On the Implications of Nonlinearity and Chaos

I picked up James Gleick’s book Chaos on the recommendation of a friend, mistakenly expecting to learn about physics. The cover misled me, conjuring visions of subatomic particles and string theory. There is physics in Chaos, and physicists playing major roles, but really it’s a book about mathematics. Specifically, nonlinear mathematics.


Nonlinear can mean different things depending on the context. For Chaos, we’re concerned with differential equations. Differential equations relate a variable and that variable’s derivative. For example:

\frac{dx}{dt} = x(t) + C_1

Nonlinear differential equations entangle the variable and its derivatives in the same term. A simple nonlinear equation would be:

x\frac{dx}{dt} = x(t) + C_2

This equation is relatively benign. C_2 is a constant, so we can separate the equation and rearrange it to a solvable form. We’re thrown this sort of thing in the first two weeks of diff eq, before moving onto harder problems.

Most conceivable differential equations are nonlinear. Certain nonlinear forms are solvable, such as the equation above. But the vast majority are not1.

This is a bit of a problem for us humans, because the universe essentially runs on differential equations. Scientists of all disciplines spent decades mistakenly assuming that unpredictable systems actually oscillated around unseen equilibria. Enough systems really do that that it wasn’t an unreasonable hypothesis—but it turns out that most of them don’t.

As the Twentieth Century progressed, things began to change. Mechanical calculators and digital computers finally let men run the numbers fast enough to see that, no, the systems weren’t doing what they’d previously thought. Edward Lorenz’s meteorological simulations are the canonical example, but biology researchers studying population changes, electrical engineers building signal processing systems, and physicists trying to get a handle on fluid mechanics discovered related phenomena around the same time.

Researchers found patterns in the noise. Lorenz discovered his attractors2. Mitchell Feigenbaum noticed period-doubling bifurcation. Benoit Mandelbrot did . . . honestly what didn’t Mandelbrot do? A quartet of physics grad students at UC Santa Cruz calling themselves the Dynamical Systems Collective (among other names) did a lot of the work, flushing out what became chaos theory and bringing it forward for publication.

Chaos started showing up everywhere. The Dynamical Systems Collective occasionally sat down in a public place and just looked for the nearest pattern of nonlinear behavior, what we now call strange attractors. Was it the dripping faucet in the coffeehouse kitchen? Even massively simplified models of dripping water are nonlinear. Was it that flag blowing in the breeze? One of the members even argued that the needle on his car’s speedometer bounced in a nonlinear fashion.

Once you notice the pattern, you’ll see nonlinear dynamics constantly. It’s easy to quell your curiosity about the world when you think everything has nice, simple governing equations. Some algebraic expression or trigonometric function, with a linear differential equation at worst. And surely that won’t be more than second order!

No. Chaotic systems are all around us. The electrons bouncing through your Ethernet cable behave nonlinearly. Do you know someone with an irregular heartbeat? That’s a nonlinear pattern. Medicine was slow to embrace chaos theory, but the human body is a massively nonlinear system.

Let me intimate that clearly: biological systems tend to be very nonlinear. They cannot be predicted with anything approaching the certainly of simple mechanical systems. And remember, there is no general solution for a simple three body problem.

No equation or set of equations can predict the location of just three lousy planets, approximated as point masses. We’ve known this since the 1880s, but it’s still beginning to sink in to the consciousness of modern civilization. Numerical integration can do wonders, but eventually the system necessarily becomes unpredictable. Only special arrangements can be described as “stable”. In fact, there’s a strong case to be made that the solar system did not form in its current configuration. This arrangement may be an equilibrium reached only after major disruptions, possibly including the ejection of multiple planets to interstellar space.

Now, try developing a semi-functional model of the human brain, neurotransmitters and all. I’ll wait.

When you begin to really think about these things, it can become truly terrifying. The size and degree of our ignorance is difficult to communicate. Engineering and science are considered hard when everything is linearized and simplified to death—the real deal makes that look nearly trivial. Economics and culture are probably even more complex3. After all, molecules don’t have minds of their own.

My biggest criticism of Chaos would probably be that the book doesn’t spend enough time emphasizing this point. There’s a lot of great factual information, but the full implications are barely sketched out. Equations are few and far between—but Mr. Gleick deserves a lot of credit for including equations at all! So despite that flaw, I would very highly recommend Chaos as an introduction to the higher mathematics which makes the world such an interesting place to live.


1Even the solvable ones can be real beasts. I still have this monstrosity bookmarked from an analytical homework problem. They told us not to attempt solving it ourselves, and I can see why.

2I took differential equations multiple times, and Lorenz Attractors were the only nonlinear form we discussed in any real detail, and even that avoided calculations.

3Yet, so far as I can tell, the sociology program at my school requires nothing more than the bare minimum in mathematics. Most of the serious work ends up getting published in econ journals.

Do You Hate Math or Just Counting?

Last weekend KU held their annual Engineering Expo, a fairly typical sort of event for trying to get little kids interested in science and engineering. If you’re unsure what that might entail, come drop by next year.

Presumably this is fun for the little kids—most of them seemed pretty enthusiastic, and I never went to such events myself. And it’s fun for most of the students putting it on. We get to hang out with our friends, tell people about our majors, and get a day off class (though could you maybe have not scheduled it during midterms this year?). Half the fun is getting set up beforehand.

Because we’re nerds, we covered most of the leftover whiteboard space in the SEDS room with various engineering equations: thermo, flight dynamics, orbital mechanics, even the definition of a derivative. This predictably elicited a number of comments, mostly from parents but also from the kids themselves, about how much they hate(d) math.

This was particularly striking to me because when I was that age, I hated math. Or rather, hated what I thought was math. In elementary school, I didn’t know very much math. Pretty much all I knew how to do was glorified counting.

Addition? Counting. Subtraction? Backwards counting. Multiplication? Fast counting. Division? Backwards fast counting. Medians and means? Counting!

This pattern holds when we move onto the secondary school subjects. Algebra is almost entirely about counting graphically. Not all of geometry is counting, but a good fraction of it—vectors, for example—is. The same goes for a lot of trigonometry.

Only when you get to calculus, a senior year subject if it’s on the table at all, do you really start focusing on relationships versus the raw numbers. Calculus was the first math class which I enjoyed. That was the first time math became a tool rather than an obstacle. (Consistently, at least.)

A lot of students abandon mathematics long before they get to the stage where it becomes useful and meaningful. This concerns me. While there are plenty of people whose brains just aren’t made for math, there’s also a lot of great minds being lost because they aren’t made for endless dull arithmetic.

Spend years struggling away with contrived problems, truth held outside of grasp, and well, yes, giving up is certainly a reasonable response. You spent the last fifty years making science and math inaccessible and now, dear politicians and school administrators, you lament that more kids aren’t studying engineering. Behold the completely predictable consequences of your actions.

But this is not just an intellectual problem, or even an economic one. It’s a political nightmare.

Learning integral calculus completely rebuilt my perception of government finance. Just gaining an appreciation for statistics made me rethink dozens of policy positions. And yet I’d bet half our sitting Senators don’t even know what “differentiation” means. This should scare us shitless when they’re trying to plan for the next year, yet alone the next ten or twenty.

I hope that we changed a few minds last weekend, or at least planted a seed of doubt.  If technological civilization is going to survive the next 100 years, we need to rethink the way our culture approaches math and science. I’m severely tempted to do pro bono math education out of fear, but a few college kids can’t do it alone.

So, for your children’s sake: Do you hate math, or just counting?