Monday, February 23, 2009
I have been fascinated with attractors ever since I realized that my life was probably governed by one. As I've explored this concept further, I've gradually come to understand that all our lives are governed by attractors that evolve over time, going through abrupt changes in form at key transition points. The attractor for one's life is not imposed from any external source but is, instead, generated by the very history--and future--of one's life.
A famous example of an attractor was discovered by the late Edward Lorenz of MIT when he was trying to find a simple model for the weather. The attractor is a plot of the trajectories of the simple three-variable system Lorenz derived from more complicated and highly detailed weather models. This site allows you to generate your own version of the trajectories on this attractor, which is known as a strange attractor because it is a fractal. I urge you to try it yourself and observe how small changes in initial starting point lead to big differences in the pathway follows by the trajectory. The trajectories traced out by the Lorenz system with the parameter values used here are chaotic--which is why the system is so sensitive to small differences in starting point, a property of chaotic systems that has become known as "sensitivity to initial conditions."
An attractor is a simple diagram that shows a system's history. In an earlier post, I described my first introduction to the Belousov-Zhabotinsky reaction, a chemical reaction that behaves in a very unusual way - instead of smoothly changing one set of chemical species into another, it oscillates between states. The attractor for a system such as this would be, quite simply, a circle. One way to graph data for an oscillating reaction such as the red-blue one, would be to plot the amount of one of the chemicals, the blue one, say, on the vertical axis against time on the horizontal axis. This way of graphing the data would produce a kind of wave going first up, then down with each oscillation of color; the up, down wave would repeat each time the color changed, producing an up, down, up, down, wavy curve across the page.
Another way to graph the same data would be to plot the amount of blue chemical at any given time against the amount of red chemical at the same time. This kind of graphing technique produces a picture of the attractor. It looks different from the first one, but is constructed from the same data; instead of a wavy-looking curve going up, down, up, down from left to right across the graph, this graph is simply a circle, going round and round with each cycle of color change.
The circle is the attractor for the oscillating reaction. Any rhythmic system (an oscillating pendulum in a grandfather’s clock, for example, or our heartbeat) has an attractor that is a circle. It is important to understand that the attractor is not imposed on the system from an external source. It is the chemical reaction itself, or more precisely the mechanism or chemical force field, that creates the attractor – and, even more to the point, it is our body which creates the attractors that govern the rhythms of our lives.
Later, I will explore the implications of the idea that our lives are governed by attractors, probably strange attractors like the one that Ed Lorenz discovered when he was trying to model the weather. The implications for how we are to live our lives are, as we will see, profound.