Chaos: The Science of the Butterfly Effect 

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

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

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.

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.

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.

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

The graphics, the explanation, the presentation, everything about this video is top class. I am just speechless.

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

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

He explained such a complex topic with so much simplicity that i am just speechless.

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!

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.

If only someone had explained science this way when I was younger.

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!

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

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!

"That's on the scale of atoms, pretty insignificant on the scale of people," said the pile of atoms.

"The printer rounded to 3 decimal places whereas the computer calculated 6" The ghost of significant figures

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

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.

Read "Chaos" by James Gleick if this stuff interests you. Probably my favourite book of all time, totally changed my outlook on everything, no joke.

I would like a 60 min documentary on the content just presented in the last 2 mins of this video.

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

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

This is one of those science videos that aren't just interesting, but also beautiful.

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.

This video is so good I keep Coming back to re-watch it time to time

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

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! :)

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."

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!

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

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.

I appreciate integrity shown when structuring embedded advertising so much that I watched to the very end. Thank you.

the way you explain things is incredible.

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

:Why aren't you successful. Me : I held a sneeze in the maths class long ago, and now I am doomed.

* Time Traveler Sneezes * Butterfly effect: * GERMANY BECOMES AN ALLY IN WW2 *

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

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.

Why am I interested in this and not in my studies?!

A person in Florida whose house has been destroyed by a tornado: ,,Those freakin’ butterflies flying in Madagascar again”

Was explained so well. Excellent job!

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

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 RU-vid), 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.

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..

This explanation explains so much about the mystery of our mysterious world. Deterministic chaos🙏

Thanks a lot for this extraordinary explanation on chaotic systems.

I understand and mostly don't understand this at the same time. How Schrödinger.

"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.

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

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

This guy is making me look at the universe in a whole different way...

and this, of course, is the choice of steins;gate

Your early dismissal of quantum affects on people took me by surprise, it took me a few ticks to figure out why. I read an article a few years ago saying that One of the inputs into the decision behind a neuron firing or not is actually based on a quantum state change. To me this makes living beings huge amplifiers of quantum state change and the apparent randomness that introduces. Makes sense to me because it seems like evolution would embrace the ability for us to differ by a little randomness in our thoughts the same way we benefit from randomness in our dna. You just reminded me that this is not a commonly held concept, so I thought I’d ask your opinion on the idea. Your videos are amazing, I just watched the multiverse one and you explained that so clearly that it’s jaw dropping. Thank you for your videos!

probably one of the best youtube channels ever. Nice job

“The three body problem” the most interesting sci-fi book you’ll ever read. Don’t read spoilers or anything about the book. Just start listening to it or reading it. Your mind will never be the same.

When you watch RU-vid to rest a bit from studying differential calculus... Then you see the PHASE SPACE.

Thank you for this video extremly well explained ! ❤

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.

Nonlinear dynamics and the concept of chaos literally changed my life and was the start of my academic journey

"A Sound of Thunder" is one of my favorite examples of this.

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

such a beautiful video, very inspiring!

Those animations are absolutely stunning !!!

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

Thank you sir,I tried to understand I think it comes to my heart and intellect. Thank you

I love your series on chaos!

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

“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!” - Chaos by James Gleick

questions like this keep me up all night

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.

I’m so glad I’m in college... YAY MATH!!! This has opened my eyes further. REMEMBER there is no such thing as hard math and easy... only math you have yet to learn and math you know!

I had built a model of double pendulum with the help of my dad for my 12th grade project I explained the butterfly effect, chaos theory relating it to the evolution of universe. Physics is really awesome !! I love the subject and I'm going to take up bsc in physics next year!

The animations are always awesome

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.

This is the best short summary of chaos I've ever seen.

For anyone interested, "Chaos" by James Gleick is a fantastic introduction (and look into the origins of) of chaos theory. Not too technical, and very engaging.

A very helpful video on the buttercream effect

Amazing explanation..great video.

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.

So basically the meaning of my life is a rounding error. Annnnnd my most upvoted comment is about the meaning of my life is a rounding error.

Just finishing Life is Strange Game and I've become really interested on Chaos Theory and Butterfly Effect.

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.

Me: *Releases hundreds of captive butterflies into the wild* Stranger 1: "That's so nice. It's a beautiful sight to release gentle creatures like that." Stranger 2: "He's obviously feeding the local birds. He'd release them in captivity if he cared about their safety." Stranger 3: "He must just be doing something like blowing bubbles to feel good." Me: *Thinking* "Go my pets! Use your butterfly effect to cause tornadoes across the world! Fly my pets! I will destroy it all with butterflies!"

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.

One of the best science chanel on RU-vid.

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

Derek, the music was just perfect. And the whole 13 minutes passed quickly. Great video. And please talk about fractals.

"I mean sure, there's Heisenberg's uncertainty principle from quantum mechanics, but that's on the scale of atoms. -Pretty insignificant on the scale of people." Or is it?

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.

This video has just blowed my mind.

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

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

Thank you, excellent explanation.

Nowadays, I think we can call it "Bat Effect"

so RU-vid algorithm let's see how you perform this time

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

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!

What I loved about Lorenz was that his initial computer simulations were an early effort to prove long range weather forecasting was possible. What he got instead was the opposite. He said that if you were to place weather sensors in a grid around the earth every cubic mile in the sky, the space in between would still not be accurately measured and therefore, impossible to map long term. If you were to put sensors every 1/2 mile, the space between would be a mystery and so forth down and down the scale. That proved that no matter how much you tried to "map" a chaotic system, the space in between the measurements will throw you off over time. Therefore, a small change: like the slight turbulence of butterfly's wing could, over time, effect the system in an unpredictable way if it's "in between the sensors" and not measured initially.