Explanation of the butterfly effect and deterministic chaos using billiards 

Congratulations on such an amazing video! I have a question: knowing that 'stretching' and 'folding' are the two factors that make a deterministic system chaotic, can you go back to the double pendulum example and identify the stretching and folding mechanisms there? You did mention that it is hard to identify the stretching mechanism on the Lorenz system; I hope that is not the case for the double pendulum. Thank you.

Great use of natural dampening effects to show why butterfly effect is rarely a worry in the real world. Sustained feedback mechanisms are more problematic.

What a wonderful video, thank you so much for your clarity and beautiful visuals!)

A great video and good explanation! I believe the two examples (double pendulum, balls on a pool table with circle boundary) the chaos is being introduced by Pi. Given than it is a number that cannot be written down completely and never repeats, even slightly different numbers multiplied by pi diverge exponentially from certain points of view.

Long time fan of 3b1b, and I still gotta say that this is very cool. I did officially learn something new today. I also gotta say, if this was not done with 3b1b's Manim library, you have done an excellent job in emulating his style

Great Job! Thanks for making this video :)

Great introduction to the subject Looking forward for more videos

Watched this last night - loved it. Hope you make more videos!

So many light bulb moments. Thank you

You could probably demonstrate the sensitive dependence on initial conditions with kneading bread dough as well! As you start kneading, push a small cluster of small, distinct objects, say poppy seeds, into the dough. If a dough being kneaded is indeed a chaotic system, they should after long enough kneading end up spread relatively evenly throughout the dough, and thus the finished bread.

Very good content. Cheer from Brazil! Keep it up

Well done, thank you!

Incredible video!

That answers so many questions I’ve always had, thank you so much

good video thank you for making it

I remember Ian Stewart saying something similar of the sort about the folding and stretching of systems which exist necessarily to lead to chaos

cool video!!

Nice video

Good video

Did you use Julia programming language?

Based

Excellent video, thank you! Could you tell which numerical methods (space discretization and time integration) did you use? Did you use explicit time integration like the 4th order Runge Kutta method or implicit time integration like the 4th order Gauss-Legendre method? Also, did you always use the same method for all shown cases? This is important, because the explicit time integrators (if not properly bounded) are known to induce unintentional numerical chaos into the simulated dynamical system due to the transition from continuum PDE to discretized PDE. One excellent example is the well-known Logistic Map, which can be generated by the 1st order explicit Euler time integration method from the Logistic Differential Equation (free of any chaos and bifurcations), being the simplest proof of numerical chaos artefacts.

Bear in mind that actual Billiards physics is far more complex than "specular reflection". First, the balls are subject to gravity and table cloth is not friction free, so the balls slow down. Second, this friction causes any moving ball (but not any photon) to roll forward. This rolling spin affects the angle of reflection when any ball rebounds at an angle off of a cushion. This rebound actually follows a parabolic arc until the rolling friction adapts to the new direction. Third, the cushions are rubber, not mirrors. The faster a ball is moving the more it depresses the cushion, which changes the "angle of reflection". Fourth, the cushions are also not friction free. Any ball hitting a cushion at an angle experiences friction which induces side spin on the ball. This spin will vary depending on the angle of incidence, the speed of the ball and the rolling spin of the ball. This in turn will have follow on effects when the ball hits another cushion or ball. Fifth, collisions between balls are not friction free. The collision angle and the spins of both balls all affect BOTH the direction and the spin of both balls after collision. These changes in direction from what specular reflection would predict are called "collision induced throw" and "spin induced throw". Sixth, of course, the cue almost always imparts some side, follow or reverse spin on the cue ball. All professional players know all of these facts "instinctively" often without being able to describe the physics. But it also seems that many professional physicists who haven't played the game don't understand these facts.

Excellent!

This is great!

Nice

8:21 is the second non chaotic thing diverging exponentially fast though? Seems to me like it is linear divergence

The last simulation is the electron random walk in confined container.

this got me thinking... is there a way to more precisely quantify chaos, or at least the attributes that make it so? can you have a system that's just *barely* chaotic, or a system on the verge of chaos but not quite there? i might just be sputtering nonsense, but i do like to think about limits

Very cool!

Awesome!

I really wonder how the median of all the points behave at 13:08.

8:20 They aren't diverging exponentially fast, they're diverging linearly fast.

Moral of the story: _Math imitates pizza._

Good talk, great videos. Drop the background music - it distracts and doesn't add anything.

I don't understand why the ball bouncing inside the rectangle is considered non chaotic, i mean, if two nearby balls hit the corner, aren't they going to diverge? Or the problem is that the divergence isn't exponential?

"Billiards are chaotic and unpredictable." > Ronnie O'Sullivan has entered the chat.

What tool did u use for this animation Hope to answer me ❤️❤️

Ok I bored, so I watch.

Period 3 implies chaos

Shouldnt it be a "chaotic" fractal pattern instead -since all forms on earth are -> fractals or are the averge and a slushy even plain chaos ?

Duffing oscillator

Very well put together!

Nice