Please no butterfly analogies.
In: Mathematics
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.
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