Please no butterfly analogies.
In: Mathematics
In general, scientists are in the business of predicting what will happen to a system based on initial conditions. This could mean predicting where a rocket will go if it’s launched, or what chemical reactions will occur if things are mixed, et cetera.
Some systems are Chaotic, which means the result is wildly different depending on tiny changes to the initial conditions. Imagine putting a marble in a box and shaking it up. You could start the marble in the (almost) same position every time, and you could use a robot to shake the box the (almost) exact same way each time, but the marble is going to end up literally anywhere in the box each time and it will be very hard to predict where it will land with even the smallest bit of certainty.
It comes down to the fact that we lack the technology to do anything *exactly* the same two times (indeed, it might be literally impossible). Some systems (ones that aren’t chaotic) won’t care about tiny tiny differences and you will be able to predict them. Other systems, the ones that are chaotic, are immune to traditional methods and so a new theory was developed to try to predict them.
Chaos theory is just the name for everything we know about predicting the behavior of chaotic systems. It’s used because there are a lot of chaotic systems that matter in the world and we want to know what they will do.
An example of a practical use of chaos theory is making weather predictions.
Holy hell, these answers are not what I would tell a 5 year old.
In science and especially physics, we expect the same result if we do the same thing, and if I do two similar physical things, I expect similar things to happen as a result. If I put a bucket filled with water on a seesaw, the seesaw moves down; if I put a heavier bucket on a seesaw, I expect it to move down faster; if I throw a ball to my friend, it goes to them, if I throw it a little bit to the left, I expect the ball to go a little bit to the left. If something is “chaotic”, I might throw the ball a little bit to the left, and then the wind takes it and makes it sail off in some other direction. I can look at how the wind changes where the ball was going to go and say “yes, I can see why it did this very different thing”, and I could’ve done a better job predicting where the ball would go if I measured the wind, but we would not ordinarily expect the outcome to be so different from such a small change, even though we can account for the changes.
ELI10 A chaotic system is something where tiny changes lead to very different outcomes. The classic example is this, [a double pendulum](https://64.media.tumblr.com/ee2dddc9163caf566f2f747e2c05edc2/tumblr_n5r8wbYqFr1tzs5dao1_640.gifv). That gif traces how the pendulum moves over time. The two start at virtually the exact same position, when I release them and let physics take over, I can calculate where they will be exactly at every second, and briefly they look the same because of how similar their initial conditions were, but that small change snowballs, and very quickly the two follow very different paths. That is chaotic.
As for how is it used, there are a variety of systems to measure how chaotic a system is. A lot of natural processes we care about are chaotic, like the weather, so understanding the math of chaos helps us understand those processes.
In complex systems, a slight change in the system’s input can give very different results.
Take playing a game of pool as an example. The cue ball and all the other balls are ideally in the same places when you start the game. If you try to hit the triangle of balls in the same place when you start the game, you’ll probably see most of the balls move very differently each time.
This is an example of chaos theory in action. The subtle difference in how hard you hit the cue ball, how the balls in the triangle are slightly in different places each time you play, and a number of other things all snowball together to give you a very different pattern of how all the balls move around the table, even though your game’s setup was so very similar to every other time you set up a game.
Where this really builds up complexity is when the balls start bouncing off the sides of the table after the first hit sends them flying. A slight difference in speed and direction will cause balls to bounce off each other at different angles and sides, or even cause totally different balls to bounce off each other.
This means that near the start, all pool games you play are very similar. Even after you hit the cue ball and it hits the triangle of balls, you’ll still be mostly the same as every other game you’ve played. It’s the complexity of all the balls bouncing around and off each other that very quickly turns small differences into bigger and bigger ones.
Chaos theory is the theory that there exist certain mathematical and indeed natural systems that are highly erratic and diverse, to the point where they may seem to be totally random, but are in fact entirely predicted by the underlying equations. A [double pendulum](https://en.wikipedia.org/wiki/Double_pendulum) is one such system, a [dripping tap](https://www.nature.com/news/2000/001228/full/news001228-2.html) is another.
The issue is when such systems exist then an absolutely tiny change to the initial conditions can have massive effects. Two almost identical double pendulums will have completely different motions for example, or the absolute tiniest difference in how tight a tap is closed will be the difference between different drip patterns and indeed whether the pattern is regular or erratic.
Chaos theory has multiple uses. For example most modern cryptography is based upon mathematical processes that produce radically different and thus impossible to guess outcomes but which can be recreated if you have the exact initial conditions. In other words you need a system that looks random but isn’t. Which is what Chaos is. Various systems in biology and economics also involve chaotic behaviour: it looks random and is exceptionally difficult to predict, but it is responding to underlying equations, it’s just that those equations are highly chaotic.
Finally chaos theory is useful at the interphase between physics and philosophy to explain how our universe can be entirely deterministic (ie the end result of various knowable physical processes) but also contain such diversity and unpredictability. Tiny changes in the initial condition of different parts of the big bang can produce all the diversity that is the universe we see. And combine chaos theory and [Heisenberg’s uncertainty theory](https://en.wikipedia.org/wiki/Uncertainty_principle) which holds that it is impossible to perfectly measure the initial conditions, because on a quantum level you change the conditions by trying to measure them, and what you’re left with is an explanation of how the universe can be completely impossible to predict, and yet fully explained by laws that we understand.
Quantum magic aside, if you have 100% knowledge about state of system, you should be able to predict its future using physics. In practice, you never have 100%, you are limited by the accuracy of measurements you can take. Usually, this is good enough. Approximate information about the system allows us to make approximate predictions. For example, we may be able to predict that a cannonball will fly somewhere between 100-110m. We would know the exact number if we knew how every single air particle it hits on its way is behaving, which we don’t. But on the whole, it’s a useful prediction.
However, if a system is chaotic, predictions based on approximate data will not just be mildly inaccurate, they will be so wildly random they are basically useless. For example, the movement of single grain of sand in a sandstorm. It’s so light and surrounded by so many particles of similar size bumping into it, a tiny variation will send it off on a completely different course. It’s pointless to even try guessing where it ends up.
Some systems fall somewhere in-between: in short term, we can make reasonably good prediction, but the accuracy steeply falls off with longer-term predictions, until the system becomes completely chaotic. Examples of such systems are weather and the movements of celestial objects.
Edit: u/bluebell_sugarslay raises good points in [his comment](https://www.reddit.com/r/explainlikeimfive/comments/mj70p9/eli5_what_is_chaos_theory_and_what_is_it_used_for/gt97r70/?utm_source=reddit&utm_medium=web2x&context=3). I maintain my example is correct, though, if my explanation is a bit lacking. If we were talking about a single grain riding an invisible predetermined roller-coaster, so to speak, the system would indeed not be chaotic despite its complexity. However, the many grains of sand and air molecules themselves in a sandstorm interact with each other constantly, which I believe does make the system chaotic.
It is used mostly in mathematics to explain systems that are highly sensible to early conditions. Imagine throwing a marble from the exact same point 1 million times. The conditions of air resistance and other variables will change the course of the marble every time. Chaos theory is just that, scenarios that by having just slight early changes lead to completly different outcomes. Its studied because it can help us understand aerodynamics and particule movement more precisely
I try a true ELI5:
you know, when you drop a ball, it will drop on a flat floor and bounce up. If you repeat this, it does not reaaaaally depend a lot on that you do everything correct. roughly the same position and roughly the same height will give you a similar bounce.
now think about a floor that has a lot of wiggles, the ball will bounce and the direction it bounces of will depend on where it drops on the wiggles. Sometimes it will bounce high up, sometimes it will jump away. If you want to repeat a bounce, you have to be quite exact that you hit the right wiggle and at the same spot. But if wiggles are still large enough, you can do this with practice.
Some things are like this but with really fine wiggles and a very small ball. now it gets tricky to repeat a bounce and even if you try very hard, you might not be able to repeat a bounce.
We still need to understand problems like these, and Chaos theory is a way to describe problems where we can’t repeat an experiment exactly because we can’t be precise enough.
There are MANY very bright folks on here, who also how to explain what they know to other people not as knowledgeable. And then they’re willing to do it, without being paid. That’s a wonderful thing. Pay attention to people like that.
Chaos science is the study of how organizes systems breakdown over time. Chaos theory states that in all seemingly chaotic events, patterns of order emerge. Think of chaos theory like a yin yang. Orderly systems are always moving towards a state of chaos, chaos always inevitably forms patterns of order.
The longer in the future you try to predict,the bigger the chance of you being wrong is.
I think thats it
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