Chapter 3 – Chaos

A late January paddling of the Okefenokee Swamp, in southeastern Georgia, is an otherworldly experience. It seems miles away from anywhere, because it is. At almost 300,000 football fields in area, there’s nothing around it. No planes fly overhead, no power boats around, only a few people, birds of many species and a number of alligators sunning themselves on the banks. The tannin-stained water flows slowly, almost imperceptibly somewhere. But where? The swamp drains into two rivers, the St. Mary River and the Suwannee River – made famous in the Steve Foster song. The St. Mary empties into the Atlantic Ocean north of Jacksonville, FL. The Suwannee River empties into the Gulf of Mexico north of Tampa, FL. Almost three miles separates these two end points but as we were paddling through the swamp, we came to the Okefenokee divide. This is the “fork in the road,” If you go to the left you travel the St. Mary’s River and end up in the Atlantic Ocean. If you go to the right, you travel the Suwannee River to the Gulf of Mexico. A millimeter difference at this decision point makes a big difference in where you end up – Atlantic Ocean or Gulf of Mexico.

Sounds a lot like some of the decisions you made in your life. Did what seemed like a little deal turned into something very big, bigger than what you imagined? You had more than one of these forks in the road. In fact, anyone who takes the time to look back on their life’s turning points will find that some of them are every bit like the Okefenokee divide. How do life’s decisions end up looking the same as the physical flow of water in a swamp? As we discussed in the last chapter, although there are different ways to interact throughout the hierarchy of life, the end result of interactions is similar. Water can’t think and doesn’t make a decision to go left or right at the divide, a decision is made for it. Doesn’t matter if you decide or someone (or thing) decides for you. At those inflection points in life, a small difference at one point can lead to a big difference later on.

We’ve now talked about hierarchy and how the universe builds up layers of complexity. We followed that with the U-ROC, the only way things change is through an interaction – an exchange with something else that changes both sides of the interaction. We then saw that simple rules for interaction can lead to either simple changes or complex changes. To complicate matters, it wasn’t clear from the rules how the pattern would turn out – simple or complex. The only way to find out what happens is to wait and watch the interactions and see what develops. In fact, it has been proven that for the complex case there is no way to write a mathematical expression that will predict the state after some number of interactions. So mathematics won’t help us predict the future state. However, we do have some things in our favor. Remember, in the examples of Wolfram, out of the 256 possible types of interactions, only 10 of them lead to complex behavior. That’s why patterns dominate the universe. Everywhere we look we see things in a pattern. The sun rises in the east and sets in the west on a regular basis. The moon phases go through a 28 day pattern from new to full and back again. The seasons go from spring to summer to fall to winter and back again.

We need patterns in order to have some type of stability in the universe. As Kasey Kasem said at the end of his American Top 40, “Keep your feet on the ground and keep reaching for the stars.” Of course, if all we had were patterns, we’d live in a world like the movie Groundhog Day. The pattern would get locked in and we’d repeat it over and over again, with little or no change. I believe that if there were only simple interactions, life would have never come into being. We need that complex behavior to drive new levels of hierarchy. Before we get into the emergence of new hierarchies, let’s talk a little more about chaotic behavior. Unfortunately, the term chaos has multiple meanings and the one we initially think of is wild and out of control. In mathematical thought, chaos has a different meaning and is more related to the example at the beginning of the section – a small change can lead to a large difference. Chaos was studied for a while before it was officially called chaos and Edward Lorenz was one of the first people to start a methodical approach to the subject of chaos and the subject of the next part of this section.

I have loved weather all of my life. When I left my first job (an epic failure) out of college, I started on the path to a Masters Degree in Applied Math at the University of Maryland. This degree required three areas of concentration, one of which had to be outside of the mathematics department. It took me only a few seconds to declare Meteorology as my area of application. My Master’s paper was on the work of Edward Lorenz. Lorenz was writing computer simulations of a simplified weather system since the early computers he was working on couldn’t handle the more complex equations. The two most important drivers of the Earth’s weather are the Sun (heat) and the Earth’s rotation. Lorenz has put together a simple apparatus that modeled this behavior. He took a turntable and placed a pan of water (more like a Bundt pan with a hollow part in the middle) on top of it. He put a heater under the pan of water and had himself a simplified weather system. As the turntable rotated, the water spun around and as the heater warmed the water it rose from the bottom of the pan and circulated. The combination of the heated, vertical rotation and the horizontal rotation lead to complex behavior that was similar to how the Earth’s weather works (at least the large High and Low pressure systems we see moving across weather maps). The first thing to note is how two simple behaviors (vertical heat rotation and horizontal rotational motion) can combine to generate complex patterns. And the patterns Lorenz saw in his simplified weather model were fascinating to him.

Lorenz created a computational model that ran on his computer system and printed out results every hour. He compared the results of his computer models with his experimental turntable system to validate the model. On one of his runs, he stopped the computer simulation after a number hours and later on decided to continue the model. He didn’t want to re-run the entire simulation from the beginning so he picked a time near the end of the first run and typed the numbers that were printed out from the first run as starting values for the next simulation run. As the computations continued and he compared the new results with the end of the first run he saw that although they matched closely early on, the results he got diverged pretty quickly. He initially thought he had entered some of the numbers incorrectly but after he verified that he had correctly entered the numbers he had to dig deeper. The only thing he noticed was that the numbers that were printed out had more digits of accuracy than he had entered. (So if the printout said 1.5434 he only typed in 1.54.) It turns out that that a small difference in input caused his model to generate wildly different results.

Lorenz was not the first, but he certainly made a study of what we now know as chaotic behavior. A small change in initial conditions can lead large differences later on. Is everything set of interactions in the world chaotic? No. Although the universe as a whole is chaotic (A recent study has shown chaos effects in the quantum world.) there are times when the chaos is small or non-existent. Something is chaotic depending on the interaction rules for that system. As we saw in the last chapter, some rules of interaction lead to repeating patterns and some lead to complex behavior. It is there complex systems that can exhibit chaotic behavior. If you add in the hierarchical effects you can have a repeating pattern at the highest level of a system but a series of chaotic behavior in the levels below. Combining these things leads to systems that have patterns most of the time, but a small perturbation (or change) leads to radically different results. This has been termed the butterfly effect for the idea that a butterfly’s wings flapping in South America could lead to the formation of a hurricane in the North Atlantic. The Butterfly Effects sounds almost comical, but remember that at the Okeefenokee divide, a difference of a few millimeters leads to a leaf going to either the Atlantic Ocean or the Gulf of Mexico – a difference of hundreds of miles.

I’d like to leave our discussion on chaos with the topic of attractors. Let’s look at two pendulums. One is a standard pendulum, a sting with a weight on the end. If we push it, it swings back and forth and slows down as friction saps its energy until it comes to rest. Having the weight hanging straight down is an attractive state. It is the state that the pendulum will ultimately return to as it swings back and forth. Now let’s look at a pendulum where the weight at the end has a magnet. At the base, right where the weight would hang, let’s place another magnet with a same polarity as the weight. That means that as the weight swings, the magnetic repulsion causes it to “bounce” around crazily. Below is a “video” of the weight when viewed from above and from the side.

Standard Pendulum:

Magnetic Pendulum:

In the standard pendulum, the weight passes through the attractive state (called an attractor) but when we add magnets the repulsion causes the weight to fly away from the attractor. In the magnetic pendulum, this is called a strange attractor since the system approaches the attractor but always misses it. Strange attractors come into being when the high level of the hierarchy is periodic but some of the underlying pieces of the hierarchy are either chaotic or the interactions from the lower level lead to complex interactions between the two levels. In the case of the magnetic pendulum, we have combined a simple, periodic system and a magnet, which is a simple system. Without the pendulum, the two magnets would just repel each other. Without the magnets, the pendulum will swing back and forth. The pendulum constrains the interactions of the magnets (without the pendulum, the magnets would repel each other but the pendulum motion keeps pulling them together) and leads to chaotic behavior.

The image below shows the movement of a slightly more complicated system where there are three magnets on the board that each attracts (rather than repels) the magnet on the pendulum. (In the graph the magnets are colored red, green and blue.) Notice how the pendulum moves between the three magnets in an apparently random manner? Each of the three magnets on the board represent a strange attractor and the pendulum visits each of them, based on the initial position and speed at which the pendulum is launched.

Systems of interactions like the ones I have been discussing come under the heading of discrete events. Discrete is as opposed to continuous. What I’m saying is that the universe is built upon interactions that are discrete and happen in a serial fashion. (That’s at the quantum level.) We see a continuous universe because we far up the hierarchy and the discrete nature is hidden. Ultimately, the force that ultimately yields the continuous world we live in is gravity. We’ll have a lot to say about gravity in the next chapter but for now, suffice it to say that gravity smoothes out the bumps of the discrete quantum world and presents to us a world where change is continuous and smooth. Why do I mention discrete events? It is because it has been mathematically proven that systems built upon interactions are non-computable. That means there is no way to write a mathematical equation to predict the exact state of the system after 100 or 1,000 sets of interactions. You have to run the 100 or 1,000 interactions and then you’ll see the state of the system. Now notice I put the word exact because reality is a little more complicated. I mentioned that gravity smoothed out the discreteness of the universe. That smoothness also allows us make predictions about this smooth system and that is how science works.

As much as scientists talk about understanding the way things work, the reality is that science is built upon the ability to predict the future. I would argue that the main reason science exists today is because of its ability to find patterns in systems and use that pattern to predict the future. How does science work? Science builds models of the system they want to study. They do that by ignoring some of the features that they feel is unimportant and won’t affect the overall pattern of the system. George Box, a statistician of some stature, made the seminal remark on this topic – “All models are wrong, some are useful.” Gravity has done science great service by smoothing things out and making it easier to build models. For example, pretty early on in mankind’s existence we started building models of the solar system and stars. The earliest models were Earth centric, with the sun, planets and stars all followed circular orbits around the earth. A circular orbit was used for religious and philosophical reasons as a circle is a more perfect shape. This model was able to make predictions of a number of events but over time it became harder to get a circular, earth-centric model to predict celestial events. Kepler came up with a different model with the sun at the center of the solar system and planets orbiting in elliptical orbits. The circular model of the solar system worked because a circle is a special case of an ellipse. Specifically, you measure how “oval” an ellipse is by a value called eccentricity. Looking at the three figures below, the one of the far left is a circle with an eccentricity of zero, the one in the middle has a slightly larger eccentricity and the one on the right has a much larger eccentricity.

Figure: Ellipses with increasing eccentricity.

I want to emphasize that Kepler’s model of the universe is wrong but it is much more useful than the earlier models that incorporated circular orbits. It is more useful in that it more accurately predicts the orbits of the planets.

Kepler’s model is indeed quite useful because while it ignores a lot of things, like the quantum behavior of the matter that makes up planets) the gravitational attraction between the sun and the planets is many orders of magnitude more important than any other effect. That means the model can predict the future paths of planets, moons and comets to a great level of detail for a long, long time. The lower levels of this hierarchy are chaotic, but it will take millions upon millions of years before they affect the motion of the planets in any measurable way. At some point in the future, the sun will use up its fuel and start to expand and eventually explode. Or we could have some interstellar object crash into one of the planets and muck up our predictions. At that point, the Kepler model will no longer be useful as the predictions it makes will be way off, But let me state it again for emphasis, Kepler’s model of the solar system has always been wrong.

We can state with conviction that the usefulness of a model in making predictions will decrease as the system being modeled becomes more chaotic. We all have our favorite weather story where the prediction just a few hours ago didn’t pan out. As a child growing up, there was a time when it rained on one half of my block, but not on the other side. No weather prediction model can figure that out. In general, the more chaotic the system, the shorter the time frame where a continuous model will be accurate enough to be useful. But what’s the alternative, do no modeling? That’s not a good way to enhance our survival. Predicting the suture grants the accurate predictor a competitive advantage so it is in our best interest to continue to build models and predict.

Science has made its name finding systems where it can build useful models. Over the past hundreds of years, science has started with the simpler problems (like the solar system) and as time progressed they got better at approximating more and more complicated systems and making more and more useful models. At the quantum level, however, scientists have resorted to statistical modeling in order to gain predictability. They count on the fact that all electrons are alike. (In this day and age, there would only be a single Facebook page that would apply to every electron.) With the assumption that all electrons are alike, you can build statistical models that yield impressive predictions. Any model that took individual electrons into account would be impossible to run in this day and age. Perhaps with time we’ll gain the computer computational power to improve the modeling of quantum systems.

We’re now ready to discuss how the hierarchies develop.