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.

You want real systems that have fold catastrophes?

Thermodynamics (phase changes) .. for which there are also very kewl graphs ... and in the biological domain: sickle cell disease (irreversible polymerization of hemoglobin but only under adverse circumstances), prion diseases (auto-catalysis of adverse conformational change), and probably some cardiac rhythm disturbances.

Oh, you're a sociologist? Mob behavior? War psychology?

I think the core concepts of dynamical systems and attractors are super useful in any domain that engages sensitivity to initial conditions.

I think the Lorentz attractor is prettier and exhibits the same sensitivity to initial conditions. The problem with the 'N' example is that it is difficult to imagine the various initial conditions that lead to very different trajectories. The meaning of the 'N' plot only becomes obvious when you mention dynamical systems- then we think "Oh, that is a phase diagram indicating the possible solutions over a single line. The solutions must be parameterized by time (or some other variable) AND the axes on the plot. Not some arbitrary parametric function."

All of this is more obvious when you look at most animations or pictures of the Lorentz attractor. Put simply, the Lorentz attractor is a set of ODEs (ordinary differential equations) describing the growth over time of the variables X,Y, and Z in terms of the current value of X,Y, and Z. Start out with x=y=z=1, turn the crank- and see the trajectory of that 'mote' of dust in the attractor. Now start that process over, keeping x=y=1, but change z to 1.00001. Turn the crank, and you get massively different results soon.

*That* isn't even the coolest part of the Lorentz attractor. Plenty of attractors do that type of thing. To me the cool part is that this VERY SIMPLE set of ODEs is equivalent to a VERY COMPLICATED set of partial differential equations that I wouldn't be able to solve in a lifetime. Le sigh. It probably took this guy like a day: http://stokeslet.ucsd.edu/mae210cdocs/lorenzderivation.pdf