Chaos Theory, Unpredictable Predictability

Every once in a while a single person, questioning long held beliefs and assumptions in one area of science, can radically change our understanding of the Universe. Edward Lorenz (1917-2008) was one such individual. A mathematician with an interest in meteorology fostered while in the US Army during World War II, after returning home Lorenz transitioned from being an Army meteorologist to a mathematician researching the field. Sometime in the 1950s he began to question whether or not the simple equations he and his colleagues used to predict the weather were physically accurate. In 1963 he published a research paper that not only overturned conventional scientific understanding of weather forecasting, but created an entirely new field of mathematics: chaos theory.

Chaos theory is one of the most complicated and confusing areas of mathematics, but it’s also one of the most incredible with wide ranging philosophical implications. This may get a bit heavy, but stick with me, it’ll be worth it.

Chaos theory involves systems (or equations) that evolve over time, and are very sensitive to their initial conditions. We’ll consider a non-chaotic system as a comparison. Let’s say you throw a ball parallel to the ground at a speed of 10 meters per second, which is pretty quick, but in line with Little League fastball speeds. Using simple equations we can determine how far the ball will go before it strikes the ground (being pulled down by Earth’s gravity). If instead of throwing the ball at 10 meters per second you throw it at 10.1 meters per second (a change of 1%) then the ball will more-or-less fall at the same point. The equations that govern how far a thrown ball will travel are not “sensitive” to their initial conditions: a small change to the input (the throwing speed) results in a small change in the output (the location where the ball hits the ground).

What if that 1% change in throwing speed resulted in the ball traveling twice as far, and a 2% change resulted in the ball hitting you in the face? Such a thing seems non-intuitive and it is not clear that such systems can even exist. What Lorentz discovered is that such systems do exist and that weather patters are just one of them. For chaotic systems, almost imperceptible changes in their initial conditions result in large differences to their end state. To illustrate how chaotic systems behave, we’ll look at one of my favorite examples: the double pendulum. A typical pendulum is a heavy weight attached to the end of a string that can swing freely, but we’re going to make it even simpler than that. Replace the string with a stiff rod, that way we don’t have to worry about the string going slack at any point. Let’s also make it so that the pendulum can only swing back-and-forth (it can’t swing at you), this makes the pendulum motion easier to visualize. We will also neglect any friction or air resistance, but will of course include gravity to make the pendulum swing. None of these assumptions make the system more or less chaotic, they simply make it easier to see what is happening.

A single pendulum is not a chaotic system, it is in fact very periodic and predictable. However, if on the weight of the pendulum we hang a second pendulum then the entire system changes. Now the first pendulum has an extra force pulling on it as a result of the second pendulum hanging from it. As the second pendulum swings it will pull the first pendulum back and forth. I’ll spare you the math involved with writing up the equations that govern this system, but I’m betting you understand that if we can write the equations governing one pendulum then we can write the equations for the double pendulum. These equations can be programmed into a computer to make pretty pictures.

We’ll start our double pendulum like this:

 

The red ball is the weight of our first pendulum, the blue ball is our second. We’ll trace the path that the blue ball takes by having it draw a blue curve, and it is this curve that will show chaotic behavior. If we release the double pendulum from this starting condition we can watch the system evolve in time.

This system involves six variables: each pendulum has a length, mass, and starting angle relative to the ground. We could add two more variables if we were to give the pendulums any kind of push when we release them, but we’ll ignore that for now.

We’ll call the double pendulum above System A, and in System B we will change the angle that the blue pendulum makes with the ground by 1%. In the next pictures, System A is on the top and System B is on the bottom.

The systems look identical, and if we were doing this experiment in a lab we might think they were identical. The equations that define the double pendulum system are perfectly defined, meaning we can take starting conditions and solve for the state of the system at some arbitrary time in the future. With the double pendulum the end state we are solving for are the locations and angles of both pendulums. But here is the catch with chaotic systems: a pair of initial conditions that are extremely close will result in completely different results if we go far enough into the future. This makes chaotic systems simultaneously predictable and unpredictable: the math says they are predictable, but in practice we can never measure initial conditions accurately enough to predict the correct response. To the naked eye the starting conditions for our two pendulum systems look identical, but how long with their behavior look identical? Let’s set them in motion (via our computer model) and find out.

There are 111 frames in the above animation, but by frame 21 we start to see subtle differences.

By the end, the path traced by the blue pendulum in each system is very different.

But what does all of this actually mean? How does this change our understanding of the Universe?

For a very long time, science was obsessed with the idea of a deterministic Universe, an idea that states that if we could understand all the physical laws of the Universe then we could predict everything that ever was or would-be. Think about that thrown ball again. If we measure the position and velocity of the ball at some point while it is flying through the air then we can – using math – predict where it will travel and where it came from. If that is possible (and we know it is) then the thinking went that scientists could do that same thing for the entire Universe. Measure the position of every atom in the Universe and you would tell the entire history – past and future – of the Universe. Impossible? Yes, of course, but it was a scientific theory that was true in principle. This was the thinking that Lorenz confronted in regards to predicting weather patters. The prevailing theory at the time was that if wind speeds, moisture levels, etc. were measured accurately enough then weather could be predicted months in advance. In his 1963 paper Lorentz shattered that notion by demonstrating that we could never measure even a single variable accurately enough to predict very far into the future.

Remember that comparison of frame 21 above. Though they are different, System A and System B look very similar at that point. Chaotic systems evolve over time and are very sensitive to their initial conditions, but if those initial conditions are close to one another then they won’t evolve along different paths until later. This is the new truth of weather forecasting that Lorentz brought about; we can have very accurate weather models for near-term forecasting (days or weeks) but long-term forecasting (months) is physically impossible in the real world, despite what the Farmers Almanac will tell you.

The implications of Lorenz’s observations on weather prediction rippled through the mathematics and scientific communities. In his twelve page research paper, Lorenz contributed to the downfall of the dream of a completely predictable Universe. Some aspects of our Universe can be predicted with accuracy far into the future, but there are limits to what types of things can be predicted buried in the framework of the Universe itself. In trying to predict everything, scientists proved that they can’t predict everything, and there is something extremely beautiful about that.

 

About Andrew Porwitzky

Dr Andrew Porwitzky is a scientist, comic book junkie, and freelance writer who is on Twitter way too much.

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