Tuesday, September 12, 2006

instability and control

This post is directly inspired by a lecture Dr. Hubbard gave yesterday, at an opening conference for the first-year undergraduate math students at the university. He spoke on dynamics, of course, specifically the motion of a pendulum with friction and a time-variable force (say, a magnetic field affecting a metal pendulum). For those interested, the time-dependent differential equation modelling the situation was: x'' = - x' - a sin x + b cos t, where x is the angular position of the pendulum, and the last two terms represent friction and the time-dependent force, respectively.

The topic was clearly a stretch for the students attending (there were a couple hundred), but we think they truly enjoyed it. There was much discussion, and a couple dozen who came down afterwards to look more closely at some of the pictures. These pictures were colorings of the (x, x') = (position, velocity) plane, with each point colored by the periodic solution to which it is eventually attracted. Such a picture might look like this:

(This particular picture was made by Sarah, using a program written by Matt, with a = .825 and b = 1.) Each (attracting) periodic solution, represented by a solid color, is a translation of the others by 2π, since x measures angular position; that is, the periodic solutions differ only in the number of times the pendulum has wound around its pivot before settling into (or nearly into) a regular periodic movement.

Here’s one salient feature of the picture, stated in theorem form: given any point on the boundary of any colored region, and any positive value ε, every other color lies within ε of the chosen point. Put another way, it requires very little change in the initial conditions to effect a great change in the outcome.

Dr. Hubbard gave a couple of practical uses for this phenomenon. When a situation is highly stable (which, one might think, means one has a great deal of control), then the outcome is nigh inevitable. One may try changing a number of things, and still the same result will occur. Hence one has less control than one might have expected. He pantomimed a skier, who when placed in a very sturdy frame has little choice about where he goes, but who when flexible and apparently unstable provides himself with all sorts of options for movement (again, a small movement can effect a great change).

I was struck during the talk by the great metaphorability of this physical observation. How many people try to clamp down on their lives with as complete control as they can muster, only to find that doing so leaves them no options as to their destination? How often do we experience our greatest joys in the unforeseen, which we must be quick to accomodate lest we lose them?

Now it is true that we must also discipline ourselves to manage the instability properly, as the skier must handle his tenuous position correctly to avoid obstacles, so I’m not arguing against planning and practice. And as always the metaphor should not be pushed too far. But this is a nice illustration of the principle that the safe route does not always lead to the best end. (Any investor could have told you that. As I said to Hannah the other day, if I wanted to be safe, I would have stayed in Tennessee. Yet here I am, again a substantial fraction of the globe away from my zone of familiarity. I pray for the wisdom and the discipline to use this instability for growth and excellence.)


Anonymous said...

I remember seeing something like this back at university. Not this example, but it was when I was very interested in bio-math. I went to a talk by a guy talking about combination therapy for AIDS patients. I forget the exact graph we were looking at, but I think it was population of the HIV in an infected patient.

It would spike to begin, then lay dormant and steady for 10-20 years before spiking again becoming AIDS. What the speaker then said was something along the lines of, "Doctors saw this dormancy as a time when nothing was happening. But, we mathemeticians realize that often stablility like this is often when the most active things are happening."

He went on to describe how targeting a single gene on the HIV made a dramatic change in the population. Then talked about the statistics of how targeting 3 different genes was the ideal.

Anonymous said...

oops, that was me (David)