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Can someone briefly explain chaos theory and any modern practices/uses of it?

In: Mathematics

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Small things can cause chaos. Easily predictable things become impossible to predict because of this. The butterfly effect. We can calculate the travel of a ball on a surface with ease using physics. Any imerfection in the surface makes this impossible over time. Hope this helps. I learned from jurassic park.

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.

Imagine a simulation that predicts the path of a ball that is thrown onto a perfectly flat floor with known initial direction. The ball will bounce a few times, but in general we can approximately predict its path. If you change the initial direction a bit, the curve of the ball and the points where it will bounce will change a bit. The point where it first hits the floor will change the least, further points more. Put mathematically, a very small change D in initial conditions (the initial speed or throwing direction) will produce a change alpha * D * L in the result, where alpha is some number, L is the length of the trajectory you follow. So if you double the change, the change in the result also doubles (approximately).

Now consider a simulation of the same ball thrown onto a floor with lots of boxes lying around randomly. Now, a small change in the balls initial throwing direction might result in it narrowly hitting a box it missed before (or vice versa) or the ball hits the edge of a box – when that happens, the path it takes from then on is entirely different than before. Mathematically, a small change in D will (on average) produce an exponentially different result exp(alpha * L) * D.

The notion of such an “exponential” sensitivity came as a surprise: indeed, some name Edward Lorenz the father of chaos theory. He was a mathematician

and meteorologist working on programs predicting the weather. He noticed that restarting his program would produce entirely different predictions in weather, and suspected a bug somewhere. What he finally realized was that the equations were so sensitive, that a minute change in parameters (say, restarting with an initial temperature of 29.0000001 instead of 29.0000000, which happens easily in numerics with finite precision) would propagate to give a very different answer for the weather in two weeks.

Summarizing, for “regular” systems, a small change in input will produce a small change in results, while for chaotic systems a small change in input can produce an entirely different result (butterfly effect, etc.). The study of such systems is called chaos theory.

You asked for modern practices and uses: while we cannot reasonably predict the exact outcome of a chaotic process, we can try to model probabilities, and describe the degree of chaos. We now know that most systems in nature are actually chaotic. We can predict repeating patterns we see and understand how stable they are. Interestingly, using statistics in describing chaotic systems sometimes works better than for regular systems. People have found that chaotic systems sometimes show almost-regular behavior, and that there are systems that feature both regular and chaotic dynamics, based on the initial conditions. These results are used to try and predict stock market prices, the weather, or cryptography.

Here you go. Jurassic Park did a quick explanation.Taught by one of the best in under 2 minutes.

https://m.youtube.com/watch?v=5cVLUPwrSmU