What do differential equations have to do with Chaos Theory?
In: Mathematics
Differential equations are a way of describing how a system’s state changes based on its current state. A great example is a simple pendulum: based on its current position and velocity, you can pretty easily predict what it’s going to do next.. you know it’s going to keep moving in the direction it’s currently going, but if it’s on the upswing it’ll be slowing down, and if it’s on the downswing it’ll be speeding up.
A system is chaotic if even really tiny differences in its state have big, hard to predict consequences. A regular pendulum is not chaotic: if you measure its position and velocity while it’s swinging, you can use that information to predict what it’ll be doing thirty seconds from now. And if your measurement of its position right now is a little bit off, your prediction of where it’ll be in 30 seconds will only be a little off.
But a [double pendulum](https://en.wikipedia.org/wiki/File%3ATrajektorie_eines_Doppelpendels.gif?wprov=sfla1) is more unpredictable. If you try to predict its future state based on measuring its current state, if your measurement of its position right now is even a LITTLE off, your prediction of where it’ll be in 30 seconds will be a LOT off. That makes it chaotic.
So: differential equations describe how systems change depending on their current state, and chaos theory is all about systems that change in a difficult to predict way. A set of diff eqs can define a simple, non-chaotic system, or you can have a set of diff eqs that describes a chaotic system.
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