Explanation of the butterfly effect and deterministic chaos using billiards
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- Опубліковано 20 сер 2021
- Created by George Datseris.
In this relatively short education video I want to explain the
butterfly effect and deterministic chaos at a fundamental level,
using the simple and intuitive concept of billiards.
Heavily inspired by 3Blue1Brown videos, and made as an entry for SoME1:
www.3blue1brown.com/blog/some1
Source code for all animations:
github.com/JuliaDynamics/Chao...
Background audio:
Reflections by Vincent Rubinetti from "The Music of 3Blue1Brown"
vincerubinetti.bandcamp.com/t...
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.
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.
Thanks! Unfortunately it isn't easy there either. To get the folding is not hard. In an intuitive level you can say that conservation of energy makes the folding, because it constrains the system into a narrow slice of all state space (the slices that has energy equal to the starting energy). For the stretching, it isn't straightforward to pinpoint a "single, simple mechanism". It all comes down to local instabilities. Once our book "Nonlinear Dynamics: An introduction interlaced with code" comes out, have a look at section 3.2. The math isn't hard, just not something I wanted to show to UA-cam on my very first video... Best, George.
@@juliadynamics6863 The points where the far pendulum rotates over close one at some low speed?
@@juliadynamics6863 I only just started a PhD in mathematical modelling of neural dynamics, and I was strongly considering learning Julia. I have just seen that your book got published this March and now I am convinced I will go with that! :) Thank you for the fantastic video by the way.
Watched this last night - loved it. Hope you make more videos!
What a wonderful video, thank you so much for your clarity and beautiful visuals!)
Incredible video!
That answers so many questions I’ve always had, thank you so much
Great introduction to the subject
Looking forward for more videos
Very well put together!
So many light bulb moments. Thank you
Very good content. Cheer from Brazil! Keep it up
Great Job! Thanks for making this video :)
This is great!
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.
Yes, you can do it with food coloring. I have seen a video of that in an old lecture but couldn't find it on UA-cam to put it in this video...
Well done, thank you!
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.
Very cool!
Excellent!
cool video!!
Awesome!
good video thank you for making it
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
Thanks! Long time fan here as well! Indeed, this wasn't done with Manim, but with Makie.jl a powerful Julia library!
@@juliadynamics6863 oh, I didn't know Makie can do videos!
@@omarhatem9598 Oh yes, it can, and quite easily as well! We just made a video tutorial on that here: ua-cam.com/video/L-gyDvhjzGQ/v-deo.html
I remember Ian Stewart saying something similar of the sort about the folding and stretching of systems which exist necessarily to lead to chaos
Nice video
Based
Good video
The last simulation is the electron random walk in confined container.
Nice
Did you use Julia programming language?
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.
Hi, thanks! None of the systems used here are PDEs, not sure where this confusion came from. Billiards are solved explicitly see juliadynamics.github.io/DynamicalBilliards.jl/dev/#Features so there is no time discretization. The Lorenz system was solved with a Runge-Kutta-Solver yes. It is extremely rare that numerical integration introduces artificial chaos. You'd have to use some over-the-top-bad method (Euler) or extremely large stepping for this to even be something to consider.
@@juliadynamics6863 Thanks for the clarification! It makes sense that the billiard ball movements and collisions are just explicit solutions. But are you saying the double pendulum system is not a PDE simulation just like the Lorenz system?
Moral of the story: _Math imitates pizza._
I really wonder how the median of all the points behave at 13:08.
8:21 is the second non chaotic thing diverging exponentially fast though? Seems to me like it is linear divergence
Damn, I think you are right... I think I missed up in the animation script and made the balls move linearly instead of exponentially fast. Sorry about that, but congrats on having such a keen eye!
@@juliadynamics6863 why would they move out exponentially fast though? Isn't the idea that they have a constant speed?
I would think a chaotic system would always have an exponential increase in distance apart
@@mukkuru Yeap, if you assume these to be billiards ball in a billiard, they would move out linearly. They point was to make them move out exponentially fast so that indeed their distance would actually also increase exponentially. It was just a hypothetical scenario of particles that diverge exponentially fast but still are not chaotic.
What tool did u use for this animation
Hope to answer me ❤️❤️
the source code for the animations is in the description, they had to program them
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
One notion of partial chaos might be how much of the state space yields chaos. IIRC, with the double pendulum, a gentle swing does not act chaotically, so you might say it isn't fully chaotic.
Transient chaos is a typical situation that would count as your "barely chaotic". It describes scenarios where time evolution is chaotic for some initial part, but then eventually stabilizes itself towards regular motion. The way to "more precisely" quantify a chaotic set is" "an invariant bounded set with at least one positive Lyapunov exponent".
thanks! youve both given me things to look up 💝
8:20 They aren't diverging exponentially fast, they're diverging linearly fast.
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.
Sure, real billiard is much more complex. But in spite of the video using a simplified version of billiard, we still get the chaotic effect.
@@BigParadox which is why using a billiard analogy was a poor choice.
@@2DXYSU Rather the opposite. Even though he uses a simplified version of billiard it still exhibits a chaotic nature. If he would have used a more realistic version of billiard, we would have seen an even more chaotic nature, provided of course that the ball would continue to roll.
@@BigParadox If you watch the video, he says that the basic principle of billiards is specular reflection and he chose billiards because people who play it intuitively understand this. But he says this is too complex for his purpose so he proposes 3 and only 3 "simplifications": no pockets, one ball, zero size ball. This displays astonishing naivety of the true complexity of billiards which actually requires far greater simplification for his purposes.
@@2DXYSU I am sure he understands how complex real billiard is. He starts his reasoning from an already (implicitly) simplified version of billiard, and then he explicitly adds some more simplifications (like removing the pockets etc.) The fact that he starts from an already simplified version of billiard should not be taken as an indication that he does not understand the complex nature of real billiard. He has already at that initial stage implicitly simplified it. It is a common way to discuss things to use implicit simplifications. If we did not do that when we discuss things or talk with eachother we would end up in enormously complicated forms of communication. If I say I will come and pick you upp at 1 o'clock and I come 1 minute after 1, you cannot conclude that I do not understand what 1 o'clock means. I made an implicit simplification when I said 1 o'clock, which all people understand. (And for your information "all" up there was an implicit simplification.)
Good talk, great videos. Drop the background music - it distracts and doesn't add anything.
"Billiards are chaotic and unpredictable."
> Ronnie O'Sullivan has entered the chat.
Ok I bored, so I watch.
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 ?
Period 3 implies chaos
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?
Indeed, if you have different angles you will get divergence, but it will not be *exponentially fast*. That is a crucial distinction. I now realize that I didn't raise this point strongly enough in the video: it is not enough to have divergence, it needs to be exponentially fast as well. In the rectangle billiard you can have at most linearly fast divergence. Thanks for pointing this out!
Best,
George
Duffing oscillator
Nice