The Real Butterfly Effect

If a butterfly flaps its wings in China today,
it may cause a tornado in America next week. Most of you will be familiar with this “Butterfly
Effect” that is frequently used to illustrate a typical behavior of chaotic systems: Even
smallest disturbances can grow and have big consequences.
The name “Butterfly Effect” was popularized by James Gleick in his 1987 book “Chaos”
and is usually attributed to the meteorologist Edward Lorenz. But I recently learned that
this is *not what Lorenz actually meant by Butterfly Effect. I learned this from a paper by Tim Palmer,
Andreas Döring, and Gregory Seregin called “The Real Butterfly Effect” and that led
me to dig up Lorenz’ original paper from 1969. Lorenz, in this paper, does not write
about butterfly wings. He instead refers to a sea gull’s wings, but then attributes
that to a meteorologist whose name he can’t recall. The reference to a butterfly seems
to have come from a talk that Lorenz gave in 1972, which was titled “Does the Flap
of a Butterfly’s Wings in Brazil set off a Tornado in Texas?”
The title of this talk was actually suggested by the session chair, a meteorologist by name
Phil Merilees. In any case, it was the butterfly that stuck instead of the sea gull. And what
was the butterfly talk about? It was a summary of Lorentz 1969 paper. So what’s in that
paper? In that paper, Lorenz made a much stronger
claim than that a chaotic system is sensitive to the initial conditions. The usual butterfly
effect says that any small inaccuracy in the knowledge that you have about the initial
state of the system will eventually blow up and make a large difference. But if you did
precisely know the initial state, then you could precisely predict the outcome, and if
only you had good enough data you could make predictions as far ahead as you like. It’s
chaos, alright, but it’s still deterministic. Now, in the 1969 paper, Lorenz looks at a
system that has an even worse behavior. He talks about weather, so the system he considers
is the Earth, but that doesn’t really matter, it could be anything. He says, let us divide
up the system into pieces of equal size. In each piece we put a detector that makes a
measurement of some quantity. That quantity is what you need as input to make a prediction.
Say, air pressure and temperature. He further assumes that these measurements are arbitrarily
accurate. Clearly unrealistic, but that’s just to make a point. How well can you make predictions using the
data from your measurements? You have data on that finite grid. But that does not mean
you can generally make a good prediction on the scale of that grid, because errors will
creep into your prediction from scales *smaller than the grid. You expect that to happen of
course because that’s chaos; the non-linearity couples all the different scales together
and the error on the small scales doesn’t stay on the small scales. But you can try to combat this error by making
the grid smaller and putting in more measurement devices. For example, Lorenz says, if you
have a typical grid of some thousand kilometers, you can make a prediction that’s good for,
say, 5 days. After these 5 days, the errors from smaller distances screw you up. So then
you go and decrease your grid length by a factor of two. Now you have many more measurements
and much more data. But, and here comes the important point: Lorenz says this may only
increase the time for which you can make a good prediction by half of the original time.
So now you have 5 days plus 2 and a half days. Then you can go and make your grid finer again.
And again you will gain half of the time. So now you have 5 days plus 2 and half plus
1 and a quarter. And so on. Most of you will know that if you sum up this
series all the way to infinity it will converge to a finite value, in this case that’s 10
days. This means that even if you have an arbitrarily fine grid and you know the initial
condition precisely, you will only be able to make predictions for a finite amount of
time. And this is the real butterfly effect. That
a chaotic system may be deterministic and yet still be non-predictable beyond a finite
amount of time . This of course raises the question whether
there actually is any system that has such properties. There are differential equations
which have such a behavior. But whether the real butterfly effect occurs for any equation
that describes nature is unclear. The Navier-Stokes equation, which Lorenz was talking about may
or may not suffer from the “real” butterfly effect. No one knows. This is presently one
of the big unsolved problems in mathematics. However, the Navier-Stokes equation, and really
any other equation for macroscopic systems, is strictly speaking only an approximation.
On the most fundamental level it’s all particle physics and, ultimately, quantum mechanics.
And the equations of quantum mechanics do not have butterfly effects because they are
linear. Then again, no one would use quantum mechanics to predict the weather, so that’s
a rather theoretical answer. The brief summary is that even in a deterministic
system predictions may only be possible for a finite amount of time and *that is what
Lorenz really meant by “Butterfly Effect.” You find the references to the papers in the
information below the video. Thanks for watching.

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100 thoughts on “The Real Butterfly Effect

  1. It should be rendered "real practical butterfly effect" since in theory you could predict anything in a determinstic world, no matter what.

  2. Isn’t the meteorologist’s conjecture of 5 days of accurate prediction with X # of inputs, then 2.5 more days with 4X inputs, etc. just that, a conjecture? Weather prediction may approximately behave like that, but Shirley the real system cant be that neat.

  3. I am working on applications of Navier-Stokes equation in duct flow using numerical techniques like CFD, DNS, and LES. Indeed, a small mistake can lead to chaos in results.

  4. this is brilliant…I always thought this was more of an entertaining theory than real observational voodoo logic,

  5. Even linear systems with infinite number of eigen values could be unstable, i.e. a small change in initial conditions could lead to a large changes in the solution.

  6. So when we produce a computer program that splits the world into 1km grids and run the program for years ahead to accurately predict the climate?

  7. That was a great summary and helps immensely in understanding why we should “predict the weather and then create it” rather than trying to predict it and wind up living with the consequences… 🤔

  8. Al Gore is going to flap his wings/gums and save us all from man made global warming and profit from his carbon trading business and live in his beachfront villa at the same time

  9. There is another kind of butterfly wing beat. One that is weak but lasts billions of years. A tiny assymmetry of mass inertia, caused by extensive magnetic fields, permanently drives the whole universe around, even without the need for dark energy.

  10. Regarding quantum theory, wouldn’t the collapse of wave functions cause something like the butterfly effect if you did in fact use quantum mechanical equations to make predictions on a large scale? I heard that collapse of the wave function is not well defined or well understood in quantum theory?? And the randomness that occurs – e.g. spin might be up or down, 50-50 odds either way, would seem to throw randomness into the results.

  11. Its just like two numbers can have infinite connections. Say 5 and 10, they can be linked together using other numbers and operators in infinite ways, 5 + 2 + 3, 10 + 4 – 9, etc.
    If there are 3 numbers and a number can be an operand as well as some kind of operator with defined but dynamic behaviour (based on operands), then even if we know three numbers, their behaviour can be so dynamic that we can't predict it, but we can keep simulating and calculating it to know about it.

  12. This completely blows my mind, does it make sense? The predictions must diverge to infinity. I will need to read the paper.

  13. Sabine, as you pointed out in the beginning of the video, given a known dynamics (i.e. equations of motion), we can of course know any future state of a dynamical system if we have perfect knowledge of the initial state. But then you say Lorenz shows that there are systems that can not be propagated arbitrarily far forward in time with arbitrarily dense samples of the initial state. Is this correct? The limit to how far you can faithfully propagate a system with an arbitrarily accurate initial state has a sort of infinite gap with the case of perfect accuracy?

    Very interesting topic!

  14. it is said that more mass create more gravitational pull so does an object placement deforms gravity?
    thanks in advance.

  15. So glad you mentioned QM at the end. The popularized "butterfy effect" makes no sense to me. I think it's an old materialist idea of determinism.
    Our best science QM says life and reality is not determined but comes out of infinite possibilities. This is our experience.
    The weather cannot be reliably predicted even days ahead. Your life cannot be reliably predicted even hours ahead. Nature cannot be reliably predicted ahead of time.
    Will we experience an asteroid, or climate change, or nuclear war, or super A.I., or alien contact, or the reelection of Trump, or all or none, or something unseen? We don't know.
    So, we make "educated" guesses. That's the best we can do. This is the nature of living in a non deterministic reality, in which all possibilities exist simultaneously until experienced. Where's my lotto ticket? Lol 🦋
    Watch "This 11-year-old girl Came out of a 4-Year Coma and What she Told Amazed Everyone" on YouTube

  16. I do not understand how from the model of the subdividing grid follows that a fully deterministic system cannot be used to make predictions infinitely into the future.

  17. An extra fine background this week! It not only matches the outfit but is suggestive of diaphanous wings. Ok, a butterfly's wings don't look diaphanous but, in fact, underneath they are. The scales which cover them and give butterflies their lustrous colours and irridescence are highly modified macrotrichia, a type of hair sported by some other insects. These "inner wings" will convey to the listener that not only will Sabine talk about the butterfly effect, but she'll get right inside the topic and penetrate its deepest meaning. Or something …

  18. The equations of quantum mechanics do not have butterfly effects because they are linear. Well the wave function is, but are we sure we don't need non-linear Born effects to to make fully use of the wave function? Can't the butterly effect arise from the Born law that pops up in pretty much all interpretations of QM?

  19. at last someone talking about the real butterfly effect… Sabine, you can even elaborate much more on that topic in future videos, specially concerning not the weather but the climate, specially in those models that claim to predict the climate in long run because "climate" "not suffer" from pitchforks, chaos etc etc in current models specially in small time-spams that flattens important things like the orbit, precession or nutation and put in a more relevant position things like clouds, cosmic rays, etc… and obviously methodological over statements in the short term like the more importance in the system of CO2 over other gasses, specially cloud covering and so on

  20. I remember some other explanation, in which a very famous Dutch meterologists went through the daily weather data and rounded it off to one decimal point. Then he went through the data again, and did not round it off. He got very different outcomes. That is how the butterfly effect was explained to me.

  21. I can't understand. Suppose a butterfly causes a tornado. Very well, but there is not only one. Now we have say hundreds of thousands of butterflies in Brazil, have we hundreds of thousands of tornadoes in New York, in addition to the flooding because of the global warming, and some other catastrophe caused by the election of Trump? And what about the nose of Cleopatra? That just doesn't make sense. We are still here waiting for the end of the world in 2012 and for the Y2K bug.

  22. This sounds like Cantor. You described a countable infinity of information being insufficient, so presumably information on the order of the continuum is needed.

  23. In other words, predictions on any deterministic system wil eventually fall victim to the law of diminishing returns because of imperfections in the dataset and/or model.

  24. I believe this butterfly effect is a property found in models of the real world, and mostly of models on digital computers. That does not mean the butterfly effect exists in the real world.

  25. Well, non linear deterministic equations cannot be like the gentleman described. The problem is this; when you solve for day 7 with a model that is good for 5 days, the errors is not in the mathematics, the maths are deterministic, if you can start with exact initial conditions and you use any mathematical deterministic equation to solve for a later date the answer is always exactly the same, this means there cannot be uncertainty at all anywhere within the model. Its can still miss the mark when explaining the world, when you finaly have infinitesimal boxes you dont nessesarly have a realistic description at all anymore, i really dont see how the maths can change from one day to another, and it cant. Take the temprature or preassure, if you go all the way to infinitesimal grid size, what does temprature and preassure even mean?

  26. At some scale temprature and preassure breaks down into atoms and radiation, and some point radiation and atoms can break down into diffrent descriptions that are appropriate for “slicing the pie” smaller. If you knew everything and the correct equations for a infinitesimal grid size you would have a completely predicable process, but the variables at 0 size is kind of a tricky business, and i dont think its even possible to nail it down given infinite time to think about it.

  27. Its nonsense that you could break down attributes such as preassure or temprature to any scale 🙂 you cant, and if you tried to take the picture from which these attributes emerge from you would discover both that the amount of information goes up a lot, and that the same problem would apply, the individual motion that creates preassure and temprature is embeded in structures that also have a kind of temprature and preassure, and so on

  28. In other words, Lorenz is suggesting that there may (without proven examples) exist systems which aren't just "chaotic", but what we may even call as hyperchaotic : the very idea of trying to increase the numerical precision and hoping that the successive approximations you get with some local solution method ("evolve forward one step in time") will converge on the true solution, can break down, and instead the "convergence" to exhibit a behavior more reminiscent of a Taylor series for a function with a singularity – converging in a local region, but exploding to infinite uselessness outside it.

  29. Since the accuracy of the measurements and subsequent numerical solution of the Navier-Stokes equations are necessarily limited to finite decimal places (finite computer word length), all memory of the initial conditions is lost after sufficient iterations, thus even a deterministic system must end up as ultimately unpredictable.

  30. It seems to me that the Taylor series will cross a size were quantum effects would come into play before it reached its limit and through everything off.

  31. The decrease in the amount of days coupled w/the increase in the number of grids reminds me of the law of diminishing marginal returns from the study of economics: the more resources you use/consume in doing something (in this case the more grids you add), the less output (re. the number of additional days) your additional resources produce. The only thing i'm confused about is the convergence to the number 10. Why wouldn't the output converge to 0 given a very high/infinite amount of input unless for some reason you cannot have a 0 amount of anything(?).

  32. We assume the theories are correct, and make mathematical inferences on them. Then we say that Nature is described by mathematics. But there is no empirical verification of that. For instance, weather models are way too imperfect for any test, and we actually really don't know why they are limited in time. So I am sorry to say you that it is not more scientific than the theory of the Big Bang, the multi-universe, the quantum supremacy, the prevision of global warming, the anthropic principle, the wormholes, and all your pet targets.

  33. ! Wrong! Wrong! Wrong! We have climate models! Climate models built on these methods! They have never been wrong! LMAO!

    Decades, centuries, our climate models operate on the God level! Just look at how many people believe our predictions! There! Is your proof.

  34. Really good explanation but you forgot to handle the other part of the effect: "Does a Tornado in Texas cause Brazilian Butterfly's to flap their wings?"

  35. There is another problem – not coming from pure math, but from the realworld computers… making the grid smaller does make the calculation more accurate first, but there is a limit, given by the limited floating-point arithmetics. So if just making the grids smaller, without giving more bits for the mantissa of your floatingpoint-calculation, first the error decreases, but at a certain point increases again. (For linear diff. eq. already a problem – far from chaotic behaviour.) So with a and finer finer grid you may also need to make your calculations better, just to circumvent the computed error. But if the 10 days are a hard mathematical limit, then at some point it will not make sense to make the floating-point calculations with longer and longer mantissas. It would be interesting to find out, how large the floating-point calculations must be to come close to the 10-days. And making the computation more accurate would then possibly be just waste of money and ressources.

  36. Thank you very much for explaining this. I have always been dubious about the butterfly effect as it is commonly described. This makes much more sense. Cheers, Russ

  37. Is there really chaos? Or just lack of knowledge?
    This video is a good example of why spending trillions to fix carbon predictions is patently stupid.

  38. This result of chaos prediction summing to a maximum of 10 time units, reminds me of the infinite series 1+2+3….= -1/12 (I think formally called the Riemann Zeta function). I wonder if they could be combined: [infinite input resolution]= -1/12 —-> [chaos non-linear function] = 10. ?

  39. Now we have clarified the term "butterfly effect". So we have a "common" and a "real" one (wouldn't "original" or "historical" be better than "real"?). Next time we talk about "common" and "real" atoms (nuclei): the one can be splitted/fissured, the others are unsplittable atoms ("greek atoms"?) which and are solid small bowls 🙂

  40. Thank you Sabine for every all the productions created. I enjoy your explanations because they fill me with wonder and amazement.

  41. 30:41 on Tim Palmer's video:

    1-(Re)define Logical AND for base 3 as (output if identical)
    2-If (x base 3) AND (y base 3) is odd in base 2, plot x,y

    You end up with Sierpinsky carpet. So Cantor's dust is directly, point by point, identical to Sierpinsky carpet. Except now it's a plane. By definition (extremely weird) you have equated the Sierpinsky carpet to all of the LAST digits of the reals and irrationals used in that attractor.

    Is that beyond weird or what?

    Does that help the mathematics?

  42. James Gleik's book was masterful and completely riveting. However, I'm glad to hear that you found the original meaning of this now-famous idea.

  43. 3:30 Assume the initial grid size is 1000 Km. How about if I started with a 2000 Km grid size instead? I'd be getting a prediction good for 10 days. And what about if I started with a 4000 Km grid size? That's 20 days' worth of prediction I'd be getting, so on and so forth.

  44. [ 5:49 In Brief ]
    This channel is a gift to researchers and lay people who looks for finer things in life,
    Dr. Sabine saved us our precious time and cuts through all the fog that lingers around what was meant about the butterfly effect and goes straight through to get the fruit of knowledge and serves it to us in a silver plate (bon appetite).

    This is what the internet is all about.

    Thank you,
    I am now a lifetime subscriber.

  45. the butterfly effect is something like a chain reaction on an atomic scale like propellant in a rocket or payload in a nuclear weapon. the first atom ignites the other eventually having accumulated so much power to put an object in orbit or annihilate a country. nothing to do with a butterfly flapping in the amazon and creating a hurricane in europe. also something going viral on the internet i consider a butterfy effect. if something doesnt make sense it probably is nonsense.

  46. Sabine, If you were to achieve and accurate predicition for the grid, say one centimeter of rain; that prediction would only apply to the grid as a whole and not each subsection of the grid correct? You would be able to measure small variations in rainfall over the entire grid without invalidating the the prediction.

  47. All mathematics relating to real world are approximations. For example you cannot determine a circumference of a circle or a length of a catenary. Only problems set by mathematics itself have a precise solution

  48. In addition to the reality of finite grid size and limited precision of measurement, don't forget the problem of roundoff error building up in calculations, though I suppose that flaw could be attributed to our human limitations. And since we are making infinitely precise measurements, we will need infinitely long registers and an infinite amount of memory in our computer. So awkward!

  49. My paper about Chaos and ice crystals, atmospheric optics:

  50. Thanks for this video!! You may be interested in papers by Marian Boykan Pour-El about the wave equation (a linear differential equation) having incomputable solutions for specific computable boundary values, given some suitable definition of these terms.

  51. Hello, would you consider making a video about Erik Verlinde's proposed entropic gravity? PBS spacetime was supposed to cover it like 2/3 years ago, but they kind of left us hanging when the hype died out. Last I heard, it was supposed to be tested using gravitational weak lensing.

  52. Serious question: Doesn't the argument assume you can keep dividing space forever? But according to quantum theory there is a smallest scale, the Planck length. If we assume that's true, how does the series change? I mean, it's not infinite anymore for one thing, and if we assume a deterministic system it's reasonable to expect that the sum is infinite. So do all the terms change or is it that just the last term has a jump up? Asking without having seen the underlying math

  53. Omni used to purport the tornado effect which I found absurd.
    It seemed to me if a tornado was on the verge of forming, the butterfly effect might well be the final straw.

  54. That opening jab was Hilarious! I haven't heard such a nice scientific joke in months! As always, you silent sword is awesome!

  55. Would have been more interesting If You went in more details about the grid and why does it cut in half the prediction time. Would love to read the paper when I get the time

  56. I thought the whole idea of Chaos was its unpredictability irrespective of measurement accuracy. But then I was never a Kingmaker.

  57. In my childhood, I first read Ray Bradbury's "A Sound of Thunder" sci-fi story, and then heard about a scientific term related to chaos description, so I concluded for myself that the story served as an inspiration. The story published in 1952 predates the scientific discussion that you have mentioned here, and I suppose, bird vs. insect talk steals the real credit… In spirit of the metaphor, but of course not literally, Bradbury's story about a small change leading to a huge difference in the future (even atmosphere composition changed) fits completely.

  58. 5D = Graviton-Axion-Dark Matter (Blue Jet-Red Sprite-gravitational wave signature) electrical charge equalization perpendicular to the magnetic pole, filtered/structured by the atmosphere and magnetosphere, gravitationally drawn to the solid-state, condensed matter (dark, or not theorized AND experimentally verifiable, yet) processing core. —–> Lightning! Quantum Gravity w/o a soupcan or an hourglass or a bow-tie. Brooklyn 2 the Universe.

    I tried to find this on youtube, but Err'body got an App, so no mo.
    der-lacheln-beherrscht-the-smile-masters an old SNL sketch, now on the NBC App
    What I imagine Boltzmann was raised on, analog not digital.

  59. Is dark matter made of butterfly wings? We want to know. Are there butterflies in the black holes? Are dark butterflies dark energy? How many seconds after the Big Bang did appear the butterflies?

  60. Very well explained! Just a small remark: I noticed you're pronouncing year numbers like 1969 as 9069. Have a listen! 😉

  61. Poincare instability within differential equations of energy and momentum are real. There is a relationship between them and Fermi-Pasta-Ullam for 1D – however FPU is in an incompressible medium while in air you have an incompressible system – and turbulence itself is beyond our ability to practically compute. Now there was a man whose voice rebuked (like the small FPU instability) winds and waves (compressible and incompressible) putting them immediately to calm. And there was a follower of his in relatively modern times whose cross fell off his neck, caught on the hull of a ship in a storm in the Aegean, and the waves and wind calmed as well. Faith is substance for belief, not a blind exercise…..

  62. This seems like an irrelevant argument. Ok, so e.g., you can only predict asymptotically to 10 days…no problemo, do what running averages do, take a measurement as a day goes by, put it in the algo, again, 10 days ahead. Now, maybe for war planners that would have the need to figure out when it would be most useful to attack some country 6 months in advance, there's no help, and ah, isn't that sad, I'd find it great to have any weather prediction that is actually accurate out to two days, because in the US, they are NEVER right. They are wrong 60% of the time on weather for the next day…

    Chaos is NOT random, so it may be solvable, idk. NP complete problems are computationally solvable to an arbitrary level of precision. Our AI isn't good enough yet, we are just talking monkeys after all. Wait for the earth movers we made for dirt, to move thinking in the same way. We are less than 5 years away, maybe less. XAI has made great strides, and DARPA is moving on the last problems of AI being its inability to learn from new data after training. Once that's solved, we all die…so let me say goodbye in case I forget later and poof, I'm gone…

  63. Wow. If we ignore quantum uncertainty and possible size limits such as the Planck length, it's proven that some systems can never be predicted beyond finite time frames, even with infinite information? This seems to contradict relativity, hence one can't ignore the quantum effects.

  64. The most intriguing feature of the butterfly effect is that it postulates a limit on our ability to measure data, plug it into a mathematical model and predict the behavior of a system. Is this just small measurement and computation errors accumulating over time, as in classical error analysis?

  65. The equations of QM are linear, but they don't by themselves describe observable reality. To make predictions a big non-linear step is required, so I think the same considerations apply.

  66. Fascinating. I think this explains the effect and then initiates a series of new questions about determinism and chaos.

    For example, would the limits of QM or plank lengths or time resolution give us a theoretical limit to the deterministic nature of reality? This is a long standing philosophical dilemma and of course religious implications concerning the limits of omniscience without direct application of omnipotence.

    I realize you are not religious but foreknowledge is one of the proofs or tests of God in the bible.

    Also what can you say about the nature of entanglement in terms of determinism? It seems clear, given recent experiments. That Einstien is wrong in assuming hidden variables or essentially preconditions determining the state of entangled particles.

    Would you say that the universe is definitively undeterministic in that the point of observation becomes the determinant of future events rather than the point of origin. Of course the point of origin determines the statistical possibilities but the point of observation determines which possibility becomes manifest. The delay determined by the speed of light and the inconcievabily of possible interventions in the interim period of delay would describe probabilistic chaos. I apologize for my speculation but some mathematician somewhere is probably working on it somewhere………

  67. Deterministic and yet with perfect information you can't predict out forever? What? What is determinism then?

    Say you can't predict beyond day 10. So you can perfectly predict day 9? Does that not mean you can predict up to day 19?

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