Please no butterfly analogies.
In: Mathematics
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.
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