Shadowing and chaos

Following on from previous posts on chaos, I'm now going to look at a bit of theory in more detail, in particular the shadowing lemma.

The issue the shadowing lemma addresses is this: given a set of differential equations which describe a chaotic system (such as the Lorenz equations), we have no way of calculating a true trajectory, since all of our numerical methods make various approximations (such as using the finite difference (x(t+Dt)-x(t))/Dt in place of the true derivative dx/dt, for example). Moreover, digital computers only calculate and store results to finite precision, so there are rounding errors at every step. In a chaotic system, these errors will grow exponentially and so the model's trajectory (when initialised from a particular state) will differ wildly from the exact system. So what can we hope to learn from the model?

The shadowing lemma provides a very encouraging answer to this problem. It assures us that, although the true system does not track the model's output when they are initialised from the same starting point, there is a trajectory of the true system (starting from a slightly perturbed initial state) that stays close to the model for an arbitrary length of time. So the model output does in fact "look like" a trajectory of the system after all. The pdf file linked from here is one of the most accessible descriptions I've found on the web (the "hyperbolic system" it refers to is a technical term which includes the standard chaotic systems of classical physics).

Stoat has a nice set of graphs showing the growth of small perturbations in the HADAM3 model (atmosphere component of the HADCM3 atmosphere-ocean GCM). As I mentioned in the comments to his post, I am a little suspicious that a small local perturbation can kick off differences across the whole globe within a day or so. Note that the "model physics" does not support pressure (sound) waves so information should only propagate at around the speed of the flow. It seems likely that the propagation speed in these experiments is instead a numerically-determined rate of one grid box per time step. Fortunately, the shadowing lemma comes to the rescue here. Although the perturbation he used would probably not grow in this way given a numerically precise solution to the fundamental equations, the shadowing lemma tells us that there is a true trajectory of the exact system which looks similar to each model run, and therefore their two sets of initial conditions form a (control,perturbed) pair whose difference really does grow as the plots show. Their initial difference would necessarily be small in magnitude, but I expect it would be globally dispersed in nature.

Now, Professor Eykholt made repeated reference to the shadowing lemma in his emails to me, which you can read on Roger Pielke's blog. (I'm amused to note that they're both happy to publish my email without bothering to ask, which rather puts Eykholt's "totally unethical" accusation into context). I struggled to find a way of interpreting his first comments so as to be somewhat relevant, and I can see now how I misunderstood them as a result. However, given that the shadowing lemma only applies to chaotic systems in the first place, it seems bizarre to attempt to use it to demonstrate that a system is not chaotic. In fact, on re-reading his emails his line of argument appears very strange indeed. It is precisely the shadowing lemma that tells us that there are initially-close trajectories of the real system which diverge in the way that the numerical trajectories do, as per the discussion of HADAM3 above. I cannot see how his statement "The shadowing lemma gives you a scale beyond which small perturbations cease to have any important effects" can be reconciled with what is generally understood about chaotic systems. Unfortunately, he refuses to communicate any further on the matter, so I'll never get to the bottom of what he is thinking.