Fold Catastrophe Model

My last post, which I had to cut short, discussed the recent paper by Scheffer et al. (2009) on the early warning signs of impending catastrophe. This paper encapsulates a number of things that I think are very important and relate to some current research (and teaching interests). Scheffer and colleagues show the consequences on time series of state observations when a dynamical system characterized by a fold bifurcation is forced across its attractor where parts of this attractor are stable and others are unstable.  In my last post, I described the fold catastrophe model as an attractor that looks like an "sideways N." I just wanted to briefly unpack that statement.  First, an attractor is kind of like an equilibrium.  It's a set of points to which a dynamical system evolves.  When the system is perturbed, it tends to return to an attractor.  Attractors can be fixed points or cycles or extremely complex shapes, depending upon the particulars of the system.

The fold catastrophe model posits an attractor that looks like this figure, which I have more or less re-created from Scheffer et al. (2009), Box 1.

The solid parts of the curve are stable -- when the system state is perturbed when in the vicinity of this part of the attractor, it tends to return, as indicated by the grey arrows pointing back to the attractor.  The dashed part of the attractor is unstable -- perturbations in this neighborhood tend to move away from the attractor.  This graphical representation of the system makes it pretty easy to see how a small perturbation could dramatically change the system if the current combination of conditions and system state place the system on the attractor near the neighborhood where the attractor changes from stable to unstable.  The figure illustrates one such scenario.  The conditions/system state start at point F1. A small forcing perturbs the system off this point across the bifurcation.  Further forcing now moves the system way off the current state to some new, far away, stable state.  We go from a very high value of the system state to a very low value with only a very small change in conditions.  Indeed, in this figure, the conditions remain constant from point F1 to the new value indicated by the white point -- just a brief perturbation was sufficient to cause the drastic change.  I guess this is part of the definition of a catastrophe.

The real question in my mind, and one that others have asked, is how relevant is the fold catastrophe model for real systems?  This is something I'm going to have to think about. One thing that is certain is that this is a pedagogically very useful approach as it makes you think... and worry.

Predicting Catastrophe?

There is an extremely cool paper in this week's Nature by Scheffer and colleagues. I'm too busy right now to write much about it, but I wanted to mention it, even if only briefly.  The thing that I find so remarkable about this paper is that it's really not the sort of thing that I usually like.  The paper essentially argues that there are certain generic features of many systems as they move toward catastrophic change.  The paper discusses epileptic seizures, asthma attacks, market collapses, abrupt shifts in oceanic circulation and climate, and ecological catastrophes such as sudden shifts in rangelands, or crashes of fish or wildlife populations. At first, it sounds like the vaguely mystical ideas about transcendent complexity, financial physics, etc.  But really, there are a number of very sensible observations about dynamical systems and a convincing argument that these features will be commonly seen in real complex systems.

The basic idea is that there are a number of harbingers of catastrophic change in time series of certain complex systems.  The model the authors use is the fold catastrophe model, where there is an attractor that folds back on itself like a sideways "N".  As one gets close to a catastrophic bifurcation, a very straightforward analysis shows that the rate of return to the attractor decreases (I have some notes that describe the stability of the equilibria of simple population models here. The tools discussed in Scheffer et al. (2009) are really just generalizations of these methods).  As the authors note, one rarely has the luxury of measuring rates of return to equilibria in real systems but, fortunately, there are relatively easily measured consequences of this slow-down of rates of return to the attractor. They show in what I think is an especially lucid manner how the correlations between consecutive observations in a time series will increase as one approaches one of these catastrophic bifurcation points. This increased correlation has the effect of increasing the variance.

So, two ways to diagnose an impending catastrophe in a system that is characterized by the fold bifurcation model are: (1) an increase in variance of the observations in the series and (2) an increase in the lag-1 autocorrelation.  A third feature of impending catastrophes does not have quite as intuitive an explanation (at least for me), but is also relatively straightforward.  Dynamical systems approaching a catastrophic bifurcation will exhibit increased skewness to the fluctuations as well as flickering.  The skewness means that the distribution of period-to-period fluctuations will become increasingly asymmetric.  This has to do with the shape of the underlying attractor and how the values of the system are forced across it. Flickering means that the values will bounce back and forth between two different regimes (say, high and low) rapidly for a period before the catastrophe.  This happens when the system is being forced with sufficient strength that it is bounced between two basins of attraction before getting sucked into a new one for good (or at least a long time).

In summary, there are four generic indicators of impending catastrophe in the fold bifurcation model:

1. Increased variance in the series
2. Increased autocorrelation
3. Increased skewness in the distribution of fluctuations
4. Flickering between two states

There are all sorts of worrisome implications in these types of models for climate change, production systems, disease ecology, and the dynamics of endangered species.  What I hope is that by really getting a handle on these generic systems, we will develop tools that will help us identify catastrophes soon enough that we might actually be able to do something about some of them.  The real challenge, of course, is developing tools that give us the political will to tackle serious problems subject to structural uncertainty. I won't hold my breath...