Lyapunov's Fractal (that Lyapunov knew nothing about)
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- Опубликовано: 10 июн 2024
- Hi everyone! I hope you enjoy my first video. I've known about Markus-Lyapunov Fractals for a few years now, and it surprised me that I couldn't find any video explaining how they work - so I thought I'd take a stab at it myself! This is also my submission for Summer of Math Exposition 2.
Basic Lyapunov Fractal Demo in Shadertoy:
www.shadertoy.com/view/fldBWr
Further Reading:
A 1990 paper by Mario Markus that introduces the fractal: aip.scitation.org/doi/pdf/10....
Chapters:
0:00 Intro
1:00 Maps
3:20 The Logistic Map
5:44 The Bifurcation Diagram
7:29 The Lyapunov Exponent
11:35 Markus's Modified Logistic Map
16:51 The Markus-Lyapunov Fractal
18:55 Overlapping Branches
20:37 3-D Bifurcation Diagram
23:18 3-D Lyapunov Fractals
23:59 Beyond the Logistic Map
Image credits:
Mario Markus picture: commons.wikimedia.org/w/index...
Lyapunov picture: commons.wikimedia.org/w/index...
Lyapunov picture (in thumbnail): commons.wikimedia.org/wiki/Fi...
Hi all! I made a basic Lyapunov Fractal demo in Shadertoy for anyone to play around with - link in the description.
Thanks! That was super well explained, in probably the shortest practical timeframe. I understood that and all the steps to get there and when I watch it again I will probably learn a few more things that I missed. I have always loved fractals aesthetically and for their profound logical order and chaos and for their infinite nature. Diving or stepping in to a fractal for a good while is a great way to grasp a little of what infinite really means. I've always had a basic knowledge of the mathematical components that create these, but this step by step with great commentary and accompanying graphics, I feel I now fully understand the process.
I rarely sub on my first view, but that's got to be worth it. *Subbed*
Too bad that the shadertoy requires GLSL ES 3.00 or newer...
> '[]' : implicitly sized array supported in GLSL ES 3.00 and above only
> '[]' : array constructor supported in GLSL ES 3.00 and above only
> '[]' : first-class arrays (array initializer) supported in GLSL ES 3.00 and above only
> '=' : Invalid operation for arrays
> '=' : cannot convert from 'const array[2] of int' to 'highp array[2] of int'
I remember Lyapuniv graphics from way back in DOS (but they took hours to compute). Now we have many times the computing power, but STILL have to buy new stuff every year, just because the industry keeps changing specs. At least the code is simple enough that it can be translated to other languages easily -- and even JS on a modern machine is faster than an ASM program used to be back then.
This fractal looks kinda like a giant plane going off into infinity, can you apply some transformations to interpret it like this and "rotate" it to look straight onto it? Fly around it?
Wukllo90
cool dude
Woah, I thought I was watching a guy who had as many subs and views as 3Blue1Brown or something, then I got to the end of the video to like it and realized that somehow isn't the case.
This was a really great video. You explained things well and didn't over or underexplain. I've watched a lot of these types of videos before but I've never come across this topic, this was a really unique and interesting video. Your animations and the effort put into this was seriously well done and appreciated. Nice job man keep it up.
Wow thanks, I really appreciate that! A big motivator was definitely the fact that I hadn't seen this topic covered in a video before
Second this!
the same here, quite impressive done 👍👌
@@desden0va thank u, wish you sucess
I was also under the same impression, lol!
That's so cool! I've wanted to know more about the logistic map after watching a Numberphile video on it, but never found anything accessible elsewhere. This video put an end to that. Thank you and keep doing more interesting things which aren't often covered :)
thanks! In this video I was going to mention the Numberphile and Veritasium videos on the logistic map but I decided not to -- those are definitely great videos though
@@desden0va it's crazy, the Veritasium video came out on the exact same day I started reading "Chaos" by James Gleick, which is the same book they recommend in that video. Of course I got recommended yours and then subbed to you, I'm always a fan of adding another dimension to plots and especially to fractals!!! 🤯😁👍
@@revenevan11 Omg that book is sooo good. I got it when I realised that both 3blue1Brown and Robert Sapolsky (that gave the Behavioral Biology lectures) recommended it.
@@desden0vaquick question, which program do you use to plot the Lyapunov exponent? I’m trying to validate/demonstrate chaotic behaviour in granular materials, however with my lack of skill in programming quite difficult to plot the results, in consequence trying to find some community that point me/help in the right direction 😊
@@marcotulioarrietarodriguez7151For this video I used Mathematica. I've also implemented it in Shadertoy -- check the Shadertoy link in the About section of my RUclips channel page and it's the Bifurcation Diagram Demo. The Lyapunov exponent is calculated and plotted in lines 185-232, though admittedly that code isn't very readable.... (and those line numbers are subject to change as I mess with the code)
Also, it depends on if your data is discrete or continuous, I've only been working with discrete data
I like math conceptually but I kind of suck at it in practice once you go past calc 1. That said, fractals are my absolute favorite mathematical concept, and its always frustrated me that I couldn't seem to grasp the concrete mathematics behind their generation. This video made it all click for me, I absolutely loved this video, I couldn't believe I'd been watching for 20+ minutes when it was over. If you'd like to make more videos like this you'll go far! I can't wait to brag about how I was one of your first subs :P
Thank you so much! I'm glad I could help you understand these kinds of fractals :)
WOW, this was amazing and I struggle to believe this is only your first video when its so thorough, accessible, and beautifully animated. Please please please make more!!
Thanks, I appreciate that! I'll definitely make more if I find another interesting topic
this video is one of the best on this platform
great job continue making this kind of videos
thank you!
This is really high quality 'teaching' material. I knew most of this already but it's so calmly explained, very simple yet not uninteresting. That's a difficult balance to achieve.
This is one of my favorite math videos. I'm glad you didn't play music and kept it simple, because the subject is really engaging on its own. I get distracted or lulled to sleep by other math videos' otherwise very nice music. This is what I want to watch on youtube! Thanks!
I've got Fractview, which has, among many other defaults, these Lyapunov fractals, but did not understand them or how they are generated. Now I feel much more comfortable with them, though I still have much more to learn. I really appreciate your posting of this video!
This is just beautiful. Mathematics and geometrics are truly the language of the Univers. Thank you so much for this video !!
Wow, superb graphics, nice calm narration, very cool maths.
Really professional production.
The way the 2 dimensional fractal looks 3 dimensional is just mind boggling.
I love watching these videos
Not because I am good at maths
Not because I like maths
But I can gain insight and appreciation to the field of mathematics. See that all systems are interconnected, and the geometric beauty of pure maths displayed. I need higher education to understand this but for what it is it was well explained
Wow, very well done. Thorough explanations that weren't over the top, nice pacing and intermix of animations. And best of all: beautiful fractals!
I would love to be able to give someone like Lyapunov computer technology like we have today. I wish that some of the old mathematicians could see their work expressed thusly.
Keep up the nice work!
This video gave me so many new questions about the logistic map and related objects! Also your voice has the soothing cadence of Joe Pera
This was so cool. The subject is awesome, your visualisations are top notch, and you achieved to explain it very well. Maths sure can create beautiful worlds.
One small tip: you can run your audio recordings through Audacity (which is free), and use the Noise reduction effect to delete the background noise of your mic/room. Instant upgrade in sound quality :)
This is the best explanation about bifurcation and lyapunov exponent ive seen, wish i knew this when doing my thesis.
Amazing work
The very last fractal is so mesmerizing and beautiful that I almost shed tears when I saw it
I love that you didn't interpolate results when fiddling with parameters in your demonstrations.
Glad this video was recommended to me! Amazing topic, explanation and demonstrations! One of the best math videos ive seen on youtube. You just got a sub, sir.
19:30 There are more explanations for the overlapping branches actually. Consider that for a certain starting value of x the same cycle is produced, namely if the starting value for the AB cycle is x0 and the starting value for the BA cycle is A(x0)(1-x0). The two trajectories will be the same, just shifted one iteration. If you use a random starting value as opposed to 0.5 for example then the overlapping branches "diffuse" into each other. This shows that the long term behaviour can depend on the choice of x0. In some cases it may fall into a stable cycle, in others it may fall into a chaotic region. Quite fascinating, isn't it?
Another observation (that doesn't generalize) is that AB and BA really produce the same plot, just mirrored along the diagonal.
Very nice video!
yep! that's kinda talked about in the paper linked in the description, it's all about the initial few iterates
When I made the video I didn't realize that the particular AB→BA example was mirroring along the diagonal as opposed to switching which branch is on top, but I've since learned that for a sequence S, swapping all As for Bs and vice-versa will flip along the diagonal (e.g. AABA→BBAB, BBAAAA→AABBBB, etc.)
This was facinating, thank you for your work! I'm glad I better understand Lyapunov's fractal now
Woah, this is the best video i´ve ever seen about someone explaining some kind of fractal, great job!
¡Qué bello fractal!
Muy buen video; un "pacing" perfecto-Ojalá sigas haciendo más videos así ❤️
Good job!
As a mathematics teacher, I really appreciate this video!
The explanation is very clear, leaving room for non-expert mathematicians to wonder and explore details.
Also, awesome animations. Keep up the good work!
thank you! :)
Amazing. I can't believe I just found your channel - as a video creator myself, I understand how much time this must have taken. Liked and subscribed 💛
Very well done! Those renders in particular are amazing- I'm very impressed!
Just last class we were learning about Lyapunov equations in modeling linear dynamic systems, & this just showed up in my recommendations... the algorithm is magic 😳
I've seen and appreciate Lyapunov fractals for artistic reasons for years, but never knew the idea behind them so thank you for explaining.
That was really fun! Thank you for all the nice renders!
And thank you for watching! I'm glad you enjoyed it!
Excellent visualizations and explanations! Great job.
Wow! This is the best presentation of how fractals work mathematically I've seen! Keep up the great content!
Thank you for walking us through to see the end at the beginning.
I love the way the SOME has brought a renaissance of math content onto RUclips. Thank you.
I was genuinely surprised as I looked at your subscriber count, because I expected it to be several orders of magnitude larger.
I watched your whole video and I’m impressed by your easy to understand explanations of this complex topic. This channel is one of RUclipss gems, that deserve much bigger audience! Keep up your great work!
A truly beautiful video. Nicely explained. Kudos!
Great work, this was the first of your videos I've seen, but I'm certain to check more of them out. This is, to me, what YT is for. Excellent explanation, high quality presentation: I wish I had this available to me when I first learned about the topic as a youngster.
Wonderfully clear presentation of a fascinating subject, with great visuals. Will look froward to future content.
Awesome video!! I love fractals and specially the logistic map, and your explanations where really well made :) Keep it up!
that’s a great video, bud! thank you for explaining this thing and making it so easy to understand ❤ you’ve got one more subscriber now.
Congratulations, what a great first video! I love mathematical topics, and this held my attention right to the end. Thank you.
I appreciate that, thank you!
This a beautiful explanation of a topic which is hard to approach. Hope you make more videos on fractal and control theory in future!!
This is very beautifully explained and illustrated. Pure aha moment. Hope you will do more!
That was fascinating and very lucidly explained. I hadn't heard of the Lyapunov Fractals before. Thanks
I think it should be noted that if the lyapunov exponent is greater than 0, it doesnt guarantee the system cant repeat, but we wouldnt expect it to. (specifically there is a 0% chance it repeats, which is different from it being impossible) We can typically avoid this problem by starting with the right initial value, specifically one where f’=0, which is why we start at 0.5 for the logistic map. But if we started with, say 0.1, there are some occasions where the system will repeat but return a positive lyapunov exponent.
yep! the circle map in particular defies my expectations of when the Lyapunov Exponent is positive compared to its bifurcation diagram. Though I suppose that's because in my implementation its initial value isn't a place where the derivative is 0. (My implementation at the end of the video is wrong, I fixed it on my Shadertoy profile, link on my channel page. I was specifically trying to recreate the one seen on the Wikipedia page for Arnold Tongues)
I love the calm nature of this video. I’m a high school student and complicated math videos with lots and lots of equation don’t make sense to me.
But I understood everything in this video as it was very well explained. Keep it up good sir! :D
This was the most exciting thing I have watched all year
thank you. I've seen a lot of presentations about fractals, but never understood what a bifurcation plot was.
Great video! Really enjoyed your explanation, the approach and the animations. Keep it up
Finally someone who is actually human and doesn’t have a room for recording in complete silence or doesn’t know how to filter background noise
I kinda appreciate it
had to start somewhere :)
Thanks for the smooth explanation and accompanying graphics! 🙏
thank you!!
Genuinely beautiful video, thanks
you can imagine my surprise to find such a good video but it's the only one on your channel. more stuff please!!
This #some2 may be the best thing to ever happen on RUclips. This is absolutely beyond phenomenal! Wow! This was presented so well, clearly, and succinctly - well done and thank you for sharing!!!
thank you very much!!
This is a solid attempt for what I assume to be your first math video ! The storyline keeps me engaged throughout the presentation.
I think you should make more videos !
Yep, first video! I definitely learned a lot along the way about the video making process, that's for sure 😅 It was fun though, and I'll definitely make more if I come across another fun topic, thanks!
Cool first video. I'm glad I got shown it, even if the math kind of flew over my head. Looking forward to more in the future ❤
I just stumbled upon your video. What luck! I had the honor of attending a lecture on Chaos Physics at the University of Dortmund in the mid-nineties, given by Professor Marcus. The Lyapunov fractals were, of course, a part of the lecture, and I remember well recreating them on the early PCs. Thanks for bringing back those memories.
That's great! Glad you enjoyed it
Man I love this stuff thanks for all the hard work!
Very well presented and beautiful visuals to match.
Great work!
Thank you!
Thank you for this. To a regular person like me who has trouble wrapping thier mind around concepts like this, I almost understood what was going on!! That's a big deal lol. If I watch it a few more times I'm sure I'd get the whole concept. Definitely a great presentation, very impressive.
Fantastic visualizations. Wish all math/physics videos could be like this.Kudos. I Subscribed immediately.
So beautiful video!!! great storytelling
Wow, what a great video! I played around with rendering of the bifurcation diagram some time ago, but never heard of Lyapunov. Keep it up!
awesome video, I love how easy you made this to understand. While I couldn't read the mathematical notation for the functions themselves, you explained it so well I could follow the whole video without feeling lost, and the fractals were some of the most unique I've ever seen. Thank you for your hard work, I liked and subbed, hope to see more of this kind of content :)
thank you very much! Lyapunov fractals are very cool, I'm surprised they're not more popular
Wow, you have done an excellent work! Fantastic explanation
Dude. That was so fricken awesome. Amazing amazing video. You created a masterpiece that will be on the internet forever. I am thrilled and so happy i found it. I'll have to watch it more to understand the math but you explained everything beautifully.
thank you!
I love this video so much! I have not understood a topic faster than your videos 🐙
For any S containing any series of only A & B values:
Swapping A & B flips the fractal along the A=B line
Any series that repeats is identical to truncating the series at the end of the first repetition. E.g ABABAB = AB, ABAABA= ABA, ABBAABBA=ABBA
A similar set of statements can be made about S containing A, B, & C as plotted in a 3D graph. I can’t think in higher dimensions, so I don’t know how to extend it beyond A, B, & C, nor would I know how to visualize it as a fractal.
Totally awesome video. Great work. Thank you.
Great content. I like that you didn't add music to the video, helped me focus.
Beautiful. And nicely explained. Thank you.
Lyapunov is IMO underappreciated. I first tripped over his name in reference to planetary system stability in terms of a many body problem.
This makes the Numberphile Feigenbaum Constant video make so much more sense.
really nice. I wrote a texture shader for this fractal waaay back in the Alias Wavefront days. small memory footprint but horribly slow to compute.
This is mind blowing but somehow understandable. Great video!
Coming from a background in control theory, it is really cool to see some of Lyapunov's ideas in different areas! I wonder if anyone has taken ideas from fractals and employed them in optimization theory, e.g., finding an ideal initial guess for an optimization problem.
I'm not a controls engineer, but I took 3 controls classes in college and they were my favorite classes! (I was an electrical engineer major). I first learned about the Lyapunov Exponent in a state-space controls class, fun stuff
i’m literally 15 with little to no maths knowledge and you explained it so well i was able to understand it … subbed at the speed of light ngl. W video W explanation W animations W brain
thanks! :D
Excellent video! You managed to cover so much so clearly and easy to follow in such a short time. Thank you, also for sharing the shader demo! I still remember this article - I read about these fractals in the German edition of Scientific American sometime in the 90s, must have been a couple years after it was first published. But at that time it was a little above my head and I could only appreciate the images :)
Thank you!
I used to have a deathly fear of anything related to Dynamical Systems, but thanks to you (and to a friend of mine that loves DSs), my fear is basically gone! 🤩
This was very informative and extremely well presented and easy to follow - even though the math is not easy at all. The only thing I would have loved to learn was Markus' reason for modifying the logistic map. Great video, I subbed!
wow this was a very well done and informative video, good job!
This is way beyond my ability to comprehend..... but the presentation graphics, calm well-spoken narration and instructional tone tells me Desdenova is great teacher.
Superb video! Thanks for making it
thank you, fascinating stuff and very well explained!
I love all the things that you can find fractals in.
Amazing! Very well done.
Gold content mate, keep it coming :)
I don't know if I really learned that much from this but it was absolutely fascinating. Thanks.
This is such a good explanation of the bifurcation diagram and Lyapunov exponent... both of which befuddled me beforehand, but this made it make sense to me
Excellent presentation!
Really well put video, glad it got recommended to me :D
Incredible explanation. I'm glad 3b1b started the SoME thing if it lead to this video. Please make more, i love your explanation style and your voice : )
Great explanation and visualization
Awesome video! Very well made!
Keep doing what you're doing bro!
thank you for the video..if only lyapunov could have seen this video of yours, i think he would have been happy beyond
Incredibile work, interesting, well animated and well explained
Best walkthrough ever 😻
I was introduced to Chaos in my Upper Division Mechanics course (Thanks, Peter Scott!). It put the "WOW!" back into Physics for me.
Awesome video. Thank you!
Absolutely amazing video! Subscribed.