The Attraction of Attractors
Strange attractors are not the only kind of attractors, but they are by far the most interesting. Other attractors are easier to describe and picture. First of all, “the essence of an attractor is some portion of the phase space,” the imaginary space where mathematicians map things, “such that any point which starts nearby gets closer and closer to” (Stewart, 2002, p. 99). There are point attractors, which, as their name suggests involve a single point. Good examples of this would be a drain in a sink or a spring coming out of the ground, in the first case, known as a sink attractor, all points converge on the attractor, whereas in the second, all points move away from the attractor or source, and it is known as a source attractor. These are known as steady-state attractors as they tend to sit still.
Then there are period attractors, which go around and around, periodically visiting the same areas over and over. They are called limit-cycle attractors, and there are two varieties of these: stable and unstable. Stable limit attractors are closed loops around which nearby trajectories converge, whereas with unstable limit attractors nearby trajectories tend to move away. These are all typical traditional attractors. Essentially, they are either a point or a circle. But what if you have an attractor that is neither a circle, nor a point, but something else? That is where the notion of the strange attractor comes in. Strange attractors are structurally stable attractors that are not classical. Stewart (2002) tells us that by using the word strange, “what they mean is ‘I don’t understand this damn thing.’ But it's also a flag, signaling a message: I may not understand it, but it sure looks important to me” (p. 109). With this in mind, let us look at a slightly more complicated attractor, the torus, before we explore strange attractors.
A torus attractor can be thought of as what you would get if you were to trace a line on the outside of a bagel or doughnut, which spiraling around might or might not wind up connecting with itself. This spiraling movement is quasi-periodic and though torus attractors are not typical, they are often observed during the transition from one state to another. They act as a kind of jumping off place to help understand chaos. (Stewart, 2002).
Stewart gives an example of swinging a cat by the tail while rotating around in a circle, but please do not try this at home.
Finally we come to the strange attractor, which, we have already been forewarned, and as its name suggests is not that easy to picture, and even less easy to understand, but we will give it a try anyway. Chaotic systems will trace out complex recognizable patterns that are confined to a region—these are known as strange attractors. Stewart (2002) tells us that we cannot observe attractors directly, what we observe is observables, their dynamics and characteristics. In this way, attractors are like archetypes, and as we shall see shortly, Van Eenwyk (1997) considers archetypes to be strange attractors. But first, back to strange attractors themselves. The dynamics of these attractors are recurrent but not periodic, although they remain close to the previous state and so we get recurrent motifs that are close but not identical to each other. They are hard to define exactly, but as former Supreme Court Justice Potter Stewart (and no relation to Ian) famously said in 1964 about hard -core pornography, in Jacobellis vs Ohio: “I know it when I see it.” Van Eenwyk (1997) tells us that strange attractors are peculiar in that “primarily their repetition does not guarantee predictability” and that “never repeating—yet always resembling—themselves, they are the epitome of contradiction: infinitely recognizable, ultimately unpredictable” (p. 54).
Predicting where on the attractor you are at any point in time is impossible, due to the stretching and folding nature. Because of this stretching and folding, some points get torn apart and end up far away whereas other far points may come back, moving closer together. Ian Stewart (2002), explains that in their broadest sense, whereas “traditional dynamics” can “sit still,” or go “round and round,” “the geometric essence of chaos” is “stretch and fold” (pp. 136-137). Thanks to computers, we can see the beautiful shapes that strange attractors trace out, which brings us to the fascinating fractal, because “it’s now customary to define a strange attractor to be one that is fractal” (Stewart, p. 207). In the next section, we will consider a few characteristics of fractals and then see the similarities with psychology.