Butterfly effect in the Lorenz attractor

Divergence of nearby trajectories in the Lorenz attractor

This code can be used to plot the starting point and the final point of a trajectory.

To showcase the divergence of nearby trajectories, we generate a collection of random initial conditions x(0), all very close to each other, simulate the system for some short time, like 50sec, and then plot the final point of the trajectory x(t_final).

The result, is that although the x(0) values are clustered together on a very small range, the x(t_final) are very far away from each other, covering the whole attractor. This phenomenon shows that nearby starting trajectories diverge over a very short time from each other.

Of course, you can replace the Lorenz system with any system of your choice.

Video explaining the phenomenon:

https://www.youtube.com/watch?v=O5b9dRd_eY0

https://www.youtube.com/watch?v=8kKkq2mk7gA

This was inspired from a graph found in Strogatz's book below.

https://www.stevenstrogatz.com/books/nonlinear-dynamics-and-chaos-with-applications-to-physics-biology-chemistry-and-engineering

https://ieeexplore.ieee.org/abstract/document/9703188

https://www.mdpi.com/2227-7080/7/4/76

https://www.mdpi.com/2673-8716/2/2/8

https://link.springer.com/article/10.1007/s40435-020-00712-0

Youtube channel: https://www.youtube.com/@lazaros_moysis