It’s a little imprecise to say that chaotic systems are unpredictable– if we know the initial speed and position exactly, the path is entirely predictable according to the laws of classical mechanics.
What the butterfly effect actually says is that slightly different starting points lead to very different ending points, which is a more subtle point! To illustrate that, here’s an animation of the same double pendulum, but with three slightly different initial conditions:
You can see that even though all three look almost identical for the first couple seconds, their paths rapidly diverge and look very different after three or four seconds. What’s happening here?
Simply put, this kind of system (a chaotic system) is extremely sensitive to the initial setup– changing the original placement and speed of the pendulum affects the outcome a lot. With time, small differences grow exponentially large.
If we could exactly measure the initial setup, we would be able to predict the system perfectly. However, because our real measurements have some error, we don’t know exactly what the initial setup is!
That means we can predict the system for a little while, but eventually our lack of precision will catch up to us and we’ll lose predictive power.
This is also why you see a 10-day weather forecast and not a 3-month weather forecast, for example (telling you exactly what the weather will be on a day 3 months from now). We think weather is also a chaotic system, and that means with the accuracy we have, we can reasonably predict the weather a little ways into the future, but not indefinitely.
Which is how we go from this (small differences):
to this (exponential effects):
And that’s what we mean by chaos theory.