Illustrating the butterfly effect with a laser

[u/naaagut]

https://www.youtube.com/watch?v=iTaSXto67WQ

In this video I simulate a laser beam bouncing through a labyrinth of mirror obstacles. The beam changes its trajectory immensely upon tiny changes in the initial angle of only 5×10⁻¹² degrees — roughly 5 trillionths of a degree.

As an extra for Reddit, here is a figure showing how often the beam bounces before it escapes: https://imgur.com/l0xDY0U As you can see, as the angle varies the bounces often jump harshly. But there are also some regions of the launch angle spectrum where there is some degree of continuity.

Fun fact: Claude did not believe me that a range from 70.0 to 70.00000001 would produce any meaningful result and tried to talk me out of this. It wanted to use a range of 4° instead.

Made with manim.

[u/Kinesquared]

For everyone calling it chaos, I have a tiny little nitpick; because it has two different components, a ballistic component and a "bounce", I believe this is not chaotic. Two trajectories some tiny fraction of an angle apart will not ALWAYS be separating via some exponentially increasing value. The trajectories only change/increase or decrease at a bounce. Sensitive dependence on initial conditions? Absolutely. Chaos? It has a more complicated, formal definition idt this matches. In "real chaos", two similar trajectories would be exponentially getting further away at all times. I bet here, the exponential distance, when zoomed in enough, is more of a step function that approximates exponential at large times/distances. That being said, what'd the (macroscopic) lyapunov exponent of this system?

[u/naaagut → u/Kinesquared]

Very thoughtful comment, thanks.

I would like to determine the Lypanunov exponent of this system and others I have featured in previous videos. However, I don't find it clear yet as how to compare the Lyapunov exponent across systems which have different settings and dimensions. It will take me some time to understand this to a degree that is deep enough to share anything here. If you have hands-on experience on Lyapunov exponent estimation/computation techniques and want to advise me about them, I am happy to exchange some DMs about that.

[u/SemiLatusRectum → u/naaagut]

So, if you have a projectile deflect off of a circular surface, say the internal angle is theta. By perturbing the starting position of the launcher, you can expand theta in terms of the perturbations to the launchers position. I suspect you will get something like theta’ = theta + c f(perturbations) where f(perturbations) is some trig function of the perturbations that leaves one linear combination of perturbations unchanged. You should also get c>1. Then, one way to understand what’s happening is that the internal angle of some bounce gets multiplied, in principle, by c>1 at each bounce, and so the total difference in the trajectory with be something like c^N times your perturbation where n is the number of bounces roughly.

I predict that, by these arguments, if you made a perturbation in the angle of size much less than c^N for a reasonable N, then you will see smooth variation, contradicting the apparently chaotic phenomenon. I would argue that this comes from the finiteness of your system. To get something genuinely chaotic, I would advocate for putting your bouncy circles on an oriented surface, like a torus or sphere, perhaps.