Stop 4: Realm of Chaos

Charlie Ahrendts

As we reach the penultimate stop on our journey through alien maths, we begin to feel a little lost in such a chaotic world. We find it hard to see any structure or patterns in the area around us. And as we talk to the aliens, we find out that as far as they know, everything is either random or chaotic on their planet. As a mathematician used to searching for patterns, this feels a little disappointing at first, but after some thought we begin to realise that as we have already seen on our journey so far there is a lot we can learn from them, and perhaps a few things they might learn from us in return…

Understanding chaotic systems is very difficult for us on planet Earth. We know that systems such as a double pendulum are governed by some rules, but we are very bad at predicting their outcomes. In the example simulation below the pendulums appear to be following the same path initially, before descending into chaos.

Image: Charleuzere

This concept of unpredictability extends to some very important fields of study including climate science, meteorology and economics. In each of these systems, a small change to the input can have a great effect on the output – often referred to as the ‘butterfly effect’. If we can learn how to work with these systems many aspects of our science would no doubt profit. 

We also recall that the aliens said that not only is their planet chaotic, it is also random in its very nature. If we toss a coin on earth 100 times, we will get around 50 heads and 50 tails. If we repeat this experiment infinitely many times, both will come up exactly 50% of the time. From this we can deduce that the probability of getting heads on a coin toss is 50%. But, in the Realm of Chaos where every event is truly random in nature, the outcome of our experiment will differ every time. Therefore, it would appear to make no sense to talk about probabilities in this, or any other scenario on the planet.

This has some interesting implications for the world of physics, and in particular for quantum physics where our current understanding is based entirely upon probability. There are three possibilities of how the aliens could deal with this issue. The first would be that they just ignore it. This would be an understandable reaction to such a situation, but not very interesting for us to explore. The second possibility is that they mirror the way we handle chaotic systems. For a long time, mathematicians were convinced that there could not be any truly chaotic or random things in our natural world. But, as our mathematics developed, the field of chaos studies emerged and is now perfectly accepted as a valid branch of mathematics. The aliens could in a similar way introduce the new field of probability theory. They wouldn’t necessarily need to fully understand it to still be able to explore quantum physics.

The third way they could handle the quantum physics problem is that they find a different explanation or model that is consistent with their fundamental axioms. Maybe we are just so used to thinking in terms of probability that we interpreted our findings under that premise – there could be another coexisting explanation we just haven’t yet worked out. 

In this context I would like to talk a little bit about philosophy. This may seem odd in a text about mathematics, but the two fields are in fact much closer than you first think. Up until about 200 years ago, many mathematicians and physicists were also philosophers and vice versa. There are many philosophical questions we could examine here, but one of the most interesting, as well as disputed, is whether the universe is deterministic. Many scientists believe that if you know the state of every single particle in our universe you could perfectly predict the exact future. There are certainly good reasons to believe so, but we also have to acknowledge that many of them stem from our understanding of perfectly predictable classical mechanics. A lot of original defenders of this theory even changed their minds once quantum mechanics was discovered. An interesting question is to ask ourselves how the inhabitants of the Realm of Chaos think about determinism? They are used to things being fundamentally random, so why would they expect the universe to be any different? 

I find it truly fascinating how such a small change in a specific field of mathematics can fundamentally change how we see the world.

Chaotic systems

Most of us will likely have a very good idea of what chaos is and what it looks like. However, when we analyse it from a scientific viewpoint, there is a lot more to chaos than you might first expect. In the remainder of this article we’ll see exactly what this means with the example of an equation that was meant to model population growth, but ended up becoming one of the most prominent examples of chaotic behaviour…

Imagine you are a researcher that wants to model a population of mice. We know that the amount of mice present next year depends on how many there are this year. Every mouse will on average birth a certain number of offspring. If this number is high, our population will grow quickly, and if it is low it will grow slowly. If we put all of these conditions together, we end up with the following equation:

where xn+1 is the population the following year, r is the growth rate of the population and xn is this year’s population. 

You might, however, have noticed a flaw in this formula. The population never stops growing. If this were true, we would have millions of mice all over the place, which thankfully is not the case. The problem with the equation is that it does not account for limiting factors such as resources, available space, or the death of older animals. In nature, populations tend to stabilise after some time. Remember this idea, it will be important soon. 

With this in mind we may decide to add a term that corrects for these limiting factors. Our new and final equation thus looks like:

We call it the logistic map (where map is just another word for function). To begin to investigate its behaviour we first need to define the maximum possible population a given environment could sustain. Let’s say this is 400 mice, which represents 100%. We are now no longer thinking of x as a number, but rather a percentage of the maximum value. To see how this works let’s do a quick calculation. Suppose our growth rate is 2, which means that if no mice die, the population will double every year. Our starting population is 100 mice, which accounts for 25% of the maximum. If we plug these numbers into our equation, we get the following:

This means that in year two, our population has reached 37.5% of the maximum, or rather 150 animals. If we repeat this process a few more times we will see that at some point the value for the following year will no longer change and thus we have reached stability. In the case of our mice, this occurs after around 5 years where the population stabilises at around 200 mice. And this in fact occurs regardless of our starting population x0. The only relevant factor that determines where the population will stabilise is the growth rate r. If r is smaller than one, the population will eventually die out, since less new mice are born than there are old ones dying.

Now you might be wondering what all of this has to do with chaos. And the truth is so far almost nothing. However, things start to change drastically once we increase r further. Up until r=3, the population stabilises just like in our example above (we had r=2). Once r gets higher than three however, we notice that the population starts to switch between two values where these two values stay consistent. If we increase r further we will find that our population goes through a four-year cycle, which means that every four years the same cycle of four different population levels occurs, and this is repeated over and over again. If we increase r even further, we reach stable systems of 8, 16 and 32… year cycles. 

This pattern continues until we reach the magic number of r = 3.56995, upon which we enter chaos. From this point onwards the population will change from year to year without any kind of predictable pattern. It is chaotic. The mathematical definition of a chaotic system states that a small change in the input to a system will cause a great change in the output. If we were to vary the initial size of the population, or the growth rate, even a tiny amount, the resulting pattern will change drastically. So, if we wanted to predict next year’s population we would need to know the exact starting conditions. In other words, there is no way to approximate the outcome. 

Bifurcation diagram of the logistic map showing the initial cycles, followed by chaos.

The logistic map was one of the first known cases where a well-defined function was shown to lead to chaotic behaviour, and as such it was often used as a source for random numbers. 

Chaotic systems are often something we are trying to avoid since their unpredictable nature makes them very hard to understand. However, we can find chaotic behaviour in many places including weather patterns, fluid flow or stock market dynamics. So, we might need to excuse the weather forecast for being wrong every now and then, they are literally trying to predict something which is mathematically defined to be unpredictable – chaos.

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