#### Am Vor Tag

Veritasium

subscribers: 14 Mio.Chaos theory means deterministic systems can be unpredictable. Thanks to LastPass for sponsoring this video. Click here to start using LastPass: ve42.co/VeLP

Animations by Prof. Robert Ghrist: ve42.co/Ghrist

Want to know more about chaos theory and non-linear dynamical systems? Check out: ve42.co/chaos-math

Butterfly footage courtesy of Phil Torres and The Jungle Diaries: ve42.co/monarch

Solar system, 3-body and printout animations by Jonny Hyman

Some animations made with Universe Sandbox: universesandbox.com/

Special thanks to Prof. Mason Porter at UCLA who I interviewed for this video.

I have long wanted to make a video about chaos, ever since reading James Gleick's fantastic book, Chaos. I hope this video gives an idea of phase space - a picture of dynamical systems in which each point completely represents the state of the system. For a pendulum, phase space is only 2-dimensional and you can get orbits (in the case of an undamped pendulum) or an inward spiral (in the case of a pendulum with friction). For the Lorenz equations we need three dimensions to show the phase space. The attractor you find for these equations is said to be strange and chaotic because there is no loop, only infinite curves that never intersect. This explains why the motion is so unpredictable - two different initial conditions that are very close together can end up arbitrarily far apart.

Music from epidemicsound.com "The Longest Rest" "A Sound Foundation" "Seaweed"

## KOMMENTARE

## Andrej Ferdinand

^{+12754}Having a bad math teacher at very young age, has the butterfly effect on the rest of your life; for example

Vor 2 years## victor guzman

^{+504}"The printer rounded to 3 decimal places whereas the computer calculated 6" The ghost of significant figures

Vor 2 years## Prem Sagar

^{+132}Having a bad math teacher at very young age, has the butterfly effect on the rest of your life; for example

Vor 2 years## Aadarsh Raghuwanshi

^{+179}this hit me hard.

Vor 2 years## Ira L

^{+184}At school I was really good at math until there was a very irritable algebra teacher with anger management issues. Her explanations were super short, if you zone out for several seconds - congrats, you understand nothing. I did ok, but I learnt to lay low. In high school and university I studied foreign literature and languages. When I was 20 - 21, during university practice, I went to my school to work with an English teacher, and I was assigned to this one class of pupils. Once I sat through their algebra lesson with that wonderful teacher. I think she didn't remember me but she felt OK to start shouting at these poor kids and shaking a dirty blackboard sponge right in their faces to make them think faster, I guess. She had also retained another beautiful habit of hers - gesticulating with both hands, but with one hand she would hold a piece of chalk, with another one - her glasses. She would use all her fingers except the middle ones, and she would raise both of her middle fingers and shake her hands to emphasize her point. She would basically give the double middle finger to the whole class😂

Vor 2 years## Bostonzo B

^{+39}Honestly that’s just an excuse and I’m sorry. My friends had the same poor math teacher at a young age for a good amount of years. One went on to be brilliant at math while the other was mediocre at best

Vor 2 years## Mohib Ullah

^{+5452}He explained such a complex topic with so much simplicity that i am just speechless.

Vor 2 years## Hugo Clarke

^{+77}It IS simple. People can't help but complicate things.

Vor 2 years## Freedom Phoenix Goat

^{+46}@ノーヴァゲイル but hes right. In fact everyting is simpil when looked at from the right angle. Just look at flatwaters overcomplicating the world because they cant grasp the fundmetals of gravity. Stupid is when you over complicated things you dont undstand because you applying the wrong models to the data.

Vor 2 years## Hugo Clarke

^{+28}@ノーヴァゲイル Reading into my comment like that is only making a case for my point. Humans need to translate things into a language they can understand. My comment would sound pretentious to anyone, myself included, but it is entirely objective.

Vor 2 years## Daekesh

^{+31}@ノーヴァゲイル Have you considered that English may not be his first language? Using an ad hominem argument means you've already lost.

Vor 2 years## Hamsterdam

^{+7}@ノーヴァゲイル did you have a bad day?

Vor 2 years## Sylvain Bérubé

^{+1082}That's pure gold. 20 years ago I had the chance to study chaotic dynamical system during my undergraduate study in mathematics. There were a few good books on the subject, I remember an interesting video too, but nothing of that quality. To the younger generation: savor and take advantage of your luck!

Vor 2 years## zjuchi

^{+25}Exactly what I was thinking during this video, we're so fortunate to have this education at our fingertips, whereas you'd have to either have to travel, pay alot of money or first apply and get accepted to receive such information

Vor year## Mr. Drew

^{+2}@zjuchi haha yup

Vor year## Ali Waheed

^{+3}Any good book recommendations on choatic systems?

Vor year## Sam K

^{+2}@Ali Waheed A professor I had recently recommended An introduction to chaotic dynamical systems by Robert Devaney. Haven’t had time to look at it, but as it’s his field of study, I take his word that it’s good.

Vor year## Naneaux PeeBrane

^{+3}@Ali Waheed The Quark and the Jaguar deals with complex systems... Chaos by James Gleick nice intro - check out the Stanford lectures on Chaos and Reductionism too

Vor year## QuesoCookies

^{+581}Bottom line: we can't predict the future because we'd need to know the exact conditions of the beginning, but we can't know the exact conditions for the beginning because the margin for error in estimating the beginning conditions is infinitely small. However, every point in time was determined from the beginning based on that initial state. So the future is determined, but there's no way for us to know what it is.

Vor 2 years## Robko против войны в Украине

^{+7}exactly

Vor 2 years## Jake

^{+15}Not necessarily, if we could see the exact conditions at any given moment in time (e.g. right now) hypothetically we could predict the future from that moment onwards. At least I think so?

Vor 2 years## QuesoCookies

^{+14}@Jake Lorenz' observation of chaos came about because he was getting different values for the same points in time when he changed the starting point of the calculations. So starting at any point other than the exact same starting point will produce different results. They could happen to produce similar results, but it'd still be impossible to know exactly how similar they might be beforehand.

Vor 2 years## Chill Guy

^{+24}@Jake But exact conditions to infinite decimals can never be known...

Vor 2 years## Chill Guy

^{+9}@QuesoCookies That's actually not true. See that part of the video again... The values were actually slightly different due to the rounding off of the printer. He mentions this. They are deterministic systems, ie., they produce the exact same results for the exact initial conditions. But not even approximate results for approximate conditions

Vor 2 years## ψ{Flame End Cyborg Guy}ψ

^{+1488}My favorite thugh on the matter is: Chaos is not anti-order. It has its own rules, secred to common eye, but quantificated, mesurable and predictable. we may not know the sequence, but we can know the rules it must obey.

Vor 2 years## Arkdirfe

^{+128}If we assume that the universe is deterministic then chaos really is just rules we haven't understood yet (and may never understand). The problem is that we, from within the system, can not figure out whether the system itself is deterministic or not because any "full" prediction of the future would involve predicting your prediction and whoops infinite recursion.

Vor 2 years## Galen Solomon

^{+3}"... and predictable." I think you missed something somewhere.

Vor 2 years## Galen Solomon

^{+4}@Arkdirfe You mean like fractals?

Vor 2 years## ψ{Flame End Cyborg Guy}ψ

^{+15}@Galen Solomon I meant rules are predictable not system itself. Statistics and rules it obeys can be seen from the function itself.

Vor 2 years## Jamie G

^{+3}the rules belong to the math topic called differential equations.

Vor 2 years## Mili Stefanova

^{+204}It's amazing how such a complex topic can be so entertaining and presented so understandable. It sparked my interest in the butterfly effect. I really loved the animations and examples of the points he made.

Vor year## Loturzel Restaurant

Keep yourself updated with the amazing Problem-tackling Youtubers Illumainugthii, knowing Better, and Some More News. These 3. Whetever its Crops, Water, Hate, LGBT, Bias, Ukraine, they cover so much and more. And before you ask: Haha, no, there was no deeper Reason for this comment, i just like to share Science, Knowlegde and Atheism in a world where many Science-Fans havent even NOTICED that Atheist-Youtubers are very similar and even often overlap with Atheist-Content - making them miss-out.

Vor year## Russwane

^{+3620}If only someone had explained science this way when I was younger.

Vor 3 years## Caveman Hikes

^{+206}I'm just happy someone explaining it this way now. YouTube has really been a blessing that way. I'm finally enjoying science and physics and math and things I hated when I was a kid.

Vor 3 years## AKABILASET

^{+5}Great I'm ever like physics

Vor 3 years## Chucky chuck

^{+35}Science is easy to understand. Math is your problem

Vor 3 years## Russwane

^{+6}@Chucky chuck geez. Tell me about it.

Vor 3 years## YourmomsnameisKaren

^{+20}They did. You were just too high to stay awake.

Vor 3 years## Micaela Royo

^{+59}I think the butterfly effect also captured so many minds because we as humans are constantly asking "Do the choices I make matter?" and "In the grander scheme of things, am I important?" and the butterfly effect gives an answer which is in most cases, yes to both questions

Vor 2 years## iiiTechnoDuckxX

^{+119}The relationship between this chaos theory, and the definition of insanity (doing the same thing over and expecting different results) is VERY intriguing.

Vor 9 Monate## aViShEk ChAkRaBoRtY

^{+2}Hmm, that's indeed a very intriguing correlation there

Vor 3 Monate## Alexander Nash

^{+3}I've always felt that you can never do EXACTLY the same thing over and over. No matter how you try there will always be some difference.

Vor 2 Monate## Krishna Pullakandam

^{+69}The graphics, the explanation, the presentation, everything about this video is top class. I am just speechless.

Vor year## ZackariasThePirate

^{+29}You will never be the same person again. The more steps you take the more different you are. Even if you go back to the start you are different, and the place is different. I love how his explanations make me visualize concepts of my own existence.

Vor year## Dana Towne

^{+94}I'm pretty sure the "butterfly effect" is from Ray Bradbury's short story 'A Sound of Thunder', where a man changes the future by stepping on a butterfly on a hunting trip 60 million years in the past. The story was first published in 1952 so it's more than 50 years old. Great video!

Vor year## Tucsonan Dude

^{+6}That's a fun story about it.

Vor year## Viktorija Jankauskaitė

^{+6}That was a good story! Bradbury was an incredible author

Vor year## JustBecause Gaming

^{+2}it was probably called the butterfly effect long before that.

Vor 10 Monate## Dana Towne

^{+3}@JustBecause Gaming , nope. I made the same observation on a literary expert's channel and he answered and said it was true. Someone else used the exact phrase a couple of years later in a different book - but not before.

Vor 10 Monate## JustBecause Gaming

^{+5}@Dana Towne according to google, lorenz called it the butterfly effect

Vor 10 Monate## SangoProductions213

^{+7001}When they talk about time travel, people almost always state that they'd make enormous changes for present day with small actions in the past. But rarely do people think that they can take small actions today to cause great change tomorrow.

Vor 3 years## Mikiness Analog

^{+243}How very "forward" of you LOL

Vor 3 years## John Behrens

^{+449}That's because they have a point of reference by virtue of knowing how the past played out. They'd hypothetically know that any differences over the course of history would be due to whatever changes they made. They don't know how the future would be changed by whatever actions they take today because as the video so succinctly explains, we can't predict the future very accurately beyond a very limited point.

Vor 3 years## Lambda Ovine

^{+72}Yeah, but when you cannot predict what will happen, does it even matter to think about that?

Vor 3 years## SangoProductions213

^{+96}@Lambda Ovine Yes. Get out there and make one small step towards a better future. You have the power.

Vor 3 years## mackk123

^{+140}If you put a bag on your head you can time travel at a rate of 1 second per second

Vor 3 years## so much to cook

^{+73}The double pendulum is unique as for explaining alternative histories, just as there is chaos sometimes the paths may align closely with each other but not for long.

Vor 2 years## AK

^{+1}Deep

Vor year## Alex Christakis

^{+7}If only I had a physics and/or math teacher that was half as knowledgeable, descriptive and well versed as you are in those famous short "lectures" of Veritasium, I would certainly have been an immeasurably more educated person. I cannot thank you enough for your uploads.

Vor year## Ethan Groat

^{+3}I'd like to mention that even if one vector/data point/object in the system were to reach the exact same state by some weird chance (like discrete numbers rounding the results at each step), the other vectors in the system would be different and so the paths each would still not be periodic. Very interesting summary of chaos theory, thank you! I like the way Jeff Goldblum explained it best still lol

Vor 2 years## x commenter

^{+324}A system trying to predict the future requires every tiny details of present which includes details of system itself. It creates a self referential paradox.

Vor 2 years## Shubham 25.3

^{+8}True

Vor 2 years## GivingItToYou Raw

^{+31}Sounds like Laplace's demon was more about an imaginary god-like intellect analyzing as an external observer.

Vor 2 years## Changer of Ways

^{+25}Stored knowledge does require matter and energy to exist and the knowledge required to comprehend the infinite expanse would require its own infinite expanse... yeah!

Vor 2 years## Rudxain

^{+30}This is simlar to the observer paradox. Using a computer to simulate an environment far from the computer would require simulating the computer itself, because just running the simulation makes tiny changes to that external environment, because total isolation of a quantum system is impossible in this universe

Vor 2 years## Anton Shablyka

^{+2}You have just described why the scenario of DEVs wouldn't work. Need to re-watch to see if they addressed this paradox there.

Vor year## yuki12

^{+10}Thank you so much! I've been so confused about the chaos theory in general and my textbooks weren't helping at all. You explained it all so well. Sincerely thank you.

Vor year## KlaxonCow

^{+4438}"Sometimes when I consider what tremendous consequences come from little things, I am tempted to think... there are no little things." - Bruce Barton

Vor 2 years## Pallabi Dutta

^{+34}The intricacies in the web of life.

Vor 2 years## sukanya

^{+13}🤯

Vor 2 years## Vinícius Humano

^{+3}yeah

Vor 2 years## Atreides

^{+9}That is certainly a tempting thought.

Vor 2 years## Yusuf Ibrahim

^{+16}i have been trying to put that feeling into words for years. thanks person

Vor year## Trayee

^{+10}Thank you Derek, after watching a series of motivational videos with no effects, this one actually made me shut down the screen and open my physics textbook!

Vor year## Periodic Table

^{+5}In the university of Alberta's science building there is a double pendulum in the main hallway in the basement where you can go and spin it yourself. Every time I spin that thing it never ceases to amaze me, making me think deeply about the nature of the universe

Vor 7 Monate## Richard

^{+5}Amazing video! I did my masters project on a parametric pendulum, and I wish I had something like this around then, shows the concepts so clearly

Vor 2 years## Lozzie74

^{+50}I look forward to the video on fractals. My kids asked about the lyric in the movie Frozen (Princess Elsa sings about the fractal shape of her ice palace in the song “Let it Go”) and I explained “the shape is the same no matter the scale” and cited that mountains are typically fractal. I’ve long been fascinated with the mathematics of fractals, first encountering the “blancmange” function in high school.

Vor 2 years## Demon Returns

^{+1}Seeing as this video was couple years ago and your comment is o my like almost a year ago I’m guessing that fractal video still haven’t been done huh

Vor year## Ben Gollapalli

Seeing your comment is 6 months ago. I wonder if it hasn't been released yet!

Vor 9 Monate## Ryan Hefley

^{+2}This reminds me of how seeding works in pseudo random computer generations, like Minecraft. If you use the exact same seed twice you get the exact same world, but any slight difference in the seed will create a vastly different world.

Vor year## James Dinius

^{+4229}Sensitive dependency on initial conditions: The exact present exactly predicts the future, but the approximate present doesn't approximately predict the future.

Vor 3 years## Joseph Burchanowski

^{+221}@kirby kirb "When you get information to the quantum level, predicting the exact future is going to be nigh impossible. Or maybe quantum stuff can be predicted. ¯\_(ツ)_/¯" There are people who believe in deterministic quantum mechanics and those who don't. As of right now, there is no way to tell which is right; hence they are consider "interpretations of quantum mechanics" where each are near equally valid until evidence comes out otherwise.

Vor 3 years## David Lewis

^{+63}so now the question becomes whether or not you can observe an exact state of a system. the easy answer would be no, but i wonder if the future has any solutions that we are not yet aware of.

Vor 3 years## Random Wyvern

^{+1}Juan Cortez Muro For the second time, are you proud of yourself fam?

Vor 3 years## avdresch

^{+9}You took that from vsauce

Vor 3 years## Kushagra Nath

^{+10}I was lucky enough to find chaos by james Gliek in my school library and i think this video might have been inspired bby it....... It is a fantastic read and you even get to know about the interesting side of math rather than the blunt equations we think math is

Vor year## Editg

^{+5}Im always blown away by his video. Wow, simple explanation on such complex concept without single equation. Great video.

Vor 2 years## SpruceOaks

^{+3}It amazes me that videos like this get 4 million views and 6,000 comments. I mean, this is great stuff and really fascinating to total geeks like me, I'm just surprised there are that many more of us out there who have discovered this content.

Vor 2 years## Enrique Sanchez

^{+11}This is what I learned in Life Is Strange. If time travel ever exists, no matter how bad you wanna change the past, like losing your family member, best friend, or prevent major tragedies, you will cause even worse consequences. Thank you, Life is Strange for teaching me better than Back To The Future.

Vor year## Jon Saboe

^{+7}Beautiful. You final wrap-up with the Lorenz Attractor caused me to think of the ACTUAL "Great Attractor" -- that structure to which our local galactic group is rushing towards.

Vor 2 years## Calm Calm

^{+1}U mean love ?

Vor year## Joseph Tran

^{+2791}"The printer rounded to 3 decimal places whereas the computer calculated 6" The ghost of significant figures

Vor 3 years## Karan Yadav

^{+90}Programmer's nightmare :D

Vor 3 years## Kane Nexus

^{+112}That's how 1+1=3 for large values of 1.

Vor 3 years## McFly0097

^{+27}at least he didn't have to program in an era of timezones

Vor 3 years## diamine665

^{+7}"Hidden figures"?

Vor 3 years## notefez

^{+2}ghosts of departed figures!

Vor 3 years## Gary Wallace

^{+3}Really enjoyed this. You should show the equations. I think by looking at the equations and the variables causing the feedback, people would have a better understanding. - just a thought

Vor 2 years## Alexandr Spiridonov

^{+7}Cool! This video reminded me of my professor of physics in the university who was obsessed with fractals and chaotic systems. Thanks for this explanation.

Vor year## Brian Zinn

^{+1}Another very well done video. I am surprised Dr. Muller did not mention Isaac Asimov's Foundation Series (which I am sure he has read), which is based on someone (Hari Seldon) figuring out the mathematics of predicting the future (called psychohistory), and how things can go wrong. I highly recommend the series, if you have not read it.

Vor year## Dakota Rodman

I'd love to see a video on the Hummingbird Effect! It's a very interesting one that I ended up reading about in the book "How We Got to Now" and I'd love to see a more visual take on it from you!

Vor 3 Monate## What lies Beneath

^{+1}Fun fact: even when studying ordinary differential equations (ode) you find somewhat similar conclusions. A Cauchy's problem is a differential equation (in this case an ode) with one "initial" point (aka the solution passes through this point). There is this math theorem that says that the local solution of the problem always exists and is unique, under some hypothesis of regularity for the differential equation (lipschitz's hypothesis). This means that if two differents solution for the same Cauchy's problem had a point in common (=have the same value at a certain time) they are the same solution (locally). In this sense two solutions for two differents Cauchy's problem never "touch" no matter how close they are at certain point in time. Of course this does not give you any information about the two solutions after infinite time (it is a local property) and most important: things always get weird when you go into infinites. Well not sure if this is even relevant for this video, but it's written already so I'll leave it here

Vor year## ɣ Δ

^{+293}This was one of the best videos you've made! Meteorologists and atmospheric science researchers don't receive much respect or recognition from the public, who often mock them whenever a forecast is even slightly inaccurate. Most people have not yet comprehended _just_ _how_ _difficult_ a forecaster's job is. Nor do most people realize just how much progress these scientists have collectively accomplished in the past 30 years or so. The job of an atmospheric scientist is _literally_ *to* *predict* *the* *future* state of the extremely complex and multivariate set of dynamical systems that constitute the Earth's atmosphere. This was one of the best and most concise educational videos about chaos theory I've ever seen and should help viewers better understand the difficult challenges that forecasters in any physical domain are tasked with on a daily basis. Once again, thank you Veritasium, for another enlightening, entertaining, and educational video! :)

Vor 3 years## Pranav Bhamidipati

^{+1}Aren't all forecasts mostly done by supercomputers?

Vor 3 years## Calvin Will

^{+5}@Pranav Bhamidipati I don't know but even if so, people must create the system first and fix the system when it is imperfect.

Vor 3 years## zwz • zdenek

^{+1}The problem is with their lack of honesty. If the job proves impossible, they shouldn't keep pretending to forecast. Weather forecast regularly fails miserably in only one day. It's so bad that simply knowing the date and looking out of the window gives better results.

Vor 3 years## santiago diez

^{+1}zwz • zdenek that was so rude and accurate at the same time

Vor 3 years## N Marbletoe

@Pranav Bhamidipati computers run models, but the models do not always agree. the weather experts can add a level of "AI" as well as explain the situation (including the uncertainties). most weather folks on the news don't go into the detail, but a few will explain all the steps they use to make the prediction. Ronchetti for example.

Vor year## Haneul Kim

^{+2}Thanks for a great video! One question, to be considered a chaotic system it needs to have an attractor?

Vor year## Siddhu R

Your videos are very good in explaining the complex topics in simple terms. Thanks a lot.

Vor 2 years## BlumhouseTV

Started not understanding any of it. Finished with confidence to explain this to people. Awesome job. Educating the world.

Vor 2 years## EpicNova

^{+1}My brain hurts watching these videos even with the simplistic style you explain them in, and I want more!

Vor 2 years## Razor Edge

^{+1}I have a question : "is the butterfly effect the same as the domino effect because the change is continuous " or "are they different because the scale of the effects increases significantly with time and stay constant in the other"

Vor 5 Monate## Faith

^{+2169}And this is why I will always smile and compliment strangers. Idk what kind words or gestures could majorly effect someone’s life.

Vor 3 years## Paul Ferris

^{+47}Absolutely. I was told that life is like a game of "Snakes and Ladders" and to respect, but also remember to help those you meet along it. Because you never know where you or they will be should you meet them again before the game ends.

Vor 2 years## prum chhangsreng

^{+70}Ok but that's not how butterfly effect work. Those action might also even cause someone to suicide. Like i said, this is "Chao". But that is not predictable, what predictavle is that compliment stranger make someone day better. It is not butterfly effect but it is a good thing to do and would make this world a better place.

Vor 2 years## YAHOOISNOTG

^{+6}@Paul Ferris This is why the movie Groundhog Day is one of my favorites

Vor 2 years## firstname lastname

^{+40}According to the butterfly effect you could also fart on them because it could have a major positive effect on their lives. But that's probably not as predictable as giving them a compliment.

Vor 2 years## random guy

^{+6}@firstname lastname YESSIR

Vor 2 years## Rybec Arethdar

^{+5}Ok, so chaos theory is fascinating, and this video does a great job of explaining it. If you want to go one better though, and simulate how quantum randomness can affect macroscopic effects over time, read on! Back when I was a CS undergrad, some students in the CIT department suggested I talk to one of their professors about simulation, because I had a habit of writing particle simulations for fun. After around two years of this, I finally did it. At the time, I was planning to write a swam simulation in Haskell. He gave me some interesting advice. He suggested I not worry about state. Normally, in simulations like this, one would use a buffering technique, so that the state currently being generated is based purely on the previous state. If you don't do this, early changes can affect later changes, blurring state between frames. I took this advice to mean that I shouldn't concern myself with this, so I wrote a simulation that advances each particle based on wherever the others happen to be _at this moment_, instead of buffering state. The result was that behavior of the simulation was far more organic, lacking artificial looking patterns that tended to show up in the state buffered simulations I had previously written. This isn't the _really_ interesting part though. Another thing I did to achieve this indeterminate state was to based advancement on time passed, rather than progressing a set amount per "frame". Frames really only made sense when trying to preserve state integrity, and since I wasn't doing that, I fell back to an older method I used to use when writing video games. So, here is how the simulation worked: I started with a list of particles. During each loop, I would advance the first particle in the list, based on the amount of time passed since I had last advanced that particle (and based on the positions of the other particles in the list). Then I would move that particle to the end of the list. (Using functional programming techniques in Haskell make this extremely easy.) This completely abandoned state integrity, which did manage to achieve very interesting results. The _most_ interesting result, however, was what happened with, on a whim, I ran two identical instances of the simulation side-by-side. Starting them at exactly the same time randomly placed the particles in the same places. To be clear, this is the _only_ place in the simulation where randomness was used, and because the RNG was seeded based on system time, starting them simultaneously seeded them identically. So, they started with particles in exactly the same positions, and the particles then started moving in exactly the same patterns. Except, imagine my surprise when I noticed the simulations begin to deviate. They started in identical states. There was no additional randomness going into the simulations. So they should have matched perfectly. That final assessment, however, was actually wrong. There _was_ still some _tiny_ amount of randomness going in. This randomness was processor scheduling. The difference between the two simulations was the _time_ passing between iterations. Desktop operating systems assign processor cycles to programs using some algorithm. How cycles are assigned depends on a lot of factors, including how many processes are waiting for CPU time, the priority level of those processors, how much input and output is being generated, and so on. This means that even running the same program twice, at the same time, won't give them identical schedules. Anyhow, in the context of the simulation, this means that while each particle started in exactly the same state as its parallel in the other instance, it _didn't_ always take exactly the same amount of time between iterations, and this allowed tiny differences due to floating point error to work their way in. But, modern processors are extremely fast, so the actual differences in time were infinitesimal, barely big enough to make any difference in the floating point time values being generated, and further, the floating point error created by these slight difference were also infinitesimal. So how, after only a few seconds, could deviation be seen? The answer is chaos theory. One might be tempted to suggest that the tiny differences would add up to big ones, but the truth is, the tiny difference _should be expected_ to just average out. If we are basing progression on time, the differences _do_ average out. The cumulative time that has been applied to any particle will always be within a few milliseconds of the total time passed since starting the program. And floating point error doesn't tend more toward one direction than another, so over many iterations that will also average out. So what was actually happening was that tiny differences (perhaps on a scale closer to quantum than macroscopic) produced from effectively random influences were having the impact of chaos. Maybe an easier way to think of this is considering each iteration as a "starting condition" for all future iterations. So, the first iteration has some infinitesimal difference between the two simulations. That difference, while initially imperceptible, results in growing deviation between the simulations. And this happens _on every iteration_. Now, this might seem like some merely interesting theoretical stuff, but it's far more than that. What degree of impact does quantum randomness have on the macroscopic world? It's easy to write it off as having literally no impact, except when we are deliberately measuring quantum effects and acting based on them. Chaos theory suggests otherwise though, and my simulation demonstrates exactly how even infinitesimal differences being added into the system on a constant basis can have a significant impact in even a fairly short period of time. So the truth is, not only would the future be unpredictable even if it was 100% deterministic, due to chaos, our universe _isn't_ 100% deterministic, because the various sources of quantum randomness are constantly injecting random new information into the universe, and this means that even if we could know the initial conditions with infinite accuracy and precision, we _still_ couldn't predict the future, because tiny state changes are constantly filtering in, and those are injecting new tiny differences that chaos will eventually amplify into enormous differences.

Vor year## Heart-Centered Channeling

interesting that just that tiny lag in CPU processor time had an observable effect. Cool story dude!

Vor 7 Monate## Shiva Samadhi

^{+10}As I’m studying this more I feel like the chaotic attractor is some sort of substructure that is being indirectly mapped by the phase space. Could be the shadow of some higher dimensional geometry that plays a role in the evolution of natural systems

Vor year## Sarah

^{+5}Five years deep into a PhD focused on complex dynamical systems and chaos, here. In short: yes, your theory has legs. I will probably spend the better part of my life working on a proof.

Vor year## wren

^{+1}@Sarah Imagine this turns into a butterfly effect of discovering something new about physics

Vor year## Sarah

^{+2}@wren Our little comment thread has already altered the pathway by which our universe has evolved since 2021...

Vor year## Tom

^{+1}Good video, even thought i get headaches thinking about them. I understand how you can be certain a system has periodic behavior but how do you "know" a system is chaotic? It could be that the repeatition only starts after the furthest point we observed. Thx already for the help!

Vor 2 Monate## Amy Lim Schwander

^{+5}Thanks for the videos - another beautiful example of rheostatic order which includes states of variability in relation to higher order predictable potential action. :)

Vor 2 years## Howard Anderson

^{+2}Years ago I did a three-body simulation of two stars and a planet and observed the chaotic behavior. One of the things I tried, because the computer was so slow, was to vary the time steps in the simulation. Longer time steps when the planet was far away from the stars and shorter time steps when the planet was close to one of the starts. The results made me very suspicious after that time of the results of "variable time-step simulations" used at that time within electrical engineering SPICE type models. I think CADENCE was doing some of that? :-) Results of the constant time-step simulations and the variable time-step simulations did not match at all in my planetary simulation!

Vor year## Danyal AG

^{+753}A Mathematician once stated the chaos of nature using the example of a pool table: " During the familiar game of pool, if a man is to calculate the collisions between the balls, the prediction of the first collision is simple enough that any college student can do it. The prediction of the fifth collision requires such things as the gravitational attraction of the two people standing nearest to the pool, while the prediction of the ninth collision is impossible, as it requires exact knowledge of all the positions and momenta of all the particles ( electrons, protons, and neutrons ) in the Observable Universe."

Vor 3 years## Jarek Nowak

^{+2}True. 👍

Vor 3 years## Casualsadi

^{+7}Chaos is a ladder :V :V :V

Vor 3 years## Connor Hammond

^{+38}Please tell me this is merely a metaphor and that people near the pool table aren't effecting the gravitational attraction on the balls enough to have any significant impact on the 5th collision. Surely not..

Vor 3 years## Jarek Nowak

^{+25}@Connor Hammond Butterfly effect. Everything affects everything.

Vor 3 years## hey wrandom

^{+22}@Connor Hammond I have personally heard this claim as well by a phycisist researcher. He said you would have to take into account the gravitational force of the people around

Vor 3 years## AMIRUL HAQE

Till today i had a very different perspective about the nature. With all my knowledge in math, physics, and engineering, i thought that almost everything is deterministic and predicable. And today, one of my friends sent this video link to me, when we were having a small discussion. This video was really thought provoking, and this changed my entire perspective. This incident itself is a chaos!

Vor year## Timothy Doyle

Was explained so well. Excellent job!

Vor 2 years## s2 on div2

Thanks for the time and effort you invest in your productions they are very much appreciated

Vor year## ShageRG

^{+57}It’s crazy how chaos relate so much to harmony

Vor 2 years## Jason Powell

^{+2}relates no opposes, just like positive & negative, this is not crazy this is balance.

Vor year## Scientific Artist

A straight line e.g. a laser beam can be miles apart at the far end, while you'd move it a millimeter from where it's starting. Non-Linear motions and especially those involving multiple objects can be far more complicated and nearly impossible to predict.

Vor year## Punman5

^{+713}For some reason I find theoretical physics like this rather unsettling. The existential implications are interesting but simultaneously disturbing

Vor 2 years## matt carrswold

^{+45}It is not a paradox. Exciting and unsettling are not mutually exclusive. A HORROR film is both EXCITING and UNSETTLING.

Vor 2 years## D

^{+10}Matt that though shows how simple humans and there emotions are, I’m sure you can’t understand that because you are just a 3D life form with neurological signals that go to a main muscle blob

Vor 2 years## Son

^{+8}@D says someone who has never studied humans and their emotions on a actual neurological scale, i presume

Vor 2 years## Lugold 876

^{+2}@Son he probably has not but you dont have to study that hard to realize that Humans are 3D life forms with a very simple mindset that turns basically everything into good and bad

Vor 2 years## Xenoghost

^{+10}@D How can you even conceive degrading humanity like that? Our bodies alone are much more than just "a 3D life form" or whatever meaningless nihilist oversimplification you're trying to push as opposed to thousands of years spent trying to understand how we work. And we are more than just our physical presences, love is indeed a thing, and it's huge. You don't know a thing. Me neither. We all know nothing.

Vor 2 years## pointman1921

^{+2}Awesome video. I keep going back to our ability to process information. In a simple environment, say a pool ball striking another pool ball, we have the ability to isolate many points of information in that "relatively small" experiment. We can essentially tell the direct future for a few seconds. We don't really call this chaos. But when we work with a larger set of data and more points of information, we suddenly label it as chaos as we are unable to predict the outcome accurately. It's not that we are unable to predict, it's that we don't have the tools/understanding to quickly identify, map out, and "do the math". The math now resides within magnitudes of complexity. If given enough time and a massive amount of data points, (plus some wicked processing) we should be able to predict the outcome just the same. It just requires more on our part. More than we currently have.

Vor 2 years## James Rosar

And then what of the earthquake? What was predictable before vs. what was will be predictable after, and the need to know the exact when of that quake to plot the sequence of events? The precise when of that event has huge influences upon all the lives touched by it. Why is any other moment so different?

Vor year## N Marbletoe

@James Rosar I imagine an earthquake goes by the math of 'self organized criticality.'

Vor year## MannyXVIII

^{+1}if you think about it this also translates to humans quite well. Trying to predict an individuals futur actions is nigh impossible, but take a large enough group of people and you can predict them pretty well.

Vor 2 years## nissrine ouabbou

Thank you for this video extremly well explained ! ❤

Vor year## mrow9999

I was taught "A Sound of Thunder," Ray Bradbury's 1952 science fiction short story about a time-traveling hunter who accidentally stepped on a butterfly and drastically altered the future. Many traced the concept back to his story. I have never researched how the concept may have been expressed further in the past. Perhaps others have some insight about this.

Vor 11 Tage## SIM-LOFT

^{+17}Q1. So could you theoretically perform a ‘Quantum Coin Toss’ to make a significant world scale decision, thus creating two deviated universes, that had 100% identical history prior to that moment?

Vor 2 years## Anuradha Nunna

Yep

Vor 2 years## James Rosar

^{+2}At least for arguments sake, as an axiomatic aid to discussion. As if time granted the opportunity for a new universe to start at any instant. We are possessed of a deep need to believe this is, has been, and always will be possible, or our agency would be lost. Either that, or the premise is in error.

Vor year## SIM-LOFT

@James Rosar you’re skipping ahead, but indeed that’s where the determinism question takes us. Q. Are we therefore drivers of our own fate, or just conscious passengers?

Vor year## SIM-LOFT

^{+1}To follow… Every instance of a being exercising True Free Will would be a deviation point, with each potential requiring its own unique base universe. Therefore, the number of unique base universes that eventually lead to Free Will is infinitely higher than unique deterministic universes. In other words, if Free Will is possible at all to exist, through evolution or creation , then … a. we almost certainly live in one of the universes that it does b. It is fundamental to the measure of existence .

Vor year## James Rosar

^{+2}We surf the expanding edge of possibility, What has happened is now fixed, and from Newton, objects in motion tend to stay in motion, until acted upon by an external force. We can be that external force, using periodicity to our advantage, for when preparation meets opportunity, luck has a way of sustaining the flows of energy we depend upon to perpetuate ourselves with.

Vor year## Zintel

^{+390}This is one of those science videos that aren't just interesting, but also beautiful.

Vor 3 years## John Smith

^{+7}Just like your comment! ;)

Vor 3 years## Buboy Chua

^{+5}True bro

Vor 3 years## Devon Josiah

^{+4}And also confusing

Vor 3 years## Hambone

^{+1}I think you would really enjoy @3Blue1Brown

Vor 3 years## Krusnna Sistla

I REALLY wish I had a Teacher like you. In India, We are after only grades. Not Excellence. Hope you Inspire more people! Good Luck!

Vor year## Carbrickscity

^{+1}Similar to the power tower paradox. But then there is order out of chaos. And even when they say things are "unpredictable", everything is "predetermined" in some way, as long as the system is finite, i.e. the observable universe. Doesn't matter how chaos it is, it has to be one of the many (but finite) possible outcomes.

Vor year## hostagna Thoma

Learning about Chaos in science is one of the most fun thing ever

Vor year## Thomas de Vos

That's why contact with one person makes total sense to me. In control, peacefully leveling and such.. and contact with another person added suddenly becomes CHAOS. I get it now.

Vor 26 Tage## Joshua Gharis

You are so great. I am thankful for the work you do.

Vor year## Vic Wat

^{+7013}"That's on the scale of atoms, pretty insignificant on the scale of people," said the pile of atoms.

Vor 3 years## Deepstaria Enigmatica

^{+276}...said the pile of atoms in command of one of the most mysterious conglomeration of atoms called the brain. We may be insignificant in the grand scheme of things but definitely in a unique kind of way.

Vor 3 years## Mathias Dagur Helguson

^{+169}and isn't THAT the butterfly effect?

Vor 3 years## G_N_Party

^{+35}Several trillion trillion atoms, yes.

Vor 3 years## Vic Wat

^{+60}@Mathias Dagur Helguson EXACTLY WHAT I WAS THINKING.

Vor 3 years## 77Avadon77

^{+6}So if the moon was conscious would it to be impossible to predict? Of course not

Vor 3 years## Suhel Mulla

^{+1}it was my M. Tech. thesis topic...I am so amazed to see how he explained it in most simplistic way... just awesome...👍👍

Vor 2 years## Aditya Chaudhari

Could you make a video on the same topic but considering the behaviour of a single object. Like, say of 2 pendulums, but you focus on the motion of only a particular one.

Vor 2 Monate## nzmarty

Great video. Beautiful graphics. I'm a solid believer in the Butterfly Effect btw

Vor year## Merilix2

The name butterfly effect indeed refers to the shape of the Lorenz attractor. I remember several years ago, a weather forecaster in tv sayd he cant predict the weather properly because the system actually was very chaotic. He gave two different forecasts instead.

Vor year## Leoni Silva

^{+1}Everything seems to be miraculous when we think about it. Thank you.

Vor year## Saswat Das

^{+403}The reason I think people are fascinated by butterfly effect is that it gives a sense that our actions have greater cause that we are special something like that..

Vor 2 years## Sumit Raturi

^{+3}People are amazed,thats it

Vor 2 years## Freedom Phoenix Goat

^{+20}If you want to argue that you can take actions that truly have not affect, that simply goes against everything we know about the universe so far. Even the flap of a butterfly wing has energy in it, energy that comes from somewhere and goes somewhere. Naturally one butterfly wing flap could never take credit for a hole tornado, yet to just because the tornado would be indifferent to you if I removed one butterfly, dosen mean its the same outcome. Put it like this, i can take one atom from you, and you would not mind, you would surely be fine even if tok a thousand (well unless they were somehow specific ones in one of you cell causing it to become a cancer cell) but there come a point where the number of atoms taken from you becomes a problem. Its not that a atom taken from you doesn't affect you, it's that it affects you to litel to make a noticeable difference by your standards, but there's still a difference.

Vor 2 years## Daekesh

^{+34}@Freedom Phoenix Goat A butterfly's wing flap, while its energy does dissipate into the surrounding atmosphere, could be the "straw that broke the camel's back." It could push conditions *just enough* for that tornado to either happen or not happen. A single atom, taken from your body, could cause you irreparable damage, if it's the right atom. Here's a poem perfectly describing the effect of the butterfly effect: For want of a nail the shoe was lost. For want of a shoe the horse was lost. For want of a horse the rider was lost. For want of a rider the battle was lost. For want of a battle the kingdom was lost. And all for the want of a horseshoe nail. Overall, the butterfly effect is about the chain reactions that happen from very small changes / influences to create very big changes in the future.

Vor 2 years## K M

^{+6}@Daekesh great illustration!

Vor 2 years## A Random Bard

^{+3}Also, it means that there is a certain amount of free will. With total determinism, our actions were decided before we were born and will be decided until we die. Effectively, we don't exist. We're just passengers to our body's existence.

Vor 2 years## You Guys

The Lorenz Attractor is applicable for social life. There's one guy who everyone, no matter how big of a jerk they are, always are friends with that guy.

Vor 3 Monate## nessie

^{+3}I was done playing the Life is Strange (the first game) and I stumbled upon this video to learn more about the butterfly effect and chaos theory. The ending I choose is Sacrifice Chloe, as sad and hard that this may sound, it goes to show that you can't keep changing fate and destiny for the people around you and for yourself, because in the end, you have to let go of the past. I like it how the blue butterfly in the end of the game represents on the right path.

Vor 2 years## Curt Panser

^{+1}Same same hahaha

Vor year## Abhishek Rawat

i am selfish let the storm happen

Vor Monat## nessie

@Abhishek Rawat dang, I forgot I commented, a year later I'm bae over bay 💀so YEAH

Vor Monat## navsenjoy

^{+3}Quite interesting. Never gave a serious thought to 'butterfly effect' untill this.... Thanks. If possible talk about concept of 'dark matter' 🙂

Vor year## Dr. C P Ramchandani

^{+3}An interesting take on the laws of physics as we know. Amazed at the inquisitive minds that study into so much detail! Do watch, especially if you are interested in Scientific knowledge that goes behind prediction of phenomenon like weather forecasts

Vor 2 years## Ky-Effect

Yes. I like chaos theory 😊 But it makes me wonder what is this vail between determinism and predictability and where does the boundary exist

Vor 9 Monate## Conrad T. Pino

^{+100}I appreciate integrity shown when structuring embedded advertising so much that I watched to the very end. Thank you.

Vor 3 years## Richard Staab

I've seen several of your videos and even recommended some to others. This is by far my favorite of your videos.

Vor year## Ayushi R

How u r able to explain such complex theory so beautifully.👍 Hatsoff

Vor 2 years## Tr7b

One can extrapolate this dynamic with a counterintuitive theory that dark matter acts as the backdrop for all chaotic systems, giving them an inherent symmetry they keeps them from flying apart.

Vor year## ChickenKorma

^{+1}The actual name of his publication was "Deterministic Nonperiodic Flow". The other title mentioned here was just for this meeting. Lorenz hadn't submitted a title for his speech so someone who probably didn't quite understood Lorenz came up with this one. And there is two things wrong with concluding what we call butterflyeffect from Lorenzs work. a) Lorenz only considered 3 variables in his model and the system was highly sensitive to changes of the initial state. In nature there are a lot more variables you might consider but a system will be robust to changes in a lot of them. So concluding that a butterfly swing could lead to a tornado is like concluding cartoon animals might actually exist because you have seen other unlikely animals in the zoo. b) It's still an attractor pulling trajectories and one with high selfsimilarity. Actually it's almost 2-dimensional. So while it's true that the longtime behaviour isn't practically predictable in an exact manner there are of course limits on it. I think what really catches in the idea of the butterflyeffect is that it implicates that each one of us could potentially have a huge impact on the world and the society with our tiny actions. But if there is anything to conclude from Lorenzs work to humanity it's this: You can make a difference but it's most likely the effect of it will continously vanish over time and you are irrelevant.

Vor 2 years## Swagmund_Freud666

^{+1}Linguists have known about this for a long time. In proto Indo European reconstructions, a lot of words could be multiple different minor variants, but which one they are actually doesn't change the result in the daughter languages.

Vor 2 years## Oni Raptor

^{+241}"A Sound of Thunder" is a science fiction story by Ray Bradbury, first published in 1952 was the first representation of the butterfly effect where the death of a butterfly in the past causes irreparable changes to the present.

Vor 3 years## The Football Planet

^{+7}good point! Eckels (is that his name im correct? steps off the path and steps on a butterfly!

Vor 3 years## Jamal Wills

^{+13}I think it’s in the same vein but isn’t really the same. If I remember correctly, the butterfly was stepped on by a guy hunting a dinosaur. They got back and language was slightly off and a different political party was in power. I think the butterfly effect would be a lot more intense. Just going back that far, just taking a single breath would change the outcome of natural history. The line that leads to the evolution of the genus Homo might not occur. Or even primates. I think our mammal ancestors were shrew like tiny little things back then. The species you come back to might not be recognizable. I speculate that the butterfly effect would be particularly hard on time travelers. I think it’s funny in time travel stories where people go back and change things, yet, just about every sperm hit the exact same egg on a global scale. Each one of those chaotic situations has like a 1 in 200,000,000 chance assuming the parents happened to get frisky at the exact same time in both timelines. I would find that odd if it happened once, yet usually that happens billions of times all over the world. Maybe once, like on the CW Arrowverse, one person might be replaced. And don’t get me started on how the entire crew of the Enterprise could turn up for the same jobs on the same star ship in an alternate history where violence and despotism won over peaceful democracy. And, of course, minor style changes to facial grooming and wardrobe.

Vor 3 years## SG7AC

^{+2}The same insect but a couple of decades before Lorenzs's paper. I wonder if just a coincidence or if Lorenzs was familiar with that book?

Vor 3 years## Sir Will

^{+1}oh my. Flashback to Freshman year in highschool almost 10 years ago

Vor 3 years## Dhairya

^{+1}Yeah man ! I heard about The Sound Of Thunder in one of Aperture's Video . Please watch the videos they are mindblowing

Vor 3 years## bats56537

^{+24}I started watching your videos because I first seen you on Bill Nye , your videos make a normal guy like me think deeper thoughts and want to learn more. So many times when I watch your videos I am confused and lost but that leads me to do more research and learn more. Thank you for making me think deeper. I was home schooled and didn't really have many opportunities to get better education so I appreciate your videos helping me to think more and learn about things I would not have thought about before.

Vor 2 years## Florida local

^{+1}@Htiek Eromezis okay Karen, chill out

Vor year## paincakes

I'm kinda greatful and infuriated that my teachers never taught physics this way.

Vor 2 years## Anastasia Alifanova

I like to use my accident when I was younger as an example of a butterfly effect the object slashed the top of my eyebrow if I had been a bit taller-shorter i could have lost my left eye It’s sorta interesting trying to think that out of all possibilities I got one small outcome

Vor year## Brian Dennehy

^{+2}The explanation of the topics and visualisation of the ideas on this channel are phenomenal. Well done to all involved

Vor 2 years## Tamás L

What software do you use to create these math based animations? Many thanks

Vor 10 Monate## Henryy SSark

^{+21}Love your stuff Derek! Thank you for getting me interested in science! After learning that science can have a interesting fun side other then just learning facts I have been working hard to get good grade and am currently striving to get into a college where I can further learn so that I can become as smart and knowledgeable as you.

Vor 3 years## southpaw

^{+2}Hell yeah brother! Get that knowledge!

Vor 3 years## Zibran Ahmed

^{+2}All the best, bro.

Vor 3 years## Geeta Kathait

Even though I am very young to understand the terms he uses but am very fascinated by how he explains. 😮

Vor 2 years## Vietcongster

^{+1}Just to clarify: Actually, there *is* a periodic orbit, that is, a loop around the attractors. There are papers that prove it mathematically, one of the methods being what we call "rigorous computation", that is, computational methods specially designed to give precise and rigorous mathematical answers. But nevertheless, any small deviation from that orbit will not be a loop, as the video stated.

Vor year## Ryan Weaver

pretty amazing when you consider the value of local understanding of larger issues. No one wants to upset everyone's applecarts at once. There's plenty of good stuff to do.

Vor 2 Monate## EveryTimeV2

You can actually predict the outcome of a coin flipping machine pretty accurately given you know the starting conditions of the coin, it might not be completely perfect but if it works 99% of the time it's still useful.

Vor 2 years## Cani Sorina

^{+2}Or is it that we are all in a simulation and this is part of the safeguards to the simulation? The fact that graphing it looks like a butterfly may just be the Easter egg of the developer.

Vor 2 years## DocSheep

^{+23}Math professor who actually studies dynamical systems here - EXCELLENT video! This is a great, non-technical introduction to chaotic systems and what makes them hard to study. The intuitive concepts behind dynamical systems can be easily obscured by intimidating technical details and computations, but you've done a fantastic job of making this topic accessible to a general audience. I'm teaching a special topics class on this in the spring and am now DEFINITELY going to show this video as part of our introduction to chaos theory. I've followed this channel for years now (and saw your video on staying relevant on YouTube), and I'm glad to see that you are maintaining your high standards for the quality of your videos. Thank you for your commitment to high standards in education.

Vor 3 years## Mihai Roman

I love your series on chaos!

Vor 2 years## FreeSpirit Karo

Amazing work, Thank you ❤ !!! Have you done a video on Fractals yet and could youpost the link here please ?

Vor 7 Monate## Daniel Tavera

The Lorenz attractor with a black holes could be related in a deep way, at the end what causes black holes can be tiny changes.

Vor 4 Monate