Strange Attractors

Like relationships between people, mathematical systems often evolve over time. But the evolution is always toward something, toward some final state. Human relationships can evolve into a strong friendship, into a committed relationship, into two strangers who no longer associate. So too can mathematical systems evolve, and the final state they move toward is known as an attractor.

The best example of this is a bowl. Drop a marble in a bowl and the marble will roll around a bit before finally settling on the bottom. That central point on the bottom is like an attractor, and the basin of attraction is the bowl itself, the area within which the attractor has influence (in this case, within a literal basin). If you were to remove the marble and set it beside the bowl on the table, the marble would no longer be in the basin of attraction and would no longer be drawn to the central bottom point of that bowl.

No matter how long that marble rolls and ricochets around the bowl, it eventually finds its way to that bowl’s “attractor”. The same is true of mathematical systems. For any dynamic system, the attractor is the end point. Of course, when I say point, I don’t mean a literal point. While some attractors are points, others are orbits, curves, and manifolds. The attractor is simply the final shape/set/what-have-you that the system settles to.

But not all systems have tidy attractors. Some systems have chaotic solutions. These systems possess strange attractors. Strange attractors are unique in that you never know where on the attractor an evolving system will be. Sometimes two points will be right next to each other, and the next time they’ll be arbitrarily far apart. The motion of systems never quite repeats and the attractor doesn’t close in on itself (thus the “chaotic” descriptor). For example, here’s the first strange attractor, the well-known Lorenz attractor:

As you can see, no solution ever exactly duplicates- they come close, but never quite. And while strange attractors can have definite figures like this, they remain chaotic. Never quite replicating what’s come before or what will come after, never quite settling into a firm shape- almost like it’s dancing around completion. Unpredictable except in short intervals, oscillating around each other without quite touching.

Like strange attractors, some human relationships never quite settle into something one might call concrete. They flow endlessly, never quite what they were, never quite what they could be. Like their timing is never right, something always pulling them apart, but throwing them back together at random moments. Unpredictable except in short bursts, even they don’t know where they will be in the future. There is no stability in their system, but there is a kind of poetry to the motions.

And I suppose that’s something.

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