Bifurcations, balls and balloons

There are some figures that are almost obligatory in any discussion of bifurcations and the related critical transitions or tipping points and early warning signals. As this post is about bifurcations and critical transitions, I will accordingly oblige.

The first figure shows how the equilibrium states of a system change with increasing forcing.

Equilibrium states and bifurcation

Figure 1. Equilibrium states and bifurcation

At low values of the force, shown on the left (A), there is a single stable equilibrium state. If the system is perturbed from this state by noise, it will return to this state. As the forcing increases, a second stable equilibrium state develops (C). If the system is perturbed sufficiently far that it crosses the unstable equilibrium, marked by the dotted line, it can flip from the upper to the lower stable equilibrium. As forcing increases, weaker and weaker perturbations are required to push the system into the lower equilibrium state. Eventually as forcing increases, the upper equilibrium ceases to be stable – the bifurcation point – and the system will tip to the lower state.

The system will tend to transition from the upper to lower state as the forcing approaches B, but will not make the reverse transition until the forcing is reduced to near C. This non-linear behaviour can make the state change essentially irreversible. As a change in the climate state or state of an ecosystem could have large social, ecological and economic costs, an early warning of impending change would be useful to adapt to, mitigate and perhaps forestall the impending critical transition. A range of early warning signals have been identified and are typically illustrated by the following figures which represent the potential across a vertical cross-section of figure 1. A ball is used to illustrate the behaviour of the system at a point far from the bifurcation (upper panel) and near to the bifurcation (lower panel). At equilibrium, the ball will rest in the bottom of one of the valleys.

Figure 2. Balls and potential valleys

Figure 2. Balls and potential valleys

When the system is far from the bifurcation the gradient away from the stable equilibria at the bottom of each valley are steep. If the ball is knocked by random noise, it moves a small distance up the slope and returns quickly toward the equilibrium state. Knocks to the the left or right result in similar movements.

When the system is near to the bifurcation, the valley is shallower. The same amount of noise knocking the ball will drive it further, leading to an increase in variance, and its return to the equilibrium state will be slower, leading to an increase in autocorrelation. As the valley is symmetrical, knocks to the left and right have a different effect, leading to skewness in the position of the ball. Increase in variance, autocorrelation and skewness have all been used as early warning signals.

The theory behind these early warning signals is solid for simple systems. Climate is not a simple system, will they still work? Several authors think so, attempting to use these warning signals to predict critical transitions in palaeoclimate data. I’ve not been greatly convinced by the published examples, for reasons I’ll expound in a future post.

Now I want to consider how the early warning signals will work in complex systems. Unlike the simple system described above with a single component, the ball, climate and ecological systems often consist of many interlinked components. Such systems might still show the classic early warning signals in some, but perhaps not all components of the system. It it easy to imagine a system with many components with two states, where once one component has a critical transition, all the others are forced to have critical transitions. The early warning signals will only be detectable in the first component.

We can explore the behaviour of more complex systems by extending the analogy of the ball in the valley. I’m going to tie a balloon to the ball (perhaps a yo-yo rather than a ball). The ball and balloon are weakly coupled, the string is sufficiently elastic to damp small shocks to the ball or balloon. What happens?

Figure 3. Ball with balloon

Figure 3. Ball with balloon

In the upper panel the system is far from the bifurcation and the ball behaves as it did before. The balloon is knocked by random noise an moves around above the ball.

As the system approaches the bifurcation, the ball exhibits the early warning signals discussed above. The balloon continues to be knocked by random noise: it shows no early warning signal.

Eventually a critical transition occurs, the ball moves into the other valley and the balloon is suddenly dragged into a new position.

We can only expect to find early warning signals if our proxy data records the component of the system that resembles the ball (or a component tightly coupled to the ball), not the balloon.

For example, in a lake undergoing eutrophication, the critical component in the system might be the oxygen concentration at the sediment-water interface, which will control phosphorus mobilisation. This component might display all the early warning signals, but the diatom community in the lake’s epilimnion might be sufficiently weakly connected to the sediment-water interface that these signals are damped out and no early warning signal can be detected in the diatoms before a critical transition.

Beautiful as the theory of early warning signals is, I suspect we will often not be able to detect them in complex systems as we have data on the wrong components, the balloons, of the system, or that there are too many candidate balls and they will generate too many false alarms and only with the benefits of hindsight will we be able to tell which component was the driver of the critical transition.

There is probably a vast literature dealing with critical transitions in complex systems, but all the papers I am aware of in the climate literature are illustrated with the simplest of models.


About richard telford

Ecologist with interests in quantitative methods and palaeoenvironments
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