Order Within Chaos in the Double Pendulum (Island of Stability Simulation) 

If you increase the resolution and calculate how long each pendulum would take to deviate heavily from the island of stability and plot that time as a colour, i can see a very cool fractal emerging...

Just randomly remembered how good your klein bottle video was, and came back to pay a revisit. I was thrilled to see the new upload. Can't wait to watch the long form video!!

What a great idea and execution, I didn't expect some to persist this long.

That is so cool, I would have never expected any stable outcomes like that, and of top of that it's also clearly shown as a well defined repeating pattern in a sea of confusion, super sweet man

Now, simulate the range of angles between "stable" and "unstable" around the periphery of the stable island, to narrow down exactly where the edge of stability is.

Could you do the same animation again but increase the floating point precision with mpmath to a 1000 decimal points? i think what you see is the 64bit accuracy limit (just an assumption)

"Andrew's Campfire comeback before GTA 6"

Literally found this channel today. And there is a new upload 😂

Absolutely awesome. Is there any rigorous theory behind? Never heard of it, and it looks like it's much better than just the vanishing of some first two derivatives in a three-parameters problem, there really is something going on here.

i love when science and art are equal in a piece of work! great one

:0 this is really neat! The beginning shape reminded me of Lyapunov Fractals (which derive from the Logistic Map). The double pendulum region idea doesn’t seem to map exactly to that of Lyapunov Fractals, but I think it might be similar, just with a non-repeating but somewhat-repetitive sequence.

IT'S BEEN 2 YEARS

That's an amazing simulation, send it to all who might or might be not interested. The longer video will be eagerly anticipated.

This animation is f*king amazing. Although, how do you deal with the problem that machines have finite precision? How do you know these islands of stability are not a consequence of the numerical computation itself?

This is super interesting. I will definitely watch your longer video.

I think watching the pendulums as points moving though phase space with trails would be cool too. Would the stable ones be orbiting some point? Or maybe redrawing the same line over and over.

this is mesmerizing to watch

Great video, it bends my brain.

This is a kool simulation. It may be helpful to add a brief about what this shows or why :)

return of the king

what's interesting to me are the "lakes" in the middle of the island, which break from the pattern even while their neighbors follow it

These pendulums in what you have named the island of stability do not pass through "sharp" points that lead to chaos. They don't have the second arm making a whole rotation and so they are behaving more like a single pendulum which is not chaotic

at around 6:20, near the top left of the island, one pendulum on the boundary that previously deviated from the island briefly appears to sync back up with the island for several swings before deviating again.

Well, that was really satisfying.

TLDR: this is a chimera state and we need to know which coupling you implemented in the simulations to figure out why it emerges. Long version: I have so many questions about this. Are the pendulums independent? If there is a coupling, is it local or global? And what type of coupling? Are the initial condition all the same or randomized? And if they are all the same, what is the accuracy of the machine number used to generate them (this being a potentially chaotic system, it should have sensitive dependence of initial conditions)? Speaking about chaotic regime, what is the projected dynamically behavior of each pendulum given its structural parameters, value of gravity and initial conditions? Also, as a side note, I would not call that an island of stability, as much it's an island of synchronicity. This is a chimera state for the system and it would be really cool to have an answer to all the above questions to see why this chimeric behavior emerges

Next up playing bad apple using double pendulum 😂

Wow that music sounds like a new dimension to another world is opening starting the multiverse war.

The greater the difference in angle between both pendulums, the larger the cross term in the Lagrangian and it's accompanying nonlinearities

If you calculate the dot product of each double pendulum and set the hue of each pixel accordingly you'll get a colorful animation!

here's my question though, why do they have slightly different periods before they diverge from the stable point? especially in the later part, you can see waves moving across the island of stability as they get further and further out of phase, indicating that in this metastable mode the initial angle has some continuous, predictable effect on the period of the pendulum. I ask because, much like the three-body problem, this strikes me as a "simplified case" where it might be possible to actually reason about the properties of an otherwise chaotic system while it stays within a known stable configuration... I mean, empirically it would be pretty easy to produce an equation that locally predicts, say, the period of a pendulum halfway between two others simulated near the centre of this island, and that equation would show that period is some product of the sum of the two angles. But... why? Why does having an otherwise stable configuration with a greater angle lead to a slower oscillation, kinematically? Clearly, even in an island of stability where the arm stays relatively straight, we can't model a double pendulum as a single one, but what's the nature of the discrepancy? I'm sure it has to do with the flexing of the middle joint "absorbing energy in the system" (although intuitively, that doesn't quite hold water, to me), but perhaps I'd hypothesize that the more the middle joint bends during each cycle, the slower the cycle is - hence a more severe initial angle leads to a slightly slower period of oscillation.

Buckle up! He's back!!! 💛💛💛💛💛

Cool video.

Cool apophenia. Now have them correlated and vary the degree of correlation.

never thought a double pendulum state would last longer than me in bed D:

I feel like I need to tell someone about this...

Is this how scientists discover new industrial materials?

Someone needs to make a paper about this, and then someone else needs to make a RU-vid video explaining what’s said in the paper.

Okay but why does it look like Sonic's head.

Now I wonder Why those specific pendulums took a long long time to go into chaos .. is there a specific reason ? Perhaps it has to do with the property of the angle measure? (101°, 115°?)

Why is the angelfish of stability so irregularly shaped, I wonder?

HOLY MOLY THIS GUY IS BACK

amazing! can you do the other stable mode?

Watching this makes me irrationally angry. I want it to devolve into chaos or order, but this half state is an abomination.

Seems to me that the "island" is mostly due to an artifact of our visual system. It is the part where our visual system is able to detect coherence. The coherence is actually there everywhere, just undetectable by our eyes. If you reran the same experiment, but recorded all angles halved for the video, you would see the island expanding significantly.

Cool!

im very why the island has a tail that extends all the way down. at least in the beginning

so glad i subbed. this is crazy lol

surely this is a programming outcome and not a mathematical one

I kinda wanna see someone make bad apple with this now sidenote, you should probably make your music timestamps start at 0:01 and not 0:00 so they don't get marked as chapters in the video

Is this in any way related to the theoretical island of stability for atomic elements?

Nice vid

Yep... the fractal.

Don't leave us again 😢

Did you add some check code that tests that no pendulum occupies the same state point as another across all time

Ur back!!!

will it go back in its original synchronized movement ?

Interesting

Oh my god welcome back my favorite brain damager guy 🥹

Why is this video so stressful

凄いな、何でこうなるんだ

It's amazing how accurately this also applies to humans, ie. their sociopolitical aspects. First, tiny deviations and misalignments, which then cause growing chaos and instability. But... No matter the sheer weight of madness and depravity, a strong heart at the center will always beat like one, like a wave, and endure. The insignificat clutter may continue to flap around aimlessly in its misguided fits, but it will be the strong ones to achieve unity.

Someone should find the function of the fractal that emerges

It looks to me like String Theory.

What is the first song. I don’t think it is Digital Abyss

No more topology and stuff??

Does the island look vaguely similar to the USA+Mexico to anyone else?

why

🙏🙏🙏🔥

Wiw ok😊

Why hello there

The island looks like Croatia

And that's what a migraine looks like 😬

Anyone else see the US and Mexico?

:O

He remembered his password guys

another copy of another video cmon