Can someone briefly explain chaos theory and any modern practices/uses of it?
In: Mathematics
Most mathematical functions we use regularly are relatively well-behaved: if you take a few inputs and their respective outputs (both of which can be multiple variables) you can predict what other input would get you in terms of output, at least in the ballpark. This is especially true if the initial outputs are almost the same: you can expect outputs which are almost the same.
But sometimes that is not the case: you slightly alter the input and you get a very different output. This is especially true in cases where you aren’t taking exact inputs but approximating them (such as what we do in pretty much every actual physical application of physics): what you initially thought were two equal inputs were not equal and their end results are very different.
However, we are still dealing with functions: they are actually predictable if the input is the same, but it’s much more difficult to approximate them. We can study them to see if there may be some interesting things we discover about how they behave, but they are harder than other functions.
One application is developing what looks to us as randomness generators: you take a sufficiently exact initial measurement, and then run that into a chaotic function. This is not truly random as if you knew the input you would know the output (a series of numbers), but it’s hard to land on the same sequence twice.
Similarly, good encryption requires functions which behave somewhat “chaotically” to an observer: you want an output which does not give you a clue of the corect input even if you only have near miss. You can technically reverse engineer most encryption algorithms with enough correct key-value pairs, but the goal is to make such a calculation completely impractical.
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