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...
I've seen a video like that before. I don't remember if it's an “island of stability” thing, or if it was based on some other property, though. I've actually got two videos on the topic in my suggested videos in this tab *right now*. (probably including the one I watched). Search “the double pendulum fractal”.
@slendgamer895 @skop6321 @atimholt I found a paper about exactly that! It shows the entire fractal for all posible angles. www.famaf.unc.edu.ar/~vmarconi/fiscomp/Double.pdf
@@JoseRodriguez-pi8cx The initial shape of the island looks more like something from a Lyapunov fractal to me.. you can see the sort of sweeping arms that you get like here: en.wikipedia.org/wiki/Lyapunov_fractal#/media/File:Lyapunov_fractal_segment.png
This is a real phenomenon and not caused by any computer rounding. Each double pendulum becomes chaotic after the lower pendulum makes its first full flip. For those in the bottom left and top right this always happens first on a leftward swing, while for those in the top left and bottom right this always happens first on a rightward swing. This means there has to be a saddle point where all four regions meet and this is the most stable point within this island of stability.
Cool! ... I don't understand why it wouldn't then have more symmetry, though. Is it just the way he's laid out?... Like maybe the increments he used happen to land closer on one side than the other and hiding the true shape of the island?
@publiconions6313 why would you expect it to be symmetric? Saddle points are usually drawn symmetrically, but that's a simplification. Saddle points also exist on physical mountain ridges and they're not symmetrical.
@@Daniel_VolumeDown rounding errors may affect the exact position of the saddle point, but the existence of the saddle point is independent of rounding errors.
@@DrUrlf indeed, that is inevitable in the infinite limit, but only in the infinite limit. That's a deep and important point about how this stuff all works.
Something you might want to keep in mind: The "Island of stability" may result from a quirk of the simulation rather than anything genuine. Floating point truncation, perhaps.
Would be cool to do this simulation using floating point numbers with absurdly large Mantissas. Extend the IEEE definition of double- and quadruple length floating point numbers to something absurd, like 4096 bits per value, and perform the experiment again. It would take MUCH longer to calculate but it would surely highlight any issues caused by floating point error
could be. But there are more accurate arithmethics made possible in software, one should try using 1024 bits instead of the original 64 bit arithmetich for eaxmples
Might be, but I saw a similar island when I looked into the double pendulum fractal a while back from sheer curiosity. Not sure if it was the same parameters though
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.
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!!
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
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.
Given enough time all the pendulums will fall into sync simultaneously. The question is, is a life time long enough to observe it? And how long would it last?
@@MarkDibley What do you mean by "fall into sync simultaneously"? They might line up position-wise but their velocities will necessarily be different so that their positions quickly diverge after.
Even chaotic systems have a recurrence time. It's incredibly long because the positions and velocities have to both recur, but they will eventually come arbitrarily close again. The amount of time that takes starts off gargantuan even for two pendulums and grows exponentially from there, but it exists. This is actually very easy to show too: you can obviously imagine running the simulation backwards. Chaotic behaviour will occur. Then - if you ran it backwards correctly - running the chaotic earlier state forwards will return to the ordered initial state. It's just mind-bogglingly improbable.
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)
Yes, this is a big simulation of floating-point rounding errors. But then, you can't really simulate physics in a computer, no matter how hard you try.
I was thinking about the same thing but I don't think rounding is causing this island. The reason I would be skeptical is because there is still quite a bit of divergence withen the island. Also I would expect islands of stability but this one seems to hold on longer than it should. I guess my suspension would be that this is pretty stable island but the pendulums become copies of each other separated by integer time steps twards the end? Not sure, I'd love to see a higher precision version and see if the island looks different. Still one of the best videos I've seen on this tho.
You can see in the island at every oscillation the pendulums have slightly shifted phases. I feel like if this was purely due to precision, they would all be exactly the same and not have phase shifts
This wordless video is worth a thousand words. I think it would also be interesting to see it in logarithmic time. Nothing lasts forever, but some things last orders of magnitude longer than others. Of course, at an accelerating time range, it would soon become impossible to discern the motion of individual pendulums, so some other rendering would be necessary. Perhaps a single “has it flipped over yet” bit, but there is probably a better way.
The stable pendulums all have a period of 3 swings (6 if you count back and forth as different). If you look at the behavior at the top of the arc, the connection between them has a pattern repeating over 1.5/3 swings. Its Above the end point Twice and then Below the end point Once.
Yo you finally posted i love your videos and I’ve been waiting for so long. They made me love math and science so much. thank you for teaching me thing I didn’t even know I liked. Keep doing what makes you happy. Have a nice day
: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.
Thank you! To visualize the stability-island's contours: let each pixel be a pendulum, where its color is a blend of red-to-white (according to the angle of the first pendulum) and blue-to-black (according to the angle of the second pendulum). Then, you can display many more pendulums on the screen, each with much finer gradations of angles. I think the overall color-pattern might look fantastic. :)
This is like atom stability where the closer you get to the perfect ratio of protons and electrons the more stable the atom is, other wise it decays over time, the more unstable the faster with the decaying here being represented by falling into chaos
Protons and electrons are always nearly indistinguishable from 1:1 in a material. Otherwise, the unbalanced charge in the system would rapidly overwhelm any other force trying to keep the system out of balance. This rebalancing is so strong that it's literally what causes lightning. It's neutrons you're thinking of.
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.
@@pianissimo7121 but of course! thanks :) I guess a naive way to understand this is thinking of a landscape with an absolute mess of extrema pretty much everywhere, and then you have one region that just looks like a nice potential with a single minima
@@pianissimo7121 Depends on how you look at it, I suppose. There is certainly enough energy in all these setups to perform a full rotation and enter chaotic behavior, so whether this happens is probably controlled by how strongly the two oscillations couple together, and how close their frequencies are. If the coupling is enough to coerce them to a common frequency, then you get synchronization instead of chaos. If not, then eventually the unsynchronized oscillations will push enough energy into one pendulum to make full rotations.
@@henryptung yes, but it's difficult to simplify. If the animations are fully accurate, then resonance between the 2 pendulums should be only at specific angles. Not a bunch, or maybe I am wrong. But whatever the reason they are not rotating, no rotation is the reason for the center to look alike.
THIS IS WHAT A MIGRAINE AURA LOOKS LIKE. So many people have tried to visualize migraine auras but honestly this animation does it so much better. The blurb in the middle that shrinks is what the migraine Aura looks like
it would be cool to see these represented on the graph maybe through a color gradient (maybe by assigning each pendulum a colored square on a grid through the combined angles of their joints, or by comparing the angles of each square to the ones adjacent to it and assigning a "similarity" value by which they are colored) and then run the simulation out over the course of many hours, because hypothetically, due the the unpredictability and randomness of the system, other "islands" could form given sufficient time.
If you don't focus on the island and instead follow the rhythm of flashes and color phasing, there are significant patterns. Like every 6th blink there is a faster double blink and at regular intervals the area is vertically divided by color. There are regular shifting shapes flashing as well, I couldn't see them well enough to describe them though.
This particular Chaos God approves- Chaos IS Order. I really like this video, naturally. Seriously though, epic choices for the music! Not so Chaotic, as it should be but still great!
@@Nictator42 Here is the new math processor benchmark system. The smaller the island, the better is the math processor. Also, it'd be interesting to try to explain why the island has the shape it has.
I'm working on a manuscript about religion, life and entropy, linking them together in the concept of heterogeneity from Mircea Eliade. This thing just blew my mind. I could watch it for hours ; all the fragility of life, concepts and meaning, represented in this simple simulation. Absolutely incredible.
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?
Artifacts from numerical imprecision usually look more like a moiré pattern or something, and usually such imprecision would destroy a clean pattern rather than create one. I admit that’s the opposite of rigorous reasoning though.
You've identified exactly what's happening. A noise pattern emerging from Floating Point Errors. We're being shown a really neat animation, but it's making people think it's real and it's anything but. It is a literal impossibility to actually simulate these kinds of physics in a computer because Real Numbers have infinite precision which CANNOT fit in a fixed space. Even if the "simulation" was coded to use floating point numbers with 10^10000000 decimal places, it's going to be WAY off with truncated numbers and noise, immediately.
Programmer here, there are systems to make exact mathematics happen in computers, usually required when handling money, bet this is built on floats though because simple and reasonably accurate, bet if it was upgraded to doubles the island wouldn't change much (unless I'm severely underestimating the sensitivity of the double pendulum, one thing's for sure, if you didn't have enough precision on 32 bit floats you'll certainly have it on 64 bit doubles, if you used a double to measure around the equator of the earth starting at -1 and going all the way around to the same place again ending at 1 you'd find that it's accurate to within a few nanometres at worst, that's smaller than a virus, on the scale of the earth)
@@CraftMine1000 Fellow programmer working in lotto for decades. You're right and wrong. Make up any old in your head right now with 50 BILLION decimal places... I'll do it for you 1.458383066394560263456......2653200001 Now cut that number in half. Hm. Where does the half of the 1 on the end go? Need another decimal of precision... now your number has 50 BILLION +1 decimal places and your new number ends with .....05 Now cut THAT number in half. You can see how there is a literal impossibility here. It is not reconcilable. There is nothing that can ever be done to change this. It is a fundamental issue and increasing precision does NOTHING to "solve"
What would it look like if all the pendulums in chaos, but swinging at the same time shared a single colour? I imagine a winding river and swirling vortex of colour.
This reminds me of a sort of self-replicating automaton much like Conway's game of life, where instead of preserving x-y coordinates across a plane, you preserve periods of the angles of the double pendulum with time
I'm excited to learn why this Island exists. I'll bet it's a rounding error with Pi. Probably happening when converting Degrees and Radians. My evidence: this 'island' appears centered on whole numbers of Degrees (101,115). 115 degrees is darned close to 2 radians.
All this did was show that your computer simulation is not completely random/chaotic, and is in fact somewhat deterministic owing to internal precision.
BEAUTIFUL! And all with just multiple replications of just 2 moving parts! If only it were possible to visualize islands of stability in more complex (n-dimensional) complex systems...
Within these islands of stability, there should be some ponds of chaos. Regions that quickly deviate from their surroundings. You'd have to zoom in (reduce the angle ranges) and increase the resolution to find them.
It would be cool to see this for a larger area I would expect to see a second island of stability for initial conditions that are dominated by the other mode of vibration
Great stuff! As someone who has written a fully featured dynamics engine (constrained multibody integrator, with resting contact, collision response, friction, driven limited joints), and who was taught ODE integration by Dr.Dormand himself, I am looking forward enormously to the long form video. Questions - How do the pendula interact with each other during collisions? What constraint, interpenetration and timestep error correction methods are used? Thank you.
You should temporarily highlight a pendulum when the arms reach opposite angles (they cross) as an indicator that they have "become chaotic" so we can see how often it occurs and which sides are slowly closing in on the island. Also, after highlighting them, darken them so only the stable ones are more visible.
This visualization is outstanding! I’ve not seen others here mention it, but using an implicit integration scheme such as one of the Rosenbrock methods (rather than a explicit R-K integration scheme) may give more reliably accurate long-term behavior.
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.
The period of a pendulum is often approximated as being constant for any initial displacement, but in actually the displacement of a pendulum slightly affects the period of the swing. I don't recall which way the relationship goes.
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.
The “chaotic” areas seem to have some order as well. What I thought was interesting is that if you focus on the image as a whole the chaotic areas start looking like characters in an alphabet
Klien Bottle video was awesome, i have subscribed to you. Your explainations with animations are good. Can you also explain the Time split light experiment vs Dual Slit Light experiment ? Eagerly waiting for your version on this topic.
Great advance. Now imagine that you can calculate the "island" without simulation, just with pure mathematics you could define something like a "fractal" of stability (this area (island) accoding to the formula is stable).
We live in an island of stability where choas becomes order against entropy where a fractal of cycles are allowed to repeat over and over. Like an engine running, so is the universe. Everything is a fractal, where patterns are both familiar and new at the same time. When you understand this you understand everything. Only then do you obtain free will where you are able to steer the results of wave function collapses.
@@delian66 I was trying to condense down a lot in my parent comment. It's hard to explain. I'm probably wrong but also could be right or parts of it could be right. If you can't collapse wave functions, you are deterministic. Like a star or rock. But anything that is conscious can collapse wave functions by being an observer. Self-awareness is knowing you collapse wave functions. It is knowing you are an active participant in reality, not just an observer. Even before now, we intuitively knew we were collapsing wave functions. We just didn't have a name for it yet. So we've been self-aware for a long time. Once you become aware of it, you will want to figure out how to control it. Somehow we are able to control their collapse. Steering them. We can somehow change the probabilities. If you decide to go check the mail, then go do it, doesn't that initiate a series of wave function collapses that leads you to checking the mail? Aren't we collapsing wave functions all the time? If we collapse them, and we are somehow collapsing them into the way we want naturally, as part of consciousness, and self-awareness is realizing that fact, then we might be able to do more advanced things if we figure out how. Let's take rolling dice for instance, full control over your ability to collapse wave functions would mean you could throw the dice and they land on the number you intended. I think maybe free will is simply when you become self-aware of what you are doing.
@@jamiethomas4079 Thanks for taking the time to expand your previous comment. As I understand it, what you call collapse, is just taking action in the world. If you exercise a lot, you can control dices when you throw them, by mastery of your fine motor skills. You can also use thought and tools to modify the dices, so that they will be more likely to land on the sides you want. Observation is the first stage towards meaningful action. It is not the end result.
“It turns out that an eerie type of chaos can lurk just behind a facade of order - and yet, deep inside the chaos lurks an even eerier type of order.” -- Douglas Hofstadter
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...
if possible I would love to see that
I've seen a video like that before. I don't remember if it's an “island of stability” thing, or if it was based on some other property, though. I've actually got two videos on the topic in my suggested videos in this tab *right now*. (probably including the one I watched). Search “the double pendulum fractal”.
@slendgamer895 @skop6321 @atimholt
I found a paper about exactly that! It shows the entire fractal for all posible angles.
www.famaf.unc.edu.ar/~vmarconi/fiscomp/Double.pdf
I’m like, 100% certain this is a julia set
@@JoseRodriguez-pi8cx The initial shape of the island looks more like something from a Lyapunov fractal to me.. you can see the sort of sweeping arms that you get like here: en.wikipedia.org/wiki/Lyapunov_fractal#/media/File:Lyapunov_fractal_segment.png
This is a real phenomenon and not caused by any computer rounding. Each double pendulum becomes chaotic after the lower pendulum makes its first full flip. For those in the bottom left and top right this always happens first on a leftward swing, while for those in the top left and bottom right this always happens first on a rightward swing. This means there has to be a saddle point where all four regions meet and this is the most stable point within this island of stability.
Cool! ... I don't understand why it wouldn't then have more symmetry, though. Is it just the way he's laid out?... Like maybe the increments he used happen to land closer on one side than the other and hiding the true shape of the island?
@publiconions6313 why would you expect it to be symmetric? Saddle points are usually drawn symmetrically, but that's a simplification. Saddle points also exist on physical mountain ridges and they're not symmetrical.
Mathematically, something something gradient?
Well it sitll might be affected in some way by rounding errors
@@Daniel_VolumeDown rounding errors may affect the exact position of the saddle point, but the existence of the saddle point is independent of rounding errors.
Finally the UA-cam algorithm understands my needs
From the eldest pendulums you can still hear stories that their land was completely uniform once upon a time.
But then more and more regions became swing states...
@@b.s.7693 damn chaos swingers ruined everything.
@@niks660097 "chaos swingers" sounds like a subversive new sub-genre of Warhammer games.
But the prophecy foresees that one day, they might return to unison.
@@DrUrlf indeed, that is inevitable in the infinite limit, but only in the infinite limit. That's a deep and important point about how this stuff all works.
Something you might want to keep in mind: The "Island of stability" may result from a quirk of the simulation rather than anything genuine. Floating point truncation, perhaps.
Would be cool to do this simulation using floating point numbers with absurdly large Mantissas. Extend the IEEE definition of double- and quadruple length floating point numbers to something absurd, like 4096 bits per value, and perform the experiment again. It would take MUCH longer to calculate but it would surely highlight any issues caused by floating point error
@@portobellomushroom5764 i like your funny words magic man
could be. But there are more accurate arithmethics made possible in software, one should try using 1024 bits instead of the original 64 bit arithmetich for eaxmples
My thoughts exactly.
Might be, but I saw a similar island when I looked into the double pendulum fractal a while back from sheer curiosity. Not sure if it was the same parameters though
What a great idea and execution, I didn't expect some to persist this long.
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.
Do you reckon it would be something kooky like a Julia set?
I'm not fully convinced that there is an ultimate edge. It could still just be a very slow divergence.
There is only an "edge" for a certain amount of "time"
It's fleeting - like my brain
@@madscientist3544 what power law would it be?
no u
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!!
Wow, thank you!
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
alright, so:
-"DOUBLE PENDULUM ISLAND OF STABILITY"
-Immediate subscribe
-What an absolutely perfect display of data. Phenomenal!
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.
Interesting observation!
Given enough time all the pendulums will fall into sync simultaneously. The question is, is a life time long enough to observe it? And how long would it last?
@@MarkDibley What do you mean by "fall into sync simultaneously"? They might line up position-wise but their velocities will necessarily be different so that their positions quickly diverge after.
Even chaotic systems have a recurrence time. It's incredibly long because the positions and velocities have to both recur, but they will eventually come arbitrarily close again. The amount of time that takes starts off gargantuan even for two pendulums and grows exponentially from there, but it exists.
This is actually very easy to show too: you can obviously imagine running the simulation backwards. Chaotic behaviour will occur. Then - if you ran it backwards correctly - running the chaotic earlier state forwards will return to the ordered initial state.
It's just mind-bogglingly improbable.
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)
Yes, this is a big simulation of floating-point rounding errors. But then, you can't really simulate physics in a computer, no matter how hard you try.
The point is to see if a pattern persists, if it does that means that the simulation has more probability of being true to actual behavior.
@@TheNadOby We're not in disagreement. I just feel it is important for people to understand what we're looking at.
I was thinking about the same thing but I don't think rounding is causing this island. The reason I would be skeptical is because there is still quite a bit of divergence withen the island. Also I would expect islands of stability but this one seems to hold on longer than it should.
I guess my suspension would be that this is pretty stable island but the pendulums become copies of each other separated by integer time steps twards the end?
Not sure, I'd love to see a higher precision version and see if the island looks different.
Still one of the best videos I've seen on this tho.
You can see in the island at every oscillation the pendulums have slightly shifted phases. I feel like if this was purely due to precision, they would all be exactly the same and not have phase shifts
That was just beautiful! Many thanks for this excellent post! There is something fundamental about the information here. Something transcendental.
I like the way you plot every single possibility in a graph, so we can see all animations simultaneosly.
This wordless video is worth a thousand words.
I think it would also be interesting to see it in logarithmic time. Nothing lasts forever, but some things last orders of magnitude longer than others. Of course, at an accelerating time range, it would soon become impossible to discern the motion of individual pendulums, so some other rendering would be necessary. Perhaps a single “has it flipped over yet” bit, but there is probably a better way.
The stable pendulums all have a period of 3 swings (6 if you count back and forth as different).
If you look at the behavior at the top of the arc, the connection between them has a pattern repeating over 1.5/3 swings. Its Above the end point Twice and then Below the end point Once.
Yo you finally posted i love your videos and I’ve been waiting for so long. They made me love math and science so much. thank you for teaching me thing I didn’t even know I liked. Keep doing what makes you happy. Have a nice day
Wow, I feel like I've accomplished my goal here. Thank you very much
never thought a double pendulum state would last longer than me in bed D:
my condolences to your sexual partners
: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.
looks similar to the burning ship imo
Very cool, and I love the music aswell! 5/8 time really suits the weird slightly off but interesting feel of these types of simulations!
Thank you! To visualize the stability-island's contours: let each pixel be a pendulum, where its color is a blend of red-to-white (according to the angle of the first pendulum) and blue-to-black (according to the angle of the second pendulum). Then, you can display many more pendulums on the screen, each with much finer gradations of angles. I think the overall color-pattern might look fantastic. :)
This is like atom stability where the closer you get to the perfect ratio of protons and electrons the more stable the atom is, other wise it decays over time, the more unstable the faster with the decaying here being represented by falling into chaos
Protons and electrons are always nearly indistinguishable from 1:1 in a material. Otherwise, the unbalanced charge in the system would rapidly overwhelm any other force trying to keep the system out of balance. This rebalancing is so strong that it's literally what causes lightning.
It's neutrons you're thinking of.
An excellent visual demonstration of entropy.
"Andrew's Campfire comeback before GTA 6"
wow, this was insanely trippy to watch. The music selection fits this so good.
i love when science and art are equal in a piece of work! great one
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.
Chaos theoy? Attractors? Maybe.
It feels to me those are the ones where neither the 1st nor 2nd do a full rotation. A full rotation immediately causes chaos.
@@pianissimo7121 but of course! thanks :) I guess a naive way to understand this is thinking of a landscape with an absolute mess of extrema pretty much everywhere, and then you have one region that just looks like a nice potential with a single minima
@@pianissimo7121 Depends on how you look at it, I suppose. There is certainly enough energy in all these setups to perform a full rotation and enter chaotic behavior, so whether this happens is probably controlled by how strongly the two oscillations couple together, and how close their frequencies are. If the coupling is enough to coerce them to a common frequency, then you get synchronization instead of chaos. If not, then eventually the unsynchronized oscillations will push enough energy into one pendulum to make full rotations.
@@henryptung yes, but it's difficult to simplify. If the animations are fully accurate, then resonance between the 2 pendulums should be only at specific angles. Not a bunch, or maybe I am wrong.
But whatever the reason they are not rotating, no rotation is the reason for the center to look alike.
That's an amazing simulation, send it to all who might or might be not interested.
The longer video will be eagerly anticipated.
this is cool. here's an idea for a future simulation: instead of varying the initial conditions, vary the number of decimal digits of precision.
THIS IS WHAT A MIGRAINE AURA LOOKS LIKE.
So many people have tried to visualize migraine auras but honestly this animation does it so much better. The blurb in the middle that shrinks is what the migraine Aura looks like
it would be cool to see these represented on the graph maybe through a color gradient (maybe by assigning each pendulum a colored square on a grid through the combined angles of their joints, or by comparing the angles of each square to the ones adjacent to it and assigning a "similarity" value by which they are colored) and then run the simulation out over the course of many hours, because hypothetically, due the the unpredictability and randomness of the system, other "islands" could form given sufficient time.
If you don't focus on the island and instead follow the rhythm of flashes and color phasing, there are significant patterns. Like every 6th blink there is a faster double blink and at regular intervals the area is vertically divided by color. There are regular shifting shapes flashing as well, I couldn't see them well enough to describe them though.
This particular Chaos God approves- Chaos IS Order. I really like this video, naturally.
Seriously though, epic choices for the music! Not so Chaotic, as it should be but still great!
return of the king
IT'S BEEN 2 YEARS
It's a lot of pendulums 😅
It all goes tits up the instant they manage to do a full rotation.
Next up playing bad apple using double pendulum 😂
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
That implies that the arrangements which lead to stable motion likely fall along a fractal curve of some kind
@@Nictator42 Here is the new math processor benchmark system. The smaller the island, the better is the math processor. Also, it'd be interesting to try to explain why the island has the shape it has.
Literally found this channel today. And there is a new upload 😂
I'm working on a manuscript about religion, life and entropy, linking them together in the concept of heterogeneity from Mircea Eliade. This thing just blew my mind. I could watch it for hours ; all the fragility of life, concepts and meaning, represented in this simple simulation. Absolutely incredible.
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?
Theoretically it would be between -1 and 1. I think.
Artifacts from numerical imprecision usually look more like a moiré pattern or something, and usually such imprecision would destroy a clean pattern rather than create one. I admit that’s the opposite of rigorous reasoning though.
You've identified exactly what's happening. A noise pattern emerging from Floating Point Errors. We're being shown a really neat animation, but it's making people think it's real and it's anything but. It is a literal impossibility to actually simulate these kinds of physics in a computer because Real Numbers have infinite precision which CANNOT fit in a fixed space. Even if the "simulation" was coded to use floating point numbers with 10^10000000 decimal places, it's going to be WAY off with truncated numbers and noise, immediately.
Programmer here, there are systems to make exact mathematics happen in computers, usually required when handling money, bet this is built on floats though because simple and reasonably accurate, bet if it was upgraded to doubles the island wouldn't change much
(unless I'm severely underestimating the sensitivity of the double pendulum, one thing's for sure, if you didn't have enough precision on 32 bit floats you'll certainly have it on 64 bit doubles, if you used a double to measure around the equator of the earth starting at -1 and going all the way around to the same place again ending at 1 you'd find that it's accurate to within a few nanometres at worst, that's smaller than a virus, on the scale of the earth)
@@CraftMine1000 Fellow programmer working in lotto for decades. You're right and wrong. Make up any old in your head right now with 50 BILLION decimal places... I'll do it for you 1.458383066394560263456......2653200001
Now cut that number in half. Hm. Where does the half of the 1 on the end go? Need another decimal of precision... now your number has 50 BILLION +1 decimal places and your new number ends with .....05 Now cut THAT number in half. You can see how there is a literal impossibility here.
It is not reconcilable. There is nothing that can ever be done to change this. It is a fundamental issue and increasing precision does NOTHING to "solve"
The greater the difference in angle between both pendulums, the larger the cross term in the Lagrangian and it's accompanying nonlinearities
It's really cool how after half the video the wave that crosses the island of stability can be clearly seen.
"Puddle of stability" might be a more apt name, there seem to be waves travelling inside this Puddle
HOLY MOLY THIS GUY IS BACK
What would it look like if all the pendulums in chaos, but swinging at the same time shared a single colour?
I imagine a winding river and swirling vortex of colour.
A brilliant visualisation. Well done.
This is fantastic! Well done.
This reminds me of a sort of self-replicating automaton much like Conway's game of life, where instead of preserving x-y coordinates across a plane, you preserve periods of the angles of the double pendulum with time
Two words: Wolfram Model
Also 4 words: New Kind of Science
Look into these you won’t regret. Start with New Kind of Science.
Check out the island of stability around this point, requires 64-bit precision.
Angle 1 = 3.3537026965490882, Angle 2 = 3.2536503336400364 (radians)
This is a pretty excellent visualization!
This is super interesting. I will definitely watch your longer video.
I'm excited to learn why this Island exists.
I'll bet it's a rounding error with Pi. Probably happening when converting Degrees and Radians.
My evidence: this 'island' appears centered on whole numbers of Degrees (101,115). 115 degrees is darned close to 2 radians.
All this did was show that your computer simulation is not completely random/chaotic, and is in fact somewhat deterministic owing to internal precision.
I would be interested in seeing a much larger simulation- Is this akin to ocean waves?
"Ah, sweet child of Kos.... Returned to the ocean. A bottomless curse, a bottomless sea. Accepting of all there is an can be."
BEAUTIFUL!
And all with just multiple replications of just 2 moving parts!
If only it were possible to visualize islands of stability in more complex (n-dimensional) complex systems...
It's me. I'm the island of stability in the chaotic world of my family and friends.
Within these islands of stability, there should be some ponds of chaos. Regions that quickly deviate from their surroundings. You'd have to zoom in (reduce the angle ranges) and increase the resolution to find them.
Now i want to see bad apple playing on it
incredibly beautiful, thank you
It would be cool to see this for a larger area I would expect to see a second island of stability for initial conditions that are dominated by the other mode of vibration
Great stuff! As someone who has written a fully featured dynamics engine (constrained multibody integrator, with resting contact, collision response, friction, driven limited joints), and who was taught ODE integration by Dr.Dormand himself, I am looking forward enormously to the long form video. Questions - How do the pendula interact with each other during collisions? What constraint, interpenetration and timestep error correction methods are used? Thank you.
This feels like a visual metaphor for how our society is a mass of people, sometimes in sync, sometimes in anarchy.
This is a complete surprise to me. Never expected something like this to happen.
You should temporarily highlight a pendulum when the arms reach opposite angles (they cross) as an indicator that they have "become chaotic" so we can see how often it occurs and which sides are slowly closing in on the island. Also, after highlighting them, darken them so only the stable ones are more visible.
resonance has caused the island double pendulums to behave more like single pendulums
I really hope you do a video to explore and explain this phenomenon further! I'll subscribe and wait with anticipation for the next video! ❤😊
This visualization is outstanding! I’ve not seen others here mention it, but using an implicit integration scheme such as one of the Rosenbrock methods (rather than a explicit R-K integration scheme) may give more reliably accurate long-term behavior.
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.
The period of a pendulum is often approximated as being constant for any initial displacement, but in actually the displacement of a pendulum slightly affects the period of the swing. I don't recall which way the relationship goes.
「安定の島」という表現が上手い
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.
They would be converging to a loop in that 4 dimensional phase space, yes
The “chaotic” areas seem to have some order as well. What I thought was interesting is that if you focus on the image as a whole the chaotic areas start looking like characters in an alphabet
Klien Bottle video was awesome, i have subscribed to you. Your explainations with animations are good. Can you also explain the Time split light experiment vs Dual Slit Light experiment ? Eagerly waiting for your version on this topic.
The way the pixels on my screen are arranged makes it change color at certain angles.
Tracing this also works great for designing island maps for tabletop games!
i wonder if one could ever re-harmonize back into stability... maybe not on their terms but in its own stable
I wonder what a high resolution outline of the border looks like.
*Looks at map of Croatia.*
"The answer's been staring us in the face the whole time!"
Should take each data point and record the first moment in time that the second pendulum does a full flip, then plot that as a heat map on this graph.
You systematized chaos. You madman.
Oooh! It's a schooner!
Someone needs to make a paper about this, and then someone else needs to make a UA-cam video explaining what’s said in the paper.
Awesome visualization
Animation is amazing.
Probably just coincidence of length of simulation step, parameters and rounding.
Chaos theory is incredibly interesting and show that prophecy is basically only matter of knowing enough of the variables
If you calculate the dot product of each double pendulum and set the hue of each pixel accordingly you'll get a colorful animation!
Great advance. Now imagine that you can calculate the "island" without simulation, just with pure mathematics you could define something like a "fractal" of stability (this area (island) accoding to the formula is stable).
I would be tempted to save computing time by just not bothering to calculate a pendulum once it has gone chaotic.
We live in an island of stability where choas becomes order against entropy where a fractal of cycles are allowed to repeat over and over. Like an engine running, so is the universe. Everything is a fractal, where patterns are both familiar and new at the same time.
When you understand this you understand everything. Only then do you obtain free will where you are able to steer the results of wave function collapses.
Assuming you understand what you wrote, In what ways did you steer the results of wave function collapses?
@@delian66 I was trying to condense down a lot in my parent comment. It's hard to explain. I'm probably wrong but also could be right or parts of it could be right.
If you can't collapse wave functions, you are deterministic. Like a star or rock. But anything that is conscious can collapse wave functions by being an observer. Self-awareness is knowing you collapse wave functions. It is knowing you are an active participant in reality, not just an observer. Even before now, we intuitively knew we were collapsing wave functions. We just didn't have a name for it yet. So we've been self-aware for a long time. Once you become aware of it, you will want to figure out how to control it. Somehow we are able to control their collapse. Steering them. We can somehow change the probabilities.
If you decide to go check the mail, then go do it, doesn't that initiate a series of wave function collapses that leads you to checking the mail? Aren't we collapsing wave functions all the time? If we collapse them, and we are somehow collapsing them into the way we want naturally, as part of consciousness, and self-awareness is realizing that fact, then we might be able to do more advanced things if we figure out how.
Let's take rolling dice for instance, full control over your ability to collapse wave functions would mean you could throw the dice and they land on the number you intended.
I think maybe free will is simply when you become self-aware of what you are doing.
@@jamiethomas4079 Thanks for taking the time to expand your previous comment.
As I understand it, what you call collapse, is just taking action in the world.
If you exercise a lot, you can control dices when you throw them, by mastery of your fine motor skills.
You can also use thought and tools to modify the dices, so that they will be more likely to land on the sides you want.
Observation is the first stage towards meaningful action. It is not the end result.
Was this inspired by the new Lorentz attractor stability islands paper posted online recently?
I wonder if given enough time, they might return to their starting arrangement
Great video, it bends my brain.
briefly, the island resembled the girl floating on a broomstick from bad apple, which gave me an idea: bad apple but it's double pendulum chaos graphs
I love your work. I would be interested if this island would survive if you keep the number of elements but in a circle grid. Thank you.
Christian: this is really, really cool
Same sensation as seeing the afterimage shapes under my eyelids after staring at a bright source.
“It turns out that an eerie type of chaos can lurk just behind a facade of order - and yet, deep inside the chaos lurks an even eerier type of order.” -- Douglas Hofstadter
this is mesmerizing to watch
You can fit a lot of social theory and anthropology into this bad boy.
Amazing 10 hour video