The Double Pendulum is Chaotic

Below is a demonstration that the double pendulum is sensitive to initial conditions.

The two pendula (left and right) are identical, apart from their initial conditions (initial conditions are the way in which a system is started). The initial conditions for the double pendulum are the starting angles of each arm, and their initial speeds. The starting angles are the same in each case, but the initial speeds are as shown.

Initial speeds, left: Initial speeds, right:
main arm = 400.0 degrees/sec main arm = 400.1 degrees/sec
secondary arm = 0.0 degrees/sec secondary arm = 0.0 degrees/sec

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Key to graphs below: Left system - red line; right system - blue line

For a short while the two pendula stay in phase, but their behaviour quickly diverges even though their initial speeds are very similar (only 0.1 degrees/sec difference in the initial speed of the main arm). This shows that the systems are sensitive to their initial conditions. Such double pendula are chaotic.
This is the so-called "butterfly effect" which is also often exhibited by the Earth's atmosphere. This makes it difficult to make medium-to-long term weather forecasts.