We've all read *Jurassic Park*, so we all know how patterns can emerge from a jumble of chaos. Out of a seemingly random assortment of stuff, orderly structures appear. But what if, instead of killer dinosaurs, they were adorable pictures of cats?

Here's the secret of the hidden order within apparent chaos. It's all thanks to the magic of strange attractors.

Even in chaotic situations, patterns emerge. We can study those patterns to figure out the wake of a jet, the rushing of water from one place to another, and whether or not dinosaurs will eat Samuel L Jackson. That's the conventional wisdom, at least. But how do you even get started looking at a chaotic system as a math or graphic problem? And how can we call something chaos, if it's characterized by a pattern?

Among the causes of chaos are little things called 'attractors.' These come in two types, tame and strange. As the name implies, strange attractors lead to a more chaotic situation than tame ones. How to tell the difference? Place two objects side-by-side near a tame attractor and their courses will be much the same. When it comes to a tame attractor, it doesn't matter if you place an object an inch or two to the left or right. Strange attractors do not have the same effect. The two objects, placed close together, will still be drawn to the attractor, but they will take very different courses to get to their goal.

Put two rubber balls fifty feet above the surface of the earth, and their journey towards their resting place on earth will be predictable and nearly the same. Put them in the stream of a jet engine, and there's no telling where they'll end up, or what paths they'll take to get there.

Which isn't to say that people haven't been mapping the paths that objects take towards attractors, both strange and tame. Some of these maps are very beautiful and very strange. Over the years the most famous ones have evolved names. The Horseshoe Map is a map of an object being squished, stretched, folded into a horseshoe, and squished again. The Lorenz attractor describes the flow of shallow fluid, with eventually folds itself into a strange, warped figure eight. And then there's the infamous Arnold's Cat Map. In the 1960s, Vladimir Arnold showed how a picture could be scrambled into chaos in a few steps, and then reassembled in its original form with a few more steps. He demonstrated this feat with a picture of his cat, earning the map its name.

Another famous pattern is the Poincaré Map. This one emerged from the study of a periodic orbit of, say, a planet. Of course, we all know what planetary orbit maps look like, since we see them in every picture of the solar system. They're pretty dull round shapes, since planets have regular orbits. The Poincaré Map imagines another, lower-dimensional, system intersecting with the orbit of the planet. Think of the planet dipping into and out of a thin stream of water. This maps the effect, and shows a beautiful, if strange, look at the chaos that emerges from the two systems interacting.

These maps, which show overall patterns emerging from chaos, are studied as a way to understand dynamical systems. Any two balls dropped through a system characterized by, say, the Lorenz Attractor might be all over the place, but they will be traveling along certain paths over and over again. A group of points stuck in Arnold's Cat Map — or some repeating pattern like it — have only a certain amount of time until they're right back where they started. The individual points undergoing the irresistible pull of a strange attractor may follow paths that differ wildly, but when we take a step back, we can see the overall structure pulling them along those paths — and we can predict it.

*Lorenz Attractor Image: **Rogilbert*

*Arnold Cat Map Image: **Claudio Rocchini*

* Poincare Map: **Composing with Chaos*

Via STCI, and Wolfram Math World very many times over.

## DISCUSSION

Let's try the Poincaré map again:

Differential equations are HARD, so instead of trying to solve a big system of differential equations Poincaré had a great idea. To simplify even further, I'll talk in terms of a three-body system composed of the Sun, the Earth, and a small asteroid.

The asteroid is so small it's not really going to affect the Earth or Sun that much, so we can just say that the Earth goes around the Sun, making a circle (almost) in a plane. In fact, we can even pick our coordinates to rotate with the Earth, so that the Sun is at (0,0,0) and the Earth is at (1,0,0) forever. Now we're interested in the path our asteroid takes.

But like I said above: differential equations are HARD! So instead of really working out the path, Poincaré chose to study where the asteroid crosses from

abovethe Earth-Sun plane tobelowit. If you know a point where the asteroid crosses then it's actually not too hard to figure out where it's going to cross from above to below again.This is the Poincaré map: a function from the plane to itself saying that if you cross at point p you're next going to cross at point f(p), and then f(f(p)), and so on. Studying how iterations of a function behave is a lot easier than solving differential equations, and can tell you a lot of the same qualitative information.