autocorrelation of the state, which are used as early warning signals. See bifurcation theory for the general theory.

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Bifurcation theory concept
Budyko–Sellers model and ice–albedo feedback concept
Dynamical system concept
Equilibrium point concept
Saddle-node bifurcation concept
Stommel box model of thermohaline circulation concept
Tipping points in the climate system definition

Tag	`92RQ4VER29`
URI	`m2://92RQ4VER29`
Kind	definition
Writing	W0 stub
MSC	86A08, 37N10, 34C23
Defines	Tipping points in the climate system
Uses	Dynamical system, Equilibrium point, Saddle-node bifurcation, Bifurcation theory, Stommel box model of thermohaline circulation, Budyko–Sellers model and ice–albedo feedback
Changed	3 November 2026

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