Let’s predict chaos.
Hahaha fooled you! Nevertheless, I advise that you stick around so we can dive into this topic that is pretty fantastic, beautiful in its own way, and essential to our very existence.
So, chaos, where to even begin? I feel like this post itself is going to be chaotic as hell, so bare with me, I shall try to explain it in an understandable manner and make it digestible, but much like my own mind and its thought processes, trying to describe them to someone, and order them is incredibly difficult, my mind is just a blazing and bubbling universe of constant questions, inquiries, theory crafting etc. etc.
Let’s start by describing what chaos is. Chaos is described as a phenomenon in which you have seemingly no order or any predictability, at least to a reliable degree, in a given system. The most famous description of chaos you most likely know and have heard is “The butterfly effect”, with the equally popular description of “Can the flap of a butterfly's wings in Brazil set off a tornado in Texas?”, and just like with chaos itself, the answer to that question is we don’t know. It’s unpredictable, at least with our current technological capabilities and understanding of the universe.
Chaos is typically described by analyzing the three following points of a given system.
it must be sensitive to initial conditions,
it must be topologically transitive,
it must have dense periodic orbits.
The first point describes how sensitive a system can be to its starting conditions, take for example the action of rolling a ball down two absolutely identical tracks at the same time, if you let the balls go in an absolutely identical manner, you would naturally expect them to go through the tracks in the same manner right? Now take one of the balls, and apply the most ever-so-slight extra push to it, that one seemingly tiny action alone is going to get magnified over time and as the balls go through the track, eventually, the balls will be going through the tracks in completely different patterns, speeds, etc. And the harder of a push you give to one of the balls, or the more energy you put into the system in any other way, the faster and more drastic those developing changes will be.
The second point describes how if given enough time, a system will cycle through absolutely every state possible, so going back to our ball analogy. Imagine that as the balls are rolling through the track, they do not lose any speed or energy ever, if you give them enough time, they will have stumbled their way through absolutely every single centimeter and corner of the track.
The third point describes how chaotic systems must repeat a specific pattern frequently, so again in our ball analogy, you can imagine this to be the pattern of the balls rolling to one end of the track, and then back to the other, so on and so forth, or anything that follows a simple and frequent pattern. It in itself is a pattern that is easily observable and repeatable in that chaotic system.
Our universe is covered from head to toe in chaotic systems, the blood flowing through your arteries is turbulent, which is another chaotic system, the air flowing through the air is chaotic and turbulent. Another very easy system of chaos you all have been familiar with since birth, is weather. It is a chaotic system as per the aforementioned butterfly effect description as well. So you might ask, how can we forecast / predict the weather? Well we shall get to that a bit later, that is also where one of my proposals lies.
Believe it or not, a pendulum swinging back and forth can also be a chaotic system. How? Well, if you add another pendulum to the pendulum, pendulumception! If you are not familiar with the double pendulum system, here is a little animation that describes it perfectly. It’s one of the simplest chaotic systems we currently know.
An animation of a double-rod pendulum
As you can tell, it’s incredibly chaotic and unpredictable, and it can also be used to easily describe the first hallmark of chaotic systems, which was sensitive dependence on initial conditions, if you took two double pendulums, and let them both go at the same time, except one of them is a micrometer higher or lower than the other, then eventually they will be swinging through completely different patterns.
Unlike what scientists might seem to think or do, I find chaotic systems to be very easily understandable, in that I have come to accept that you are not supposed to try and force our logic and deterministic ways of understanding into it, I mean it literally makes 0 sense. It's like embracing the mystery and beauty of the universe's complexity, rather than trying to force it into a box of complete understanding. Chaos can even be found in the world of elementary particles like atoms, take Erwin Schrödinger’s wave equation, which governs the behavior of atoms or other elementary particles. It does not try to predict their exact position constantly, since that is not the point nor is it possible, it predicts their evolution through time in a range of states / numbers, the PROBABILITY, as long as the particle is unobserved. Maybe we need to apply the same concepts to chaotic systems? Since to me, chaotic systems of all scales, and the behavior of elementary particles in the quantum realm have a lot of similarities as you might now see as well.
As far as I have seen, we have tried to predict chaotic systems with rigid numbers and equations, and try to predict them perfectly into the future, which is obviously not really possible. Maybe to help us better understand very chaotic and unpredictable systems like the double pendulums, or another classic example, three-body systems like for example three planets orbiting each other, unlike two planets or stellar bodies orbiting each other which is predictable, three of them orbiting each other is very unpredictable. You can try imagining it yourself and see how your brain starts to hurt and says “no I don’t think I will”.
If it weren't for all of this chaos and indeterminism, such a complex and vast, rich universe that we have right now would be literally impossible, in a paradoxical way, so I don’t get why scientists are so desperate to understand it, yes we could apply it to our lives and advance ourselves as a society, but scientists seem to be so eager to figure it out that they stumble over themselves and in my opinion have overlooked some quite big pieces of the puzzle. As I described before they try to apply rigid, predictable numbers and concepts that adhere to logic and determinism, to describe a system that is literally by its very nature and existence, illogical, chaotic, and indeterministic.
In order to fully understand it, you would exactly have to accept that you don't understand it, that you can't understand it, just like how I describe the world of quantum to people, by saying that if they felt like they didn't understand anything and that it made absolutely no sense, then that's great and exactly means that they understand it. It DOES make no sense and is illogical by nature, as described before, it has to be.
Let me describe you how I imagine the three-body system in a thought experiment, and hopefully I can break it down so you can understand it better too. Unlike what it did to Isaac Newton, gave him headaches and kept him up at nights, I found it to be very easy to understand. It's quite logical actually once you read my description. So take the fabric of space and time and the concept that mass and matter can bend it in certain ways, depending on many factors, as you know if you have read any of my previous blog posts about cosmology. Imagine those three planets coming closer and closer to each other, you can throw absolutely every other variable in the system out, and only have the fabric of space, and the planets themselves, and gravity. That's it, and even with such a simple system, you can get chaos. Because when the planets are coming closer to each other, they start to pull more and more on the sides facing each other, than they do the sides which are facing further away, a concept which we already know to be an incredibly extreme version of it is spaghettification in or near black holes. So because of just that one small variable, you can get chaotic and unpredictable motions in a three-body system, if all of the planets would pull on every single atom of each other with the exact and opposite force, then you could incredibly easily describe it with basic Newtonian physics, specifically Newton's third law, that every action has an equal and opposite reaction. Described by this equation: T = W = m g. Which MIGHT be familiar to you if you paid attention in physics class in the latter couple years of high school.
Want to know the only solution I can come up with right now to perfectly predict chaos? It's sort of the most obvious one. You would have to know the state of EVERY single nucleus of the atom, EVERY single electron, EVERY single atom, EVERY single molecule, EVERY single interaction they had with each other and with themselves, etc. etc. Scaling it all up until you get the entire planet, taking into account everything else that is outside as well like any matter floating around outside of the planet, or the shape of the planet, the density, etc. etc. And you would have to be able to understand and gauge all of those variables down to an infinite decimal place of time and speed, only then can you reliably predict such chaotic systems always. But obviously doing that, at least right now, is impossible.
To go back to my previous example of Schrödinger’s wave equation, in order to formulate his famous wave equation, Schrödinger added imaginary / complex numbers into his equation, symbolized by the symbol i, and then suddenly it made sense. Suddenly it became this wave equation, instead of just a heat conduction equation. Maybe we need to do the same with chaotic systems? Instead of trying to add rigid numbers and concepts as previously mentioned, we need to add imaginary and complex numbers into the system, because as should be evident by now, the universe works with complex numbers, and not real numbers. Only by giving up math’s connection to reality and logic, could it ironically guide us to a deeper and more fundamental understanding of reality. Maybe we should be doing the same exact thing with chaos theory? Give up its connection to reality and logic. You will never be able to describe their exact states or predict them perfectly into the future, but maybe you can get a bit better understanding and predictability for them, at least to a certain degree.
To also go back to us being able to predict a chaotic system like the weather, at least some distance into the future. It turns out that some chaotic systems are easier to predict than others, as you may have also picked up during this blog post. So another one of my proposals is to assign something called a “Prediction score” to chaotic systems, so stuff like weather and planetary orbits would get a high prediction score, and stuff like turbulent flow in fluids or any other medium, three-body systems, double pendulum, etc. would get a low prediction score, in that you cannot reliably predict and evolve those systems out into the future. I feel like that would help both scientists and regular people understand how much they should trust in a prediction of a given chaotic system, or whether they should trust it or attempt to predict it at all.
To go back briefly to the term “Butterfly effect” too, to show you where exactly it comes from, let me show you another famous chaotic system called the “Lorenz attractor”
An image of the Lorenz attractor
Looks familiar right? The Lorenz attractor is a set of chaotic solutions of the Lorenz system. In popular media the "butterfly effect" stems from the real-world implications of the Lorenz attractor, namely that several different initial chaotic conditions evolve in space in a way that never repeats, so all chaos is unpredictable. So to make it a bit more understandable, you could think of it as the most chaosy of chaotic systems, and to apply my predictability score proposal, this would receive the lowest score possible.
And I think that’s it! So in conclusion, there is no way to predict chaos reliably, especially not some chaotic systems like the aforementioned Lorenz attractor, however there are definitely ways where we can better understand, and accept chaotic systems, if we simply abandon their connection to our linear and dumb logic and reason. I hope you enjoyed this post and found it informative and easy to understand! Also side note, the reason it suddenly took me a minute to come up with another blog post, is because not going to lie, I ran out of ideas what to write about lmao. Nothing jumped up at me as interesting or engaging enough to write about, until I found myself last night pondering about chaos, and making progress on it, which I now laid here. Hopefully there wont be too many of such moments in the future, you’re also welcome to give me ideas as to what to write about!
