Lyapunov's Fractal (that Lyapunov knew nothing about)

[@desden0va]

Hi all! I made a basic Lyapunov Fractal demo in Shadertoy for anyone to play around with - link in the description.

[@timboatfield]

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*

[@achtsekundenfurz7876]

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.

[@grubzer1369]

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?

[@mehmetbozarslan3744]

Wukllo90

[@sunmagician27]

cool dude

[@TheStringKing7]

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 :)

[@desden0va]

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

[@revenevan11]

@@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!!! 🤯😁👍

[@germaindesloges5862]

@@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.

[@marcotulioarrietarodriguez7151]

@@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 😊

[@desden0va]

​@@marcotulioarrietarodriguez7151For this video I used Mathematica. I've also implemented it in Shadertoy -- check the Shadertoy link in the About section of my KZbin 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

[@rileyrileyrileyriley761]

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.

[@desden0va]

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

[@houstonbova3136]

Second this!

[@sleepyelk5955]

the same here, quite impressive done 👍👌

[@fss1704]

@@desden0va thank u, wish you sucess

[@justsomeguythatlikesart]

I was also under the same impression, lol!

[@lexinwonderland5741]

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!!

[@desden0va]

Thanks, I appreciate that! I'll definitely make more if I find another interesting topic

[@tricksshotsfinity4415]

this video is one of the best on this platform great job continue making this kind of videos

[@desden0va]

thank you!

[@David_Lloyd-Jones]

@@desden0va but normal people close the window when they make an audio recording...

[@Jacob-zp5ci]

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

[@desden0va]

Thank you so much! I'm glad I could help you understand these kinds of fractals :)

[@ramonbril]

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.

[@pondcurtis9725]

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!

[@sqrtyraccoon]

This is just beautiful. Mathematics and geometrics are truly the language of the Univers. Thank you so much for this video !!

[@dproduzioni]

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!

[@desden0va]

thank you! :)

[@walternullifidian]

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!

[@andrewharrison8436]

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.

[@MC_Kun]

This is the best explanation about bifurcation and lyapunov exponent ive seen, wish i knew this when doing my thesis. Amazing work

[@LaGuerre19]

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!

[@itsowenstylez3857]

Woah, this is the best video i´ve ever seen about someone explaining some kind of fractal, great job!

[@lotizorro]

¡Qué bello fractal! Muy buen video; un "pacing" perfecto-Ojalá sigas haciendo más videos así ❤️ Good job!

[@JessWLStuart]

Wow! This is the best presentation of how fractals work mathematically I've seen! Keep up the great content!

[@toasteduranium]

I love the way the SOME has brought a renaissance of math content onto KZbin. Thank you.

[@frankconley7630]

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.

[@desden0va]

thank you!

[@eleanorblake697]

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

[@nezby3945]

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

[@aventurileluipetre]

This was facinating, thank you for your work! I'm glad I better understand Lyapunov's fractal now

[@PowerhouseCell]

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 💛

[@sumandproduct]

That was really fun! Thank you for all the nice renders!

[@desden0va]

And thank you for watching! I'm glad you enjoyed it!

[@ai_is_a_great_place]

This #some2 may be the best thing to ever happen on KZbin. This is absolutely beyond phenomenal! Wow! This was presented so well, clearly, and succinctly - well done and thank you for sharing!!!

[@desden0va]

thank you very much!!

[@FrankElsner]

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.

[@desden0va]

That's great! Glad you enjoyed it

[@0range2un]

Wow, that's nice work! Great visualization and explanation, thank you, I really enjoyed this video!

[@desden0va]

thank you!

[@0range2un]

@desden0va I wish you inspiration and success in your life, you are talented a man

[@josephyoung6749]

I've seen and appreciate Lyapunov fractals for artistic reasons for years, but never knew the idea behind them so thank you for explaining.

[@firenzarfrenzy4985]

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

[@ResonanceHub]

Awesome video!! I love fractals and specially the logistic map, and your explanations where really well made :) Keep it up!

[@macronencer]

Congratulations, what a great first video! I love mathematical topics, and this held my attention right to the end. Thank you.

[@desden0va]

I appreciate that, thank you!

[@MetaMaths]

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 !

[@desden0va]

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!

[@VANQUlSHED]

you can imagine my surprise to find such a good video but it's the only one on your channel. more stuff please!!

[@picksalot1]

That was fascinating and very lucidly explained. I hadn't heard of the Lyapunov Fractals before. Thanks

[@Nymayn]

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 :)

[@carlosserrano4048]

Thank you for walking us through to see the end at the beginning.

[@negativebudgetproductions4908]

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!

[@desden0va]

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.)

[@metaljake89]

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.

[@bhavanamusunuri5781]

This a beautiful explanation of a topic which is hard to approach. Hope you make more videos on fractal and control theory in future!!

[@marquamfurniture]

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.

[@barbietripping]

This was the most exciting thing I have watched all year

[@jeffreyguilmot8772]

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!

[@Sebwolf54]

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 KZbins gems, that deserve much bigger audience! Keep up your great work!

[@baremetaltechtv]

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 :)

[@desden0va]

thank you very much! Lyapunov fractals are very cool, I'm surprised they're not more popular

[@DirtyDubstepp]

I feel honored to be one of your first one thousand subs, keep making content like this and the sky is the limit for this channel. Nice work!!

[@richardpike8748]

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

[@dzl999]

Wonderfully clear presentation of a fascinating subject, with great visuals. Will look froward to future content.

[@theairaccumulator7144]

When you started talking about the As and Bs, I instantly understood how the fractal works. Amazing video.

[@kiwischolz9811]

Thanks for the video and very detailed explanation of those fractals. This brings back some memories. Back in school, my maths teacher introduced lyapunov fractals to us and also invited Mario Markus to give a brief talk about them. I also tried to implement them myself back then, but I didn't fully understand the math and algorithm involved, so I never could recreate the actual fractals. Still managed to create some interesting results. Markus seemed to be a real kind person. He even offered to take a look at my implementation and help find my mistakes, but I never actually managed to write back to him (mainly because he gave me this weird thing called 'email address', which was still a quite new concept back in the late 90ies 😅). Now I might dig out my old code and try to fix it.

[@desden0va]

Wow, that's a great story, thanks for sharing! It definitely took me a while to understand how these fractals work, and I still don't understand all their nuances... Good luck to you if dust off the old code!

[@FrankElsner]

He is a very kind person indeed.

[@aleacecconi3786]

Thanks for the smooth explanation and accompanying graphics! 🙏

[@desden0va]

thank you!!

[@baongocnguyenhong5674]

The very last fractal is so mesmerizing and beautiful that I almost shed tears when I saw it

[@foo_tube]

This is very beautifully explained and illustrated. Pure aha moment. Hope you will do more!

[@novakonstant]

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.

[@celeph]

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 :)

[@desden0va]

Thank you!

[@stevestarcke]

A truly beautiful video. Nicely explained. Kudos!

[@nathanenright3079]

Man I love this stuff thanks for all the hard work!

[@ballpien]

Excellent visualizations and explanations! Great job.

[@aviatrixastralis17]

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 😳

[@TheAndreArtus]

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.

[@patrickgono6043]

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!

[@ИмяФамилия-е7р6и]

Excellent visualization and presentation of the material. Subscription from the first acquaintance is rare behavior for me. Thank you! I think about the noise on the background, it has already been written. But. For me personally, it is even more pleasant than the music that they like to insert in such video lessons. And this is despite the fact that I adore music, he he

[@desden0va]

thank you! I definitely think music is nice but can be distracting... and luckily I have since learned how to process the audio to avoid background noise

[@ameteuraspirant]

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 ❤

[@fibbooo1123]

Very well done! Those renders in particular are amazing- I'm very impressed!

[@johncoppinger2241]

I don't know if I really learned that much from this but it was absolutely fascinating. Thanks.

[@haf3113]

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

[@desden0va]

thanks! :D

[@dmytrk]

Omg, such amount of work to make those visualizations🤯

[@desden0va]

definitely had some late nights 😅

[@treelibrarian7618]

I have wanted to understand this family of fractals since I was a child, and never found anything complete and comprehensible till today - thankyou so much 🙂

[@desden0va]

I'm glad my video helped you understand! :)

[@Curbiture]

Very well presented and beautiful visuals to match. Great work!

[@desden0va]

Thank you!

[@andreaalflavendett]

Wow, you have done an excellent work! Fantastic explanation

[@maximearmand8903]

I love this video so much! I have not understood a topic faster than your videos 🐙

[@dAni-ik1hv]

This vid is REALLY good, and I feel that someone with no knowledge of this topic (given basic calc understanding) could easily understand this with very little research, so overall a really amazing video.

[@desden0va]

thank you! that's what I was hoping to accomplish here

[@dylanberger8701]

I love that you didn't interpolate results when fiddling with parameters in your demonstrations.

[@mapleigue]

Great video! Really enjoyed your explanation, the approach and the animations. Keep it up

[@TheDRAGONFLITE]

This is mind blowing but somehow understandable. Great video!

[@kirawr8064]

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

[@desden0va]

had to start somewhere :)

[@edgeblur]

Fantastic visualizations. Wish all math/physics videos could be like this.Kudos. I Subscribed immediately.

[@tapioms]

So beautiful video!!! great storytelling

[@lukacs1130]

A very lucid explanation of a complex mathematic process. You show promise as a video expositor. Good Job! (my background is in physics and information theory. I know quite a bit about the more commonly displayed fractals but had not seen the Lyapunov), Mike Lukacs

[@UserAnonymus1995]

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 : )

[@mliittsc63]

thank you. I've seen a lot of presentations about fractals, but never understood what a bifurcation plot was.

[@oosmanbeekawoo]

Yes! Your first video is very good! The one important thing I understood from you is how all of Chaos is based on this one x := x(1 - x) formula. What are its origins, what does it mean and who and how was it discovered.

[@desden0va]

I would love to have given more historical context, especially regarding how Lyapunov created the Lyapunov Exponent, but sadly I didn't have time... I think Veritasium's video on the Logistic Map gives more historical info about that equation though

[@lunafoxfire]

That's awesome! I never would have thought about doing that alternating r values thing but it sure makes some cool pictures. Definitely makes me want to code this up and play with some functions.

[@desden0va]

Go for it!

[@houstonbova3136]

Pics or it didn’t happen. JK but if you do please share the repo I’d love to poke around with and maybe run some extended GPU renders

[@machineman8920]

Fantastic. Ever since i discovered python i was fascinated by what can you do with self referential loops, i remember plotting chaos of the newton's method for the first time and being in awe about my trashy notebook glued together plot, definitely gonna explore this stuff fiddling with the parameters is always the best part. Thanks!!!!

[@enbyarchmage]

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! 🤩

[@noahbliss1589]

Just cited you in my research proposal, thanks for teaching me about Lyapunov exponents!

[@desden0va]

wow that's great, thanks!

[@calvinkielas-jensen6665]

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.

[@desden0va]

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

[@623-x7b]

Great content. I like that you didn't add music to the video, helped me focus.

[@noahbawdy3395]

I love all the things that you can find fractals in.

[@eddboy6063]

358 subs? What a hidden gem this content rocks

[@wynnschaible]

I've been working artistically with Lyapunov fractals for about a decade but this is the first i've understood what they're about and why!

[@lunafoxfire]

wow the stuff on your channel is awesome :D

[@wynnschaible]

@@lunafoxfire Thanks very much LL! Plenty of it in various forms!

[@TheDanubeDepleter]

Genuinely beautiful video, thanks

[@mehmetedex]

holy moly sweet ravioly I am inspired thanks to you sir respect for your hard work

[@samcavanagh7993]

wow this was a very well done and informative video, good job!

[@colinbrown7947]

I've wanted to know how these work for a while, this is great!

[@desden0va]

Thank you!

[@BritishBeachcomber]

2:59 The correct terminology is Convergent, Cyclic, Divergent, and Chaotic.

[@NoNTr1v1aL]

Absolutely amazing video! Subscribed.

[@ghosthermes]

What a powerful visualizer

[@MV-vv7sg]

Very cool, thanks for this amazing video!! Cool to know even more crazy fractals come from plotting more properties of the bifurcation curve. Bifurcation leading to Mandelbrot set was pretty amazing, but the Lyapunov exponent fractals when moving between different A,B sequences is even cooler. Now I just want to see the Lyapunov Markus Fractal in the complex plane too (3D Fractal) AND how it might change when you have more than 2 values of ‘r’. How will it all change when you have 3 or 100 values? You’ve opened up a rabbit hole I wish I had the computing skills to explore!

[@douglasstrother6584]

I was introduced to Chaos in my Upper Division Mechanics course (Thanks, Peter Scott!). It put the "WOW!" back into Physics for me.

[@tommyb6611]

thank you for the video..if only lyapunov could have seen this video of yours, i think he would have been happy beyond