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Visualising Chaos: The Lorenz attractor
Two trajectories that start almost at the same point and end up somewhere completely different.
The Lorenz system is three simple differential equations, first written as a model of convection in the atmosphere, and one of the classic examples of chaos.
I solved it in Python for two starting points that differ by only 0.001 and animated both trajectories together. For a while they move as one, then they separate and trace completely different paths on the same butterfly-shaped attractor.