Heuristic

Lorentz attractor

When things are chaotic, the little details make big differences. If we can use this to control and overcome the forces of nature, we can use it in other ways, too. But how much does it really help us?

Sometimes it’s the way things start that really decides how things will go.

When mathematician Edward Lorenz said a butterfly flapping its wings might ultimately cause a hurricane on the other side of the globe (alluding to Ray Bradbury’s “A Sound of Thunder”), he was describing how little changes can cause big differences in the long run. In many cases, our models of predicting the future vary a lot depending on their initial conditions. The way a population of lynx grows may vary widely by how the population began. When a river is about to flood, one person’s actions can save an entire village. These examples of the “butterfly effect”, an important tenet of chaos theory, help us figure out these things that are highly sensitive to how they begin.

We like to use “chaos” to refer to constant disorder, disarray, or other disses, but, speaking scientifically, we strictly refer to “chaotic systems” as those that satisfy three conditions. (1) you start massively influences what happens, (2) very similar starting conditions can give very different results, and (3) even if you start from very different locations, you’ll eventually end up near the same place. We’ve already touched on (1) and (2), but not so much for (3). By (3), we mean different starting points in a chaotic system will give at least similar results as they keep going repeatedly.

“What?” you’re probably thinking “Why should something chaotic give results that are similar? Shouldn’t something random give a lot of different results?” Ah, but they should! What if I told you to pick a number, and then double it repeatedly. No matter which number you pick, you’ll end up with a widely different result that is very distinct from all the other results. But some might say this isn’t “chaotic” because, no matter what you choose, you’ll end up at infinity (or negative infinity). That’s why we need this third condition.

These conditions show why our scientific theories of chaos aren’t exactly what we usually mean when we say something is “chaotic.” In the world of science, chaotic doesn’t mean completely unpredictable, random, uncertain, disorderly or just plain crazy. It’s something really interesting that shows us how to look at the world. With chaos theory, we can look at complex systems in more accurate, refined, and fun ways.

“But wait a second,” you say. “The examples you gave are just examples of things that are very very random. A leaf falling in the wind just falls unpredictably because of the wind, so how can you say its starting positions cause this?” Ah, perhaps they are! When you look at a leaf falling in the wind, how can you tell that its path will cause a hurricane in Florida or if it’s just going nowhere? There’s no way to tell them apart from another when we observe them. But, if we had the ability to create a theoretical model, we would notice chaotic systems differ from random ones in that chaotic systems would differ greatly between two starting positions that are close to one another.

Now, you might continue and tell me, “Well, a system that changes largely based on how it begins could just be something that we don’t fully understand. What if we just haven’t made our mathematical models good enough to understand them?” Surely, if Edward Lorenz discovered the concepts of chaos theory in the 1960’s, it could have been that we were making tremendous scientific achievements and updating our understandings of things, rather than actually discovering a new phenomena itself. Feigenbaum’s constant tells us how fast a fluid can move from smooth to turbulent or, more generally, how a non-chaotic system becomes chaotic.

We can graph μ and x, which are two numbers that multiply to give a fixed value. By looking at different μ values that are close to one another, we can calculate δ, the Feigenbaum constant.

The Feigenbaum constant δ comes out to about 4.669 (or maybe we should just round up to 4.7).

And, since these chaotic systems have these special conditions, chaotic systems are unique, rather than a simple “update” of our current models of looking at things. But, enough about math and science for now, what about how chaos theory works in what we do? Can we use chaos theory to explain our own actions? Let’s take a look.

People like to debate ethical questions, such as: if you had the choice to kill baby Hitler and prevent everything in his future from happening, would you? It might seem immediately obvious to do so and change the course of history, but how do you know a different future (that might be even worse) would not happen instead? In order to answer a question such as this one, we would need to understand what could happen in different situations and scenarios of history, and, before we could do that, we would need an near-perfect knowledge of cause-and-effect of 20th century German history. So how do we do it?

Scholars might use human agency as a primary cause in the course of history’s events. David F. Lindenfeld cites on Henry Turner’s “Hitler’s Thirty Days to Power” as an example of how “empowering and constraining causes of specific human actions” helps us explain history. Lindenfeld takes this idea a step further by drawing an analogy with chaos theory in that we see very different large-scale outcomes by little small-scale differences if we can look at all the different causes and courses of history we could take. Killing baby Hitler is a small event that wouldn’t ever be recorded in a history textbook, but things could be massively different on a large-scale over the decades of history. We could look at everything that happened and figure out how things would be widely different by such a small action as killing baby Hitler to make a decision. There would no WWII, no Holocaust, no Cold War, nothing as we know it. But, if chaos can make sense in history, can we really use it in our own lives?

As I mentioned at the beginning, we easily ignore details in our everyday lives. And a lot of the things in our lives are unpredictable and unforeseeable. It’s very easy to apply the butterfly effect of chaos theory to what we do in optimistic self-help advice. We can tell ourselves that the little things we do everyday will have big effects on the future (e.g., reading for 30 minutes a day, waking up earlier, working out regularly, etc.), even when those effects seem unpredictable or random. But, while it may be true those things make big differences when we do them over time, it doesn’t make sense to say they are examples of the butterfly effect. Our lives might be uncontrollable and unpredictable, but, when we make big changes in the future, they’re not simply the result of different initial conditions (as chaos theory would suggest), but, rather, from making those decisions constantly over time. In these ways, chaos theory isn’t so helpful for our individual lives.

Chaos theory has emerging as an transdisciplinary, institutional, especially thanks to science historian James Gleick’s “Chaos: Making a New Science.” With applications and techniques crossing numerous disciplines, it could be part of a greater paradigm shift in our understanding of science (and the rest of the world). It likely won’t be as huge as an entire scientific revolution, but we can be sure it’s more than just a temporary trend. And, even though we don’t know what the future will look like, I like to think these small things will make a big difference.