Back in the dark days of the last century when I was a university undergraduate, chaos theory was my first love. Quantum mechanics was, to me, a mess of contradictions, but I felt like I might actually understand something about chaos. More the fool me, I guess.

I got sucked into optics, and, like all summer romances, nonlinear dynamics—the broader field in which chaos theory sits—became a sepia-colored memory. Until this year at least. Unexpectedly, I had to teach a course on it. And that led me to pay more attention to current research, which managed to generate this article on Ars Technica. Yes, my brain is a nonlinear dynamical system that defies prediction (unless you are my wife; she probably saw this coming).

When we teach nonlinear dynamics in classes, the equations are always naturally dissipative. That means there is some mechanism for energy loss in the equations. However, we always balance these with an external energy input. For instance, a set of equations describing an idealized climate allows energy to be radiated out into space, but that is balanced by energy from the Sun. However, there are many places where balance doesn't exist: a coin toss, a roulette wheel, or a pinball machine are classic examples.

A new paper published in Physical Review X highlights some of the interesting differences between a roulette wheel and a balanced version of the roulette wheel. The paper shows how an abstract geometry can describe all the interesting physics changes, going from something that looks like standard chaos to something that looks a bit like a fairly simple pendulum-type problem. But to truly highlight how cool all of this is, first you must embrace the chaos (er, chaotic systems specifically) before ever reaching roulette.

A chaotic leap into chaos

Abstraction is the key to understanding chaotic systems (I will use "chaos" and "chaotic systems" in place of the more cumbersome nonlinear dynamical systems, even though they are not technically correct). This is great for the scientists in the field but absolutely terrible for the rest of us who are trying to grasp what is actually going on. So this is going to be a rambling walk through the forest of chaos.

At the heart of chaos lies a problem: the equations that describe some physical systems are unsolvable. A perfect example of one of these unsolvable equations is a simplified model of convection roles in the atmosphere. Three seemingly simple equations, called the Lorentz equations, describe the position and movement of a little block of air. But in mathematical parlance, the Lorentz equations cannot be solved analytically, which means that, following the rules of mathematics, it is impossible to write down a solution to the equations that describes the motion of air.

Well, you say, luckily I have a computer, and, 10 minutes later, I have a numerical solution to the Lorentz equations. The movie below shows the result of doing just that for three different starting conditions. The starting conditions (large stationary spheres) are so close together that they look like one sphere in this movie, and all three solutions to the Lorentz equation (the moving spheres, trailed by little dots) trace a complex and beautiful curve. But the trajectory of solutions is completely different, and the end points are completely different.



So, three starting points that are very similar lead us to predict not just different speeds at which the air is moving, but that the air column is rotating in a different direction. In other words, any particular solution to the Lorentz equations doesn't tell us a lot.

Making sense of chaos

Instead, we have to take a more global view. For many physical systems, if you examine the equations carefully, you will find several sets of values that correspond to fixed points—if you start at the fixed point, you will stay there. Depending on the conditions, the Lorentz equations have either one fixed point, three fixed points, or no fixed points.

So, a fixed point is a location at which nothing ever changes. For a swing, there are two fixed points: one with the swing hanging directly down, and a second with it vertical, above the pivot point. Obviously, you never see a swing at the second fixed point because it is not stable. Even the smallest deviation from vertical will send the swing careening around.

Likewise, depending on the conditions, the stability of the fixed points of the Lorentz equations changes, too. In the movie below, solutions for the Lorentz equations starting from different locations are shown. Here you see trajectories that do not go to a fixed point, but they don't run away, either. Instead they tend to orbit one of the fixed points. So, the fixed points are stable. Outside of that, and you orbit the fixed point, slowly spiraling inward. In other words, fixed points are key to producing stable reproducible behavior.

What do these solutions represent physically? Well, convection often results in a rotational flow. The air is heated from below and cooled from the top, so it flows up one side and down the other, creating a cylinder of rotating air. The two orbits represent the two different rotational senses: clockwise and counter-clockwise.

So, if we go back to our movie with the unpredictable behavior (the first movie), we can see that the air might start circulating clockwise but will suddenly switch to counter-clockwise, and later flip back again. There is no warning of the transitions; they happen rapidly and seemingly at random.

But, for the stable fixed points, the solutions quickly move close to the fixed points and then never move away. In terms of the physics: air that starts rotating in a particular sense will keep rotating that direction. It might speed up or slow down, but, it will slowly converge on a steady rate of rotation. None of the solutions runs off to infinity, or collapses to zero (unmoving air). We have a constant input of heat at the bottom, we have constant cooling from above, and the air is doomed to circulate forever. This is what we mean by keeping the conditions fixed.