What everyone gets wrong about the butterfly effect
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- เผยแพร่เมื่อ 18 มิ.ย. 2024
- Chaos theory is weirder than you think. Try everything Brilliant has to offer for free for 30 days: www.brilliant.org/simonclark
When people talk about the butterfly effect, they miss a large part of what makes chaos theory so incredible. Lets talk about that, and what it tells us about Earth's atmosphere.
My book Firmament: geni.us/firmament
Strogatz textbook on chaos: geni.us/strogatz
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Music by Epidemic Sound: nebula.tv/epidemic
Some stock footage courtesy of Getty.
Edited by Luke Negus.
What does the butterfly effect say about chaos theory? What is the science of the butterfly effect? Where does chaos theory come from? Is the butterfly effect real? Who discovered the butterfly effect? All this and more in this science video essay about the butterfly effect and how chaos theory is weird. Weirder than you think. I made it through the whole video without making a single Warhammer reference!
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Very, very nice presentation. Viewers should know that Lorenz had two assistants, Ellen Fetter and Margaret Hamilton, who contributed a lot to this discovery and research. Hamilton was so bothered by chaos that she left Lorenz's lab and moved over to NASA where she was responsible for the on board flight software for the Apollo program. Her software got the astronauts to the moon and back, including Apollo 13. (Co-author of An Experimental Approach to Non-Linear Dynamics and Chaos, Tufillaro et al.)
Thank you for this extra bit of information !!
Ah yes, moving from chaos theory to the "determinism" of calculating 3D gravity trajectories through space. Where the billions of small untracked asteroids in the solar system are just slightly perturbing all predicted vehicle motion paths..
Awesome video Simon! Two fun facts:
1) Lorenz probably did not do the programming himself (as this story was often told), and was most definitely by Margaret Hamilton. She programmed all of the weather forecasting and other meteorology software for Lorenz, and quite possibly identified this before Lorenz did. In a story as old as time, her name was not included in the papers that Lorenz published on Chaos theory (cf Burnell, Franklin etc.) so exact attributions has been made impossible. And this is the same Hamilton who led the team for writing the programs for the Apollo guidance systems.
2) Public consciousness of Chaos Theory came about thanks to Jurassic Park with hot sexy Ian Malcolm talking about chaos theory and the inherent unpredictability of nature runs undercurrent through all of Jurassic Park (in the movie and moreso in the book), precisely because Micheal Crichton is a big time climate change denier, and points to chaos theory as the reason why climate models are wrong.
One additional book recommendation is Chaos by James Gleick, which is a really good overview of the entire history of the field, including the story of how Lorenz discovered this in the first place.
As a huge space geek, I'm now obligated to read as much as I can about Margaret Hamilton. Thanks! Also...
I would have *never* guessed that the person behind Jurassic Park rejected global warming. It seems so fitting with that work's theme of "just because we can, doesn't mean we should."
This makes so much more sense as an explanation for chaos theory than anything I've ever heard. The fact that we can't reliably and perfectly predict the future (in this case with weather) is, for the most part, because we can't measure something infinitely small - and that infinitely small difference causes wild divergence. Thats honestly awesome.
While the game's use of the butterfly effect and chaos theory is I'm sure not particularly scientific, there's a quote in Life is Strange that I really like. "I wish I could stay in this moment forever... but then, it wouldn't be a moment." This video explaining that the state of the universe at any given moment is never what it's been like before, or what it will ever be like again, makes one of my fave quotes from one of my fave games even more meaningful, so thanks Dr. Simon!
Yesss
I know it was just an additional point in the whole lesson, but I don't think there's a single better way to teach why weather predictions are still so inaccurate and hard to make, despite the computing power and our knowledge about how weather works having improved so much over the years. Not what I expected to learn from clicking this, but always happy to be surprised by getting taught something new
It would be cool if this gets pinned
"inaccurate"?
3 days is pretty good - 90% accuracy
This is a super refreshing take on chaos theory, well done as always Simon!
Ray Bradbury wrote A Sound of Thunder in 1952 about a time traveler stepping on a butterfly in the age of the dinosaurs and changing the present and Lorenz wrote the attractor in 1963?
Is the “butterfly effect” term popular because of Ray Bradbury?
I fly paragliders, we rely on convection to stay airborne. In my experience so far it does feel like no two days are exactly alike. While weather forecasts indicate favourable conditions, the available lift and "feel" of the air is slightly different every time.
It's fascinating to hear there's a mathematical basis for this!
Now I can finally understand the horus heresy and why Horus made a deal with Chaos. I guess Chaos Theory is pretty Tzeentch
Lol Like it ❤
And yet order and stability so Nurgle too?
@@satyr1349 Nurgle is less about stability than it is about inevitability.
Jimmy Space really should’ve seen it coming, his name was literally Horus Heresy and wanted to take all 40,000 warhammers.
@@ZzzyaxTheCuber That's just Horus Hearsay.
Another very important takeaway on Lorenz's attractor is, while the single data point is chaotic, the statistics is not.
Say some part in the phase space means clear day, some part is fair rain, and some part is tornado, no matter the initial condition, the chances to have a tornado in a year is fixed. A butterfly in these system might change if it happens at May 1st or November 3rd. but it won't change the chance of having on average 2 tornadoes in a year to vastly different one like 20 tornadoes a year.
This is exactly what he basically said, when he said "it's locally (single data point) chaotic, but globally (the statistics over many points) stable".
What a great unintentional demonstration of the implication of the same set of equations on the strength of the forcing (the sigma term): the stronger the forcing, the further the states visited from the original (weaker forcing). Translation: extreme weather increases as global warming increases, so the more fossil trade leads to GHG increase, the more weather disasters happen and the more costly they will tend to be.
2:23 love the _To Pimp a _*_Butterfly_* on the monitor. Cheeky 😁
The comment I was looking for!🙌🙌
This is why I love experts in other fields I haven't a clue about.
The ability to explain the world and universe in completely different and in depth ways. Never ceases to be fascinating.
"the seagull effect" ....yeah I can see why they changed it
the way you explained chaos theory especially at the end felt very much like poetry and i really do love when science communicators makes me both teach me something new and make me love the world
Thank you for reminding me about that. I've heard that explanation somewhere before once or twice and afterwards everytime I saw somebody mention chaos theory and butterfly effect I was like "wait something is missing here, but I can't grasp on what that was" and now I know!
There is an inherent beauty of the model and finding out it never touches is so fascinating! I'm going to look up that textbook to read more as I already have your book and it's phenomenal! I always learn so much from you!
I think it's one of your best videos so far. It had everything: clearly explained, beautifully illustrated, and going into more detail than most. I love that you showed the math behind it and the historical context. And you got me quite emotional as well. Thank you!
BTW, what software did you use to generate those animations?
Simon! I just read Firmament! Cracking read; and as a fellow physicist (though, I have turned to the dark side and now teach mathematics) the joke about biology "left as an exercise for the reader" had me chuckling! I have recommended the book to my students.
Strogatz's nonlinear dynamics was the book that made me a physicst. Great to see it recommended here.
It's a great video, Dr. Simon, keep up the good work!
One thing that I noticed about your video is about the explanation for 9:46, of "lines never cross."
Although it is a really good beginning explanation (and also a motivating one for anyone to start studying this subject), we need to note that this property is actually quickly generalized to every differential equations with "well-behaved" right hand side function. It usually termed as "Existence and Uniqueness of Differential Equations," which guarantee the solution to both exists and unique.
What usually set things apart in chaos theory, in my opinion, with E&U property obeyed, is that solution forced to "fill up" the space to make sure the solution never cross. The consequence of this is that the butterfly shape will actually have some sort of volume in it, which later can be calculated and shown to have fractional dimension (in case of Lorenz, it will be around 2.062, by Julien Clinton Sprott, denote as Kaplan Yorke Dimension).
Which make chaos theory usually puzzling from the perspective of geometer (as far as I know) because representing surface with fractional dimension is quite a nightmare (and I don't think there is analytical technique for that, for now) and it usually studied by approximating the attractor using Mobius strips and any other surfaces.
However, this video is really good, motivating videos for someone to start embrace the subject of chaos theory. Not only it is can be explained simply in lay terms, the frontier research of this field actually quite "accessible" to most scholars. Meaning, someone can start learn, simulate, and also make contribution in the field of chaos theory sooner than they thought..
All in all, I wish you, Dr. Simon, and everyone good luck!
Beautifully explained, and it even has a very wholesome message lol.
I spent a fair bit of time at uni creating logistical maps, really enjoyed this blast from the past 😂
If I'd stayed on to do a PhD, NLD + chaos was definitely the area I would've gone into. Our lecturer was great ❤
There is another layer of complexity in weather forecast : at our scale the atmosphere is a continuous system, as such the precision of our measure is not all that matters, how finely grained the measurement are is also very important.
This is the essence of Lorenz talk on the butterfly effect. To quote Lorenz talk in which he introduced the term butterfly effect (in a session devoted to the Global Atmospheric Research Program, at the 139th meeting of the American Association for the Advancement of Science, in Washington, D.C.; on December 29, 1972) :
1. Small errors in the coarser structure of the weather pattern-those features which are readily resolved by conventional observing networks-tend to double in about three days. As the errors become larger the growth rate subsides. This limitation alone would allow us to extend the range of acceptable prediction by three days every time we cut the observation error in half, and would offer the hope of eventually making good forecasts several weeks in advance.
2. Small errors in the finer structure-e.g., the positions of individual clouds-tend to grow much more rapidly, doubling in hours or less.
This limitation alone would not seriously reduce our hopes for extended-range forecasting, since ordinarily we do not forecast the finer structure at all.
3. Errors in the finer structure, having attained appreciable size, tend to
induce errors in the coarser structure. This result, which is less firmly established than the previous ones, implies that after a day or so there will be appreciable errors in the coarser structure, which will thereafter grow just as if they had been present initially. Cutting the observation error in the finer structure in half-a formidable task-would extend the range of acceptable prediction of even the coarser structure only by hours or less. The hopes for predicting two weeks or more in advance are thus greatly diminished.
4. Certain special quantities such as weekly average temperatures and
weekly total rainfall may be predictable at a range at which entire weather patterns are not.
"Every moment is unique. You're never going to get another chance to experience the world exactly as it is now. So take that chance, live every moment..."
Words worthy of Carl Sagan.
I am a long-time subscriber to your channel and watch your video's by finding them in my personal feed or going specifically to your channel page. LOVE your content! However, many times I miss your video's because it's hard to recognise them just based on the thumbnails. I feel the thumbnails are more adapted to people that search for a subject using the search bar, to find an answer to a question. Maybe this is an conscious choice, that's fine, but I just wanted to mention that it's easy to miss the video's when scrolling through the personal feed because your thumbnails don't really have a "channel-identity". If that makes sense.
Liked the original thumbnail more, bit sad it didn't perform as well as you would've wanted.
Yeah, this video did perticularily bad. I alao liked the old thumbnail - the new one is misleading in terms of what this channel is about. Misleading might be the wrong word but you get what I mean.
this is really cool, a more physical look at chaos theory, ive only seen some maths stuff on it, really looking forward to this at uni
I learned some chaos theory as an elective back in uni, and this video was a treat to watch. Thanks Simon !
I did not expect this video to be this insanely inspiring. Just clicked on it to support you as I thought I already knew alot of the butterfly effect. But this perspective really made my day. Thank you.
Nonlinear Dynamics was one of the most interesting courses I took in college. I highly recommend the self-sufficient book "Nonlinear Dynamics and Chaos" by the legendary Steven Strogatz to anyone who is interested.
Amazing video, Simon! Really well done!
I think a pile of rice with dropping 1 grain at a time and explaining the chance of a collapse of the pile with each grain drop is a better description of chaos theory. It goes in with the critical phase change concept which is easier in my opinion. It also allows you to demonstrate how all elements are understood yet predictability breaks down.
Clear and concise. Thanks!
Nice Video. And from this it is only a small step towards the first and second mode of predictability of Lorenz. While you cannot say exactly where the actual position of your element is in the phase space after a couple of weeks, you can still make statements of the shape of the phase shape depending on the external forcings (so the shape and the angle of the attractor). And with this we have the reason why we can just make statements about whether for a short time, but about cliamte for a much longer one. And a nice story additionally to it: It was not Lorenz who ran the machines, it were his two (of course female) assistants: Ellen Fetter and Margaret Hamilton.
I actually cried during this video. Chaos theory is one of the most beautiful things in the world.
That's fascinating!!
Thanks Simon, you rock.
This is awesome! I've recently taken a class on nonlinear dynamics, and we studied these models in detail. It's super interesting!
amazing visualisations!
Loved this video, clear and poetic description/illustration of chaos theory!
Just a quick correction @5:34: ρ is the Rayleigh number, and σ is the Prandtl number. Also, later in the video the Reynolds number is mentioned a few times -- for those confused and/or interested, in this simple convection model, the Rayleigh number is approximately the square of the Reynolds number.
you improved your video quality a lot. keep it up
Beautiful, informative, Thought-provoking. Thanks Simon for helping me be distracted from writing my paper!
Computer models of systems subject to chaos not only are sensitive to initial conditions but are also sensitive to floating point introduced imprecision the longer the model runs.
Beautiful video! ❤
You have done the impossible. You have made me understand chaos theory!!
Hi Simon, another science video that reaches the head, heart and guts. That's hard to do. Doing it understandably is harder still. You have a rare talent. I try to teach the basics of climate to schoolkids, and the way you frame concepts is a great example and lesson - thank you!
great video !!
Wow, very informative 🎉🎉 love the vid ❤
Amazing video!
Great video!
Thanks for the 'Most Excellent ' explanation, I love it
nice video clarifying what is meant
Very interesting, thanks!
You keep this up and your channel is going to explode in popularity soon.
What are you talking about mate. This channel is already plenty big enough o_O
Wow, the new studio is looking good! For a while there, it really looked like you were outside ;)
What a brilliant and wonderful video. Getting some Brian Cox vibes from this one! Excellent work!
I’m off to programming this out in python and matplotlib right away!
Really moving conclusion. Love it
It means nothings out of place and never has been and never will be . Otherwise everything would be different. The tiniest change in anything necessitates a change in everything
Great video
So Twain was right about history not repeating itself, but often rhyming.
12:20 actually hit really hard, wth
Even the six decimal places on the printout wouldn't be "real," but rather a conversion of the binary used behind the scenes. And switching to a different floating point standard, or a different machine, or even just converting back and forth to observe, would introduce stealth changes in values that might quickly show a big effect.
this video is beyond amazing
Dr. Simon's clever way to tell us to go out and touch grass.
I love this explanation. As a meteorologist I can now clearly see the chaos I'm dealing with.😂
why is the tpab cover there lmao
When people mention the butterfly effect, I always ask them how much a butterfly weighs and how much one cubicmeter of air.
Very interesting subject. From thought experiments about time travel, I realized that the "butterfly effect" is a real thing. I came to the conclusion that any change no matter how small will have extreme effects over time. This also lead me to understanding of branching realities.
“You cannot step into the same river twice” Heraclitis
I'm slightly confused. Isn't there a finite number of components that constitute the atmosphere, so, given enough time, wouldn't it run out of new states and have to repeat an old state? How can there be infinitely many configurations?
Most variables are continuous rather than discrete so not really
The count of molecules in the atmosphere is vast, so the number of possible states are mind boggling vast. The Earth, and thus the atmosphere, will not last long enough to even experience a tiny fraction of the possible states.
This thought just crossed my mind: “if what you say in 12:16 this is true, and so is the “Borsuk-Ulam” theorem, then Deja Vu in atmospheres are bound to happen?” 🤔
Ah, the good old "a man never crosses the same river twice," bit, but backed up by math.
Great explanation, though there is a small issue with your poetic conclusion about the atmosphere never repeating a state. The Lorenz equation describes a closed system, while the atmosphere gets constant interference from outside: anything from volcanoes, solar winds, interstellar radiation to life being life. Even when considering every possible variable in the universe as a whole, radioactive decay and quantum effects are inherently random inputs to that system, slightly nudging that butterfly curve around and causing intersections.
So even if you do somehow manage to get all the variables perfectly correct, its state is still impossible to predict.
How did you make the visualizations? Theyre really beautiful!
Never thought id be interested in meterology
super cool
Now plan to spend the rest of the day watching a butterfly flutter by.
Finally someone explained the movie for me.
Hi Simon, thanks for the great video! A question, could you compare this idea of aperiodicity to the Poincaré recurrence effect? I would be grateful if you could highlight the difference in a comment, but I would also be very interested in a followup video or just a short answer in a QnA type video:))
I asked ChatGPT, and receieved this answer: In summary, the Lorenz attractor illustrates a chaotic system where trajectories do not cross, and the system displays sensitive dependence on initial conditions. Despite this, the Poincaré recurrence theorem still holds for such systems in a broader sense, ensuring that states will eventually recur within the finite measure of the attractor's phase space. The recurrence times might be long, and the exact nature of recurrence might be complex and difficult to observe directly, but the principle remains valid.
Does this mean that even a chaotic system still might recur to one of it's previous states due to the Poincaré recurrence theorem, even though it's phase trajectories don't cross?
o shit those animations were dope
I don't get the fascination with uniqueness. Every piece of rock is unique, but that doesn't mean I'll look twice before I walk over it.
So what exactly makes you think that seagulls would not be the perfect animal for an analogy about chaos?
33 sec ago upload lmao never this close to uploads
Thoroughly enjoyed this video! Thank you, Simon and Luke :)
Super interesting, enjoyed it a lot, like all your other content.
Very well done, Simon!
Woah, a video about literally the most fascinating topic in my opinion and then just incredibly beautiful, thank you❤
I've been stabilising the worlds weather by squishing all the butterflies.
Really cool video, I really love the visualizations!
Wow i've been hearing so much about chaos theory and even talked about it without really knowing how it works. But now i finally understand, somewhat. On a surface level. Thanks, awesome video, so glad you've decided to continue your work here on youtube!
Is there any link to, or a name for, the proof that the line never crosses itself?
Lovely. Cool. No woo-wiol. Well done!
But the atmosphere isn't really deterministic is it? Because it is a part of a larger (perhaps deterministic) system. Couldn't something like a sunflare change the current conditions just enough to make a repeat of states possible?
2:49
Instant like
So my question is: If the atmosphere will never reach the exact same state twice and there is only a finite number of possible states: do we have a built-in deadline?
But why should there be only a finite number of states? This is a continuous model so there are infinite numbers it can be.
Fascinating video! Would love to see more like this that stray slightly off the strict "climate change" focus of the channel into more "atmospheric science" type stuff as well 😊
Math quickly makes my eyes go crossed and the effort itself becomes chaos in my mind so I need a less-mathy book. Is Chaos: Making a New Science book by James Gleick ok? Is there any overlap between what was discussed in this video and the book?
Hello, I understand your struggle. Genuinely, go and use brilliant.
I'm not saying it'll suddenly be easy but imo you're probably more of an intuitive thinker who didn't have a teacher that made maths 'click' for you.
Brilliant's method of isolating a concept, teaching it's sub-topic logic, then very slowly increasing the complexity until you're solving the problem is great. The better part of this method is you begin to look at the problems intuitively rather than as numbers, which is valuable of you want to explore physics
Towards the higher end of stuff you'll eventually find it lacking but that's just because you will have developed a lot, best of luck and happy learning!
what is the meaning of two "30 degrees N" text fields at 5:50?
tl;dr: Hey Simon, are σ, ρ, β "Finely Tuned" due to the anthropic principle or am I misunderstanding something?
Something I wasn't quite clear on are the parameters that have to be just "finely tuned" for this Butterfly Effect to occur. Are the Rayleigh number and the Prandtl numbers physical constants? Pretty sure the height of the collumns isn't a constant.
So I'm wondering why are these 3 parameters finely tuned? Does anyone know if there's a reason? Or is this a case of the anthropic principle where there could never have been a world with humans to discover this if these paramers weren't fine-tuned?
Sorry, I've been ruined by cosmology. xD
A good question, but no, they aren't "finely tuned".
σ, the Prandtl number*, is the ratio between the molecular diffusion of heat versus momentum, and therefore a property of the fluid. For air close to the Earth's surface, σ is about 0.7; for room-temperature water, it's about 7.
ρ, the Rayleigh number*, is basically the ratio of the strength of the temperature forcing -- the difference in temperature between the ground and the upper atmosphere -- which acts to destabilise the fluid and generate motion, to the molecular diffusion of momentum, which acts to stabilise the fluid and suppress motion. This is therefore an external parameter rather than a property of the fluid, and depends on the exact circumstances of the convection. For atmospheric convection, this number is huge -- around 10^15--10^20 depending on how you estimate it.
β, which doesn't have a name, is a function of the aspect ratio of the convection cells, and in the real world is an emergent property of the physical system specified by σ and ρ. However, in Lorenz's simplified model, it becomes another control parameter. Based on theoretical investigations, β would be expected to be of order 1 (i.e. not very small or very large!).
Lorenz's numerical experiments used σ = 10, ρ = 28, and β = 8/3. There's nothing particularly special about these numbers, and the sensitive dependence on initial conditions is quite a generic feature for other values of the three parameters. Solutions of the full, unapproximated equations of motion also exhibit sensitive dependence in initial conditions for high enough ρ -- look up "turbulent Rayleigh-Benard convection" if you're interested!
*Note that Simon accidentally referred to these two the wrong way round.