the issue is that you then lose the clear boundaries, and it's a little harder to see the fractal.. it still would probably be a pretty cool visualization though.
Fascinating. I wonder how much the stability patterns are dependent on the chosen integration method. That is, I'm wondering how much of what we're seeing is the behavior of the differential equation and how much of it is due to the chosen method and error estimate tolerances, or even floating point accuracy.
I assume the Shadowing Lemma applies to this system, but I could be wrong. The Shadowing Lemma says that every computed orbit stays within some small distance of a true orbit, i.e. small errors don't matter.
I was thinking the same thing. I would be interested to see one particular time step compared across multiple different integretion schemes, e.g. what would time step t=2.0 look like with delta-t of 0.1, 0.01, 0.001, etc.
@@qzamboni The thing here is that according to chaos theory, adding a small error to a step would lead to huge discrepancies down the line so not sure if the Shadowing Lemma would apply here. That said it's the first time I hear of this so I could be wrong.
Copying what someone else said, it's probably an island of stability, where nearby pendulums converge into the same shape rather than falling into chaos
I am thinking about ways to "isolate" the central fractal from the surrounding noise. Performing computation on the noisy regions is, in a sense, "wasted", since a noisy region never seems to further change from being just white noise. In the Mandelbrot set, after each iteration, if the iterated value escapes the unit disk in the complex plane, we can safely discard it, as we know the value of this point will diverge into infinity. Usually corresponding starting point is colored white. We could do a similar thing here, and find a range of values for angles and velocities that we know for certain is in the chaotic region, and discard the pendulums that enter that region, then color the corresponding positions white. I believe this should eventually filter out ALL (or most) of the noise outside of the main fractal, even outside this filter region, since unstable positions will, by pure chance, "trickle into" the filter region, like water circling into a sink drain.
@@quantumsoul3495 I don't think so. The small island of stability at x=101°, y=+115° appears to be a basin of a stable attractor, where the pendulums get more and more similar by all approaching the same single limiting cyclic motion, instead of drifting apart in their behavior. Even with limited numerical precision of the simulation, they will stay similar, as long as the corrective attraction of the attractor is greater than the floating point errors.
i wonder if you could instead use the difference between neighboring pixels, and pixels which are surrounded by other values which are all sufficiently different could be discarded. in this way i think you could also adjust the desired noise amount? idk how to express this idea rigorously or even really coherently lol
@@goji_crafter I think that won't quite work for what I intended, which is filtering out the "useless" white noise (useless in the sense that it has no structure). Because even in the non-white-noise regions, you can still get quite dense changes in behavior. Such as the central oval, which starts as a fairly stable region and gradually fills up with higher frequency noise from the top-left and bottom-right. But notice how it never quite becomes _pure_ white noise, unlike the outside regions. This tells us there's some subtle underlying stable behavior which prevents degradation into white noise. Visually, you can see the boundary between the two different kinds of noise, which still shows the outline of the central oval.
Oh, thank you. Immediately shared with whoever is around. It is interesting if the island of stability is the same with different pendulum angle"resolutions".
This is an incredible addition to the collective understanding about Chaos Theory. My god what a visual treat. Your previous one was badass, I’m blown the fuck away by this one. Very well done I really appreciate this kind of thing.
I was looking at this graph and I was wondering why it is not symmetrical around all 4 corners and the answer is because of gravity. And a double pendulum with no gravity is not chaotic. I was wondering how does the transition between no gravity and small and large number of gravity will look like
In this case, changing gravity is equivalent to just changing the timescale of the system, or its length. Want to see what this would look like with stronger gravity and the same lengths? Play it on ×2 speed
Have to wonder how much of what we're watching is real and how much is an artifact of the floating-point representation being used. The nature of the system means that any finite rounding will tend to dominate the simulation over a relatively short period of time, which suggests that if infinite-precision calculations were possible, the picture that comes out would look quite different to what we're observing. What is known about the magnitude of this effect in the context presented here? My math isn't strong enough to have a good intuition beyond the sense that rounding *might* have an outsized impact.
It should be possible to make an "accurate" picture without infinite-precision calculations if you control the error terms well enough. By which i mean that at any given time we can calculate a picture that is as close to the real one as we want (this might fail here because of the imprecisions). it might be very inefficient though.
@@Galinaceo0 The problem is that with a chaotic system (basically, any system where the result loops back into itself without diminishing returns), the lack of precision always dominates the output after a certain number of iterations. Any region of the image that doesn't converge to some stable value will always look very different after many iterations, even if the error is 1.0x10^-200 or something else very, very small. The question that I can't answer is the amount that this instability would affect the "interesting" parts of the image highlighted in the animation. Would there still be an interestingly-shaped island in a "real" version of this with infinite precision, and would it have a similar shape? No idea but it is interesting to contemplate.
@@cthonianmessiah That can be fixed by recomputing the whole thing from the beginning with more and more precision each time you want to see more into the future. The problem however is, as i said, that this is extremely inefficient.
@@Galinaceo0 Yes, you hit a wall due to nonlinear behavior and no matter how much compute you throw at the simulation, the progress basically stops. This is, for example, why weather forecasts stop at 7-10 days even as available compute resources increase exponentially.
@@cthonianmessiah Well in weather simulation it's actually impossible to measure everything we need to compute it to a given precision. for double pendulums, as long as we got enough memory and time it is always possible to compute.
The bulbs of the Mandelbrot converge to periodic orbits. The center region seems to be where the angle is small enough that they swing in unison. I would be curios what the ratio of oscillation period is in the stable region.
I remember seeing similar effects back in the 1980's on my ZX Spectrum, you used the random number generator function to plot pixels and ended up with interesting patterns. Of course it was simply down to the method used to generate random numbers being a psuedo-random function.
I wonder what it would look like if we considered starting positions with initial angular velocity as well. Sure, the space would be 4d, but if we only consider a slice of pendulums with the same total energy we would get a 3d shape. Interesting how trajectories might look there. (Since energy is preserved all of them would lie in the slice.)
This is known as phase space. Some issues are that the constant energy 3D surface is curved in 4D, but also we can't even see into 3D volumes very well.
fun thing to do here: turn the video up to various resolutions and then take a screenshot of any of the little regions of pure noise. Its really neat to see how many different colors you can find, with most of them averaging out to dull earth tones.
Interesting how there is a definite boundary between the perturbative regime and the chaotic region where it's just noise. You can see the two oscillatory modes in the perturbative region, each with its own frequency. The yellow/green one is the lowest frequency mode, and the red/blue one is so fast it looks purple. The low frequency mode looks stable at relatively high amplitudes, which really is surprising to me. And then there is that tiny island where it also happens to be stable.
Yes, very close to the center is the perturbative regime where you can study the pendulum quite easily. It turns out there are two modes, one of which oscillates between yellow and green in this color code, and the other oscillates between red and blue. The surprising part is how sharp the boundary is between that quasi-stable regime and the chaotic region. You do get a Newton fractal, because the four colors must always touch.
I believe that if you change the coorsinate system we're gonna see some interesting things too. Like polar coordinates or something like that. There's probably a system (maybe projecting on an object ?) where we recognise pattern since it's deterministic. Maybe the part you zoomed onto is gonna be round or squared ? You did a great job simulating that shit ahah
@@UniHorned Yes. Also, the chart shows that within certain ranges of starting positions where both pendulums start with similar angles, they maintain a periodic pattern. Outside of that, their motions become more chaotic.
You inspired me to try it myself with that last video and I got a pretty nice result by computing the average absolute central difference (dx_angle1, dy_angle1, dx_angle2, dy_angle2) / 4. Then the chaotic values are bright and the ordered ones are black. I'm also pretty sure it's fractal in nature so you could do a zoom in which might be cool.
There is something about this video, that explains order within chaos, fractals, and other forms in which the Universe is revealing its inner working concretely
Imagine extending it to 3D where the Z axis represents a change in the ratio of the outer mass to the inner. And then 4D, where the W axis represents a change in the ratio of the outer arm length to the inner arm length.
In case you're curious, the biggest one I found is close to the y-axis, between -30 and 0 degrees. Note, though, that every island has a mirrored version due to the nature of the double pendulum, so there are two of them.
What if the angle are relative to each other? Right now they are absolute degrees, but I wonder we get better resolving on the stable regions if the angles are different. I wonder how mass impacts this, also, it wouldn't surprise me if there is a Julia/Mandelbrot set embedded somewhere in here
Very cool. I suspect the "stable" part (however one would formalize that) is actually connected at every point in time, maybe even simply connected. Similarly to how the Mandelbrot set is actually simply connected.
What about a gradient based on the relative velocities of the ends of the first and second pendulums? It’s a function of the angles certainly so we wouldn’t learn any more really but we would get a different cool fractal.
This would seem to me to be a two dimensional slice of a 4d object which would include the initial velocities of each pendulum, this slice being where the initial velocities are zero. Would it be possible to do this animation for a slice where the combined potential and kinetic energy of each pendulum is the same by adding whatever clockwise velocity is needed to one of the arms (the clockwise velocity can be negative), with the double pendulum that is pointing straight up at the start having zero added velocity?
I'm curious how this fractal compare to the one for time to first flip. The barrier between mess and not mess doesn't appear to be the same as the one between flip and no flip (at least not around theta2=180 and theat1=0)
Thank you for this wonderful video. Any hints how this was calculated? Maybe you have something on github? One very interesting thing is why this graphs isn't symmetrical via x/y axes...
Interesting. What happens if you define the second axis as the second pendulum’s angle relative to the first one, rather than to the absolute reference frame?
idea: add a thing that shows the precentage of all pendulums that are each color. example 66.3 % blue 21.2%yellow 41.8%green 11.1% red it would be cool to see how it changes
I want to see a simulation of some pendula from that weird island of stability. We should see really stable motion even with small perturbations of initial conditions, but why does this happen, what are the staring angles that do this? I assume it's some form of resonance between the two
would be interesting to see some of the pendulums with starting conditions in that small island of stability. also is the fractal shape accurate, or is it due to bias from the floating point error that accumulates
Very curious what this would look like if you use a gradient instead of 4 separate quadrants
Definitely needed
the issue is that you then lose the clear boundaries, and it's a little harder to see the fractal.. it still would probably be a pretty cool visualization though.
Im pretty sure someone already made that in a different video
It would just look less defined.
There is significance to the divisions
Fascinating. I wonder how much the stability patterns are dependent on the chosen integration method. That is, I'm wondering how much of what we're seeing is the behavior of the differential equation and how much of it is due to the chosen method and error estimate tolerances, or even floating point accuracy.
I assume the Shadowing Lemma applies to this system, but I could be wrong. The Shadowing Lemma says that every computed orbit stays within some small distance of a true orbit, i.e. small errors don't matter.
I was thinking the same thing. I would be interested to see one particular time step compared across multiple different integretion schemes, e.g. what would time step t=2.0 look like with delta-t of 0.1, 0.01, 0.001, etc.
@@qzamboni The thing here is that according to chaos theory, adding a small error to a step would lead to huge discrepancies down the line so not sure if the Shadowing Lemma would apply here. That said it's the first time I hear of this so I could be wrong.
@@qzamboniI'm pretty sure the double pendulum doesn't have any hyperbolic invariant sets
I had the exact same thought
In the bottom right corner: Are we seeing here the shape of chaos or the shape of floating point inaccuracy?
The question is does the Shadowing Lemma apply to this system.
Copying what someone else said, it's probably an island of stability, where nearby pendulums converge into the same shape rather than falling into chaos
What even is chaos?
ugh….this youtuber doesn’t check his comments. i guess we’ll never know for sure.
I had the same question. 😀I wonder how this animation would look different with 64 bit floating point math?
It's very cool that some islands of stability appear in some places, like the one you are showing in the bottom right, it's super counterintuitive !
All the explanation at the start was excellent and very interesting and then I forgot what everything meant as soon as it started moving 😂
I am thinking about ways to "isolate" the central fractal from the surrounding noise.
Performing computation on the noisy regions is, in a sense, "wasted", since a noisy region never seems to further change from being just white noise.
In the Mandelbrot set, after each iteration, if the iterated value escapes the unit disk in the complex plane, we can safely discard it, as we know the value of this point will diverge into infinity. Usually corresponding starting point is colored white.
We could do a similar thing here, and find a range of values for angles and velocities that we know for certain is in the chaotic region, and discard the pendulums that enter that region, then color the corresponding positions white.
I believe this should eventually filter out ALL (or most) of the noise outside of the main fractal, even outside this filter region, since unstable positions will, by pure chance, "trickle into" the filter region, like water circling into a sink drain.
Doesn't it all eventually fall into white noise?
@@quantumsoul3495 I don't think so. The small island of stability at x=101°, y=+115° appears to be a basin of a stable attractor, where the pendulums get more and more similar by all approaching the same single limiting cyclic motion, instead of drifting apart in their behavior.
Even with limited numerical precision of the simulation, they will stay similar, as long as the corrective attraction of the attractor is greater than the floating point errors.
@Adam-zt4cn I disagree
i wonder if you could instead use the difference between neighboring pixels, and pixels which are surrounded by other values which are all sufficiently different could be discarded. in this way i think you could also adjust the desired noise amount?
idk how to express this idea rigorously or even really coherently lol
@@goji_crafter I think that won't quite work for what I intended, which is filtering out the "useless" white noise (useless in the sense that it has no structure).
Because even in the non-white-noise regions, you can still get quite dense changes in behavior. Such as the central oval, which starts as a fairly stable region and gradually fills up with higher frequency noise from the top-left and bottom-right.
But notice how it never quite becomes _pure_ white noise, unlike the outside regions. This tells us there's some subtle underlying stable behavior which prevents degradation into white noise.
Visually, you can see the boundary between the two different kinds of noise, which still shows the outline of the central oval.
Oh, thank you.
Immediately shared with whoever is around.
It is interesting if the island of stability is the same with different pendulum angle"resolutions".
I turned up the resolution of the video on my phone and the UA-cam app crashed
Nahhh💀
The 4K version of this video has an average bitrate of 126.6 Mbps! A challenge to Google's VP9 encoder.
"ow! my bitrate!"
I was going to drop a comment about the encoder not knowing what hit it when the sim starts up :)
Thanks for that info, Shota World
lol
Literally crashes my UA-cam app lol
Always wanted to see a visualization like this, good work!
This is an incredible addition to the collective understanding about Chaos Theory. My god what a visual treat. Your previous one was badass, I’m blown the fuck away by this one. Very well done I really appreciate this kind of thing.
Chaotic processes leave behind fractal structures :D
So chaos lives in non-integer dimensions, makes sense
@@Bombito_the unknowable becomes more knowable with each passing day
It would be cool to have another plot with a pendulum from a given pixel, so we can see the relative timescale and chaotic nature up close
I was looking at this graph and I was wondering why it is not symmetrical around all 4 corners and the answer is because of gravity. And a double pendulum with no gravity is not chaotic. I was wondering how does the transition between no gravity and small and large number of gravity will look like
In this case, changing gravity is equivalent to just changing the timescale of the system, or its length. Want to see what this would look like with stronger gravity and the same lengths? Play it on ×2 speed
@Rotem_S incorrect. It is a non linear differential equation and it wouldn't simply double the speed.
@@mahditr5023it's a linear effect on a nonlinear differential equation
This video is Taco Bell for compression algorithms.
Have to wonder how much of what we're watching is real and how much is an artifact of the floating-point representation being used. The nature of the system means that any finite rounding will tend to dominate the simulation over a relatively short period of time, which suggests that if infinite-precision calculations were possible, the picture that comes out would look quite different to what we're observing.
What is known about the magnitude of this effect in the context presented here? My math isn't strong enough to have a good intuition beyond the sense that rounding *might* have an outsized impact.
It should be possible to make an "accurate" picture without infinite-precision calculations if you control the error terms well enough. By which i mean that at any given time we can calculate a picture that is as close to the real one as we want (this might fail here because of the imprecisions). it might be very inefficient though.
@@Galinaceo0 The problem is that with a chaotic system (basically, any system where the result loops back into itself without diminishing returns), the lack of precision always dominates the output after a certain number of iterations. Any region of the image that doesn't converge to some stable value will always look very different after many iterations, even if the error is 1.0x10^-200 or something else very, very small.
The question that I can't answer is the amount that this instability would affect the "interesting" parts of the image highlighted in the animation. Would there still be an interestingly-shaped island in a "real" version of this with infinite precision, and would it have a similar shape? No idea but it is interesting to contemplate.
@@cthonianmessiah That can be fixed by recomputing the whole thing from the beginning with more and more precision each time you want to see more into the future. The problem however is, as i said, that this is extremely inefficient.
@@Galinaceo0 Yes, you hit a wall due to nonlinear behavior and no matter how much compute you throw at the simulation, the progress basically stops.
This is, for example, why weather forecasts stop at 7-10 days even as available compute resources increase exponentially.
@@cthonianmessiah Well in weather simulation it's actually impossible to measure everything we need to compute it to a given precision. for double pendulums, as long as we got enough memory and time it is always possible to compute.
The bulbs of the Mandelbrot converge to periodic orbits. The center region seems to be where the angle is small enough that they swing in unison. I would be curios what the ratio of oscillation period is in the stable region.
very cool! would love to see a version of this where the points are colored by the difference between their angles and their neighbors
I always wondered what the wavy shapes were as you zoom in and out of pictures of a monitor screen
That's called a moiré pattern.
“Still think there’s nothing to chaos theory?”
-Half Life 1
I remember seeing similar effects back in the 1980's on my ZX Spectrum, you used the random number generator function to plot pixels and ended up with interesting patterns. Of course it was simply down to the method used to generate random numbers being a psuedo-random function.
adding damping could be cool. it would start off ordered, then become chaotic and then become ordered again right? ending with a single color
It would always end with both pendulums pointing down due to simulated gravity.
I wonder what it would look like if we considered starting positions with initial angular velocity as well. Sure, the space would be 4d, but if we only consider a slice of pendulums with the same total energy we would get a 3d shape. Interesting how trajectories might look there. (Since energy is preserved all of them would lie in the slice.)
The double pendulums usually start with no velocity, with the only force acting on them being the g-force
This is known as phase space. Some issues are that the constant energy 3D surface is curved in 4D, but also we can't even see into 3D volumes very well.
3:56 purple
fun thing to do here: turn the video up to various resolutions and then take a screenshot of any of the little regions of pure noise. Its really neat to see how many different colors you can find, with most of them averaging out to dull earth tones.
I'd recommend looking through Henon & Heiles seminal paper, "The Applicability of the Third Integral of Motion".
Am I correct in assuming this model doesn't simulate energy loss through friction?
Interesting how there is a definite boundary between the perturbative regime and the chaotic region where it's just noise.
You can see the two oscillatory modes in the perturbative region, each with its own frequency. The yellow/green one is the lowest frequency mode, and the red/blue one is so fast it looks purple.
The low frequency mode looks stable at relatively high amplitudes, which really is surprising to me. And then there is that tiny island where it also happens to be stable.
If you look very closely at the top of the oval with the highest quality, you can see a long island with what looks like a mix of all 4 colors
That little stable island is so bizarre! I love this!! I wonder if we could map the change over time in the third dimension what shapes would come out
Gorgeous! Thank you! Your map deserves Mandelbrot-status
Loving it. Is there a mathematical something that describes the stable area?
Yes, very close to the center is the perturbative regime where you can study the pendulum quite easily. It turns out there are two modes, one of which oscillates between yellow and green in this color code, and the other oscillates between red and blue.
The surprising part is how sharp the boundary is between that quasi-stable regime and the chaotic region.
You do get a Newton fractal, because the four colors must always touch.
@@Ricocossa1they don't touch all in the start, why do they start touching?
If each of these frames were to be put together into a 3D shape, would one get four three-dimensional fractals?
I believe that if you change the coorsinate system we're gonna see some interesting things too. Like polar coordinates or something like that.
There's probably a system (maybe projecting on an object ?) where we recognise pattern since it's deterministic.
Maybe the part you zoomed onto is gonna be round or squared ?
You did a great job simulating that shit ahah
Since there is no friction, the pendulums will move forever. What exaxct state / time are you simulating?
Very cool, great job mate
Mesmerizing!
Very informative, and very long
Exactly what I need
fine visualization 👍👍👍
amazing demonstration of entropy
woaaah nifty evolving fractal you've got there
My youtube aplication crashed after like 20 seconds into the simulation
i understand literally nothing in this video but its interesting
Me to
I found it to be really well explained.
@@NickCombs The beginning, sure
@@UniHorned Yes. Also, the chart shows that within certain ranges of starting positions where both pendulums start with similar angles, they maintain a periodic pattern. Outside of that, their motions become more chaotic.
Only comment on this vid I understood
You inspired me to try it myself with that last video and I got a pretty nice result by computing the average absolute central difference (dx_angle1, dy_angle1, dx_angle2, dy_angle2) / 4. Then the chaotic values are bright and the ordered ones are black. I'm also pretty sure it's fractal in nature so you could do a zoom in which might be cool.
Do you have an upload anywhere of the result of that?
Отличная работа!
so THIS is what a portal inside a portal looks like. neat!
It’s a fractal isn’t it? Start the zoom videos.
UA-cam compression did you dirty
There is something about this video, that explains order within chaos, fractals, and other forms in which the Universe is revealing its inner working concretely
Anyone else notice that the noise in the crevices of the shape have a yellow tint to them?
Imagine extending it to 3D where the Z axis represents a change in the ratio of the outer mass to the inner. And then 4D, where the W axis represents a change in the ratio of the outer arm length to the inner arm length.
Is this the only island of stability?
I think I can see several more, look around the bottom near the -30 on the X axis for example
In case you're curious, the biggest one I found is close to the y-axis, between -30 and 0 degrees. Note, though, that every island has a mirrored version due to the nature of the double pendulum, so there are two of them.
I see two at about 30 degrees x axis, ~145 and ~160 y-axis, I guess that's the mirror of what IsZomg is saying.
What if the angle are relative to each other? Right now they are absolute degrees, but I wonder we get better resolving on the stable regions if the angles are different. I wonder how mass impacts this, also, it wouldn't surprise me if there is a Julia/Mandelbrot set embedded somewhere in here
i would love to see a version of this where it goes until all the pendulums run out completely (even if it takes more than a few hours)
Does anyone know what frequency the really fast red-blue region and the slow yellow-green regions are oscillating at?
It would be interesting to see this mapped on a hilbert curve
Ok now get a double pendulum and put it in that spot.
Oh my sweet computation time
Seems a subset of the three body problem, with some stable paths and other unstable paths.
How much does the result change with a finer mesh? Or does it not change at all other than the resolution of what you see?
Very cool. I suspect the "stable" part (however one would formalize that) is actually connected at every point in time, maybe even simply connected. Similarly to how the Mandelbrot set is actually simply connected.
You just gotta zoom in far enough.
cant wait to write a shader for this
What about a gradient based on the relative velocities of the ends of the first and second pendulums? It’s a function of the angles certainly so we wouldn’t learn any more really but we would get a different cool fractal.
Thank you so much for sharing this! What should I learn to make an animation like this of my own?
processing probably
The cool topology guy!
FRACTLS❤
i checked for the slope of the lines that swipe the small island on the bottom right and its -2.13, i had hopes it would be -e
Beautiful, how finals quadrants are the "mix" of colors from the opposites.
its so messed up that this is so different yet not the first time I've seen it
This would seem to me to be a two dimensional slice of a 4d object which would include the initial velocities of each pendulum, this slice being where the initial velocities are zero. Would it be possible to do this animation for a slice where the combined potential and kinetic energy of each pendulum is the same by adding whatever clockwise velocity is needed to one of the arms (the clockwise velocity can be negative), with the double pendulum that is pointing straight up at the start having zero added velocity?
Can we zoom into this fractal please?
youtube video compression’s worst nightmare
Maybe, I said maybe, this patterns can occur because of the running hardware precision limitations.
Is there another "island" in the inverse cornoer as well? Due to the nature of the simulation it would be reasonable to assume so.
I don't know why, but something tells me that I'm going to need the highest bitrate possible for this video.
I wonder what happens if you take the point at the end of the pendulum and map its X and Y axes to hue and brightness respectively
Would be nice to see if they all would be connected (standing on same table, for example, so would sync with time)
Suggestion: add a few contour lines showing equipotential surfaces
I wonder about the difference in angle values. Maybe a gradient?
I'm curious how this fractal compare to the one for time to first flip. The barrier between mess and not mess doesn't appear to be the same as the one between flip and no flip (at least not around theta2=180 and theat1=0)
Please tell me you only simulated half of the pendulums, and then mirrored their motion to the other half
Thank you for this wonderful video. Any hints how this was calculated? Maybe you have something on github? One very interesting thing is why this graphs isn't symmetrical via x/y axes...
I think even more pixels wouldn't be bad.
Grooviest beats eva!
a battle between chaos and entropy, good VS evil type aah🗣
I would love to see the behaviour of the pendulum in the isolated stable region
Is there gravity? Resistance? Is energy introduced? I don't understand the simulation aspect.
Very curious what this would look like if you mask out the instability region (the noise) from the stabil region, and how does it crals inward
Hey! Does the pattern brakes later? What happens, when blue thin waves become less then 1 pendulum?
This is with zero friction, correct?
Interesting. What happens if you define the second axis as the second pendulum’s angle relative to the first one, rather than to the absolute reference frame?
idea: add a thing that shows the precentage of all pendulums that are each color. example
66.3 % blue
21.2%yellow
41.8%green
11.1% red
it would be cool to see how it changes
thats a nice mug
i wonder if anyone will get that
Looks good, but what is the relation between the pendulums?
I want to see a simulation of some pendula from that weird island of stability. We should see really stable motion even with small perturbations of initial conditions, but why does this happen, what are the staring angles that do this? I assume it's some form of resonance between the two
Use inverse kinematics to initialize all the starting positions that would be really cool
would be interesting to see some of the pendulums with starting conditions in that small island of stability. also is the fractal shape accurate, or is it due to bias from the floating point error that accumulates
Have you/someone tried, whether it translates to reality? Eg., whether is that region around 100 deg stable in our univesrse?
It's symmetrical. You only have to simulate one half, and then rotate it 180 degrees around the 0,0 point.
it's not symmetrical, it would be if there were just two colors to represent positive and negative difference of angles but this is not that video
@@mateuszodrzywoek8658The point still holds. You could simulate half, rotate 180, and invert the colors
That's actually what they did
maybe they did that... You wouldn't know😉
I didn't get any of this.
But it was beautiful.
can u calculate the magnetic isteresis o a dominion?