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.