The main stuff
So chaos theory is a pretty interesting topic (to a nerd like me) to talk about, especially since it pops up in so many places that most people don't know or care about. But if you're reading this, I have either forced it upon you, or you have a genuine interest in this stuff.
Introduction
So what is chaos theory? Well in scientist jargon, its "the study of apparently random or unpredictable behavior in systems governed by deterministic laws." What that basically means is, you have a system that's supposed to work normally and follow the laws of motion and gravity and all the things you learned in high school physics, but instead it just goes crazy and, well, chaotic.
Predicting the unpredictable
The main thing about these chaotic systems is that they start out seemingly predictable (to give you false hope), and then take a left turn and go completely insane and unpredictable. "How do you predict this, if its all random?", asked someone a long time ago. Well, there's 3 things that determine how long you can predict it:
How much uncertainty can be tolerated in the forecast? If you want Planck-length-perfect measurements, then your predictions aren't gonna hold true for too long, so keep a fair margin of error in your prediction, but not so much that it contradicts the second point.
How accurately is its current state measured? If you're measuring chaotic motion of a double pendulum rounded to whole numbers of meters, again, your predictions aren't gonna last very long. So be accurate too, but not so much that it contradicts the first point.
Lyapunov time. This is one of the more complex things that my brain couldn't comprehend, but I'll do my best to explain it. Due to the uncertainty in forecasts, there is always a (small) error in the prediction made from the actual state of the system after some time. And as we all know, that small error can get really big over time, hence why the system becomes unpredictable over time. Lyapunov time is the the time for the error to be multiplied by a certain factor, like 2 or 10. Basically it measures the rate at which things get chaotic. Now, this is necessary because if the Lyapunov time is big, then your predictions will be accurate for a longer period of time and everything will be calm. But if it's small, then things are gonna get crazy, quick.
Chaos theory in action
Butterfly effect
This one is actually quite popular and you may have heard of it, but in case you haven't, here's what it is. The butterfly effect states that the flap of a butterfly's wing could, eventually, form a hurricane on the other side of the world. Which is a prettier way of saying, small actions can have big consequences. That's not just life advice, it's science.
Pinball
Your favorite arcade game, unless of course it's not. Pinball is an excellent example of chaos theory, because everything should be predictable. You know how rolling works, and how collisions work, and how to hit something to get it to a certain place. These are all things from high school physics. And yet, no one knows where the ball will end up and the game gets more and more and more unpredictable and random over time.
Double pendulum
Imagine a pendulum. It swings back and forth in a regular motion (called simple harmonic motion). Now, stick another pendulum to the free end of the first pendulum. This contraption is called the double pendulum. And it is nothing at all like the normal pendulum. Sure, it starts out normal, but after just a short amount of time, it starts flopping around all over the place and follows no pattern at all. It becomes chaotic. The double pendulum is actually one of the most popular demonstrations of chaos theory, probably because not everyone has a pinball machine.
Conway's Game of Life (source: trust me bro)
I don't have any sources to back this up, so I might be completely wrong, but Conway's Game of Life, which is a "zero-player game" that every nerd has been freaking out about since the 1970s, seems like a really good representation of what chaos looks like. It has really simple rules, and it seems really predictable, but the longer you let it go on, the more random and chaotic it gets. Sound familiar? Of course, this is just my understanding of it, and I could be completely wrong, so take it with a grain of salt.
Other applications
Chaos theory pops up in places you wouldn't expect it to be, like economics or chemistry. What does an advanced mathematical concept like chaos theory have to do with money or funny-smelling chemicals? I don't know, but what I do know is that chaos theory has helped robots walk. It has helped people keep stuff secure with cryptography. Chaos theory has thousands of different applications, in thousands of different fields. It has led to many many developments that make our lives better, in some way or another. In short, chaos theory is really useful.
The end
So what have we learned? Chaos theory lives by pretty simple rules. If you change the starting point a little, the outcome should change by a lot. Chaotic behavior starts out normal and becomes crazier over time. To predict chaotic behavior, just be sensible with your measurements and hope you have a long Lyapunov time, or else good luck keeping up with the chaos. And finally, chaos theory is used pretty much everywhere you don't expect it to be, along with all the places you do expect, of course.
This was a pretty short project because I don't understand the advanced stuff and I couldn't be bothered to watch YouTube videos about it, but I did some "research" and this is what I've got to show for it. I did have fun doing this, though, and I hope to explore this more when I get older and have more intellectual ability, because it seems like a really cool topic that definitely excites the nerd in me.
Anyway, thanks for reading!
Sources
There's not a lot of sources because many websites were just repeating what Wikipedia said, or were too complicated for me to understand, or both. And I didn't bother going past the first page of search results, so it's possible I missed out on a lot.
chaos theory | Definition, Examples, & Facts | Britannica
ELI5: What exactly is Chaos Theory? How does it apply to everyday life? : explainlikeimfive