# Is the butterfly effect real?

Is the butterfly effect real? It is a well known statement that a butterfly can, by flapping her wings in a slightly different way, cause a hurricane somewhere else in the world that wouldn't occur if the butterfly dídn't move her wings in a slightly different way.

I can't imagine that this is true.

• Comments are not for extended discussion; this conversation has been moved to chat. – David Z Jan 30 '16 at 12:15
• For one, such tiny things like electron-based computation (or even quantum!) in CPUs are changing lives of many humans, and steer sometimes huge machines (think spacecraft). While you might argue that it's not random, unlike the butterfly wing flaps, said wing-flaps are not random at all. It's just something that butterflies do for their purposes in response to their environment. Also +1 for good question. – loa_in_ Jan 30 '16 at 21:56
• In general this is just a hypothetical example of what a dynamical system is. – galois Jan 30 '16 at 22:29
• I've rolled back the major edit because you seem to be trying to use it to engage in a conversation that way, and that is simply not part of the Q&A model used on Stack Exchange. – dmckee Jan 24 '18 at 22:13
• Alright, Mr. dmckee. I just wanted some mathematical answers, though, but I guess they are too complex. – descheleschilder Jan 25 '18 at 0:54

First the statement is a poetical way to express how in chaotic systems, small changes can trigger drastically different results. This statement does not attempt to relate butterfly movement to large scale weather changes.

I would say we have elements to say it is not, for example, waves are a chaotic behaviour of the sea surface, but we have never found tsunamis born out of this; you can always trace a tsunami back to certain earthquake or similar large scale event. You can think of a number of chaotic systems and we have not been able to link their behaviour with larger scales effects.

Finally, if small changes in systems could trigger large scale changes, our science would look very much like witchcraft, because we would have to admit that sudden large events can appear without apparent cause, at least until we found the small butterfly pattern causing them. But even tornados, which are hard to predict, are known to be preceded by large changes in atmosphere pressure and other large scales features.

• Granted, we never get tsunamis from sea waves chaos, but we do get rogue waves sometimes. – Ruslan Jan 26 '16 at 15:56
• I guess we will have to wait for scientific explanations of them. My bet is on fluid behavior under certain macroscopic conditions, rather than small effects (plankton effect?). :) – rmhleo Jan 26 '16 at 16:01
• @rmhleo The point that you are missing is that frequently deterministic chaos takes place not in closed systems but in systems with driving forces (frequently periodic). In this circumstance whether a driving force adds to or detracts from a certain phenomenon at a particular time can depend critically on the initial conditions and it is the accumulation over many periods that leads to such drastic differences. For this to occur the system must also be nonlinear. All of this was known to Poincare at the end of the 1800's and was rediscovered by Lorenz and others in the 1960's-1970's. – Lewis Miller Jan 26 '16 at 20:11
• @LewisMiller trying to move that to reality, there is an abundance of phenomena that can be understood in terms of non-chaotic framework. This leads me to think that chaotic phenomena, deterministic or not, must be short spanned in time and unstable, and must have negligible effect on scales larger than those where it "lives". That is the position I am inclined to. – rmhleo Jan 26 '16 at 20:44
• You're still missing the point. Imagine a device, where a drop of water can take two different paths. Which path it takes is determined by a tiny irregularity in the piping - one side is slightly more favoured than the other. However, the two pipes lead to two sides of a dam - as long as the water on each side is balanced enough, everything is fine, but as the disbalance accumulates, the dam breaks and everybody dies. And yet, this would have not happened if the irregularity wasn't there - that's the tiny change producing massive differences in results. – Luaan Jan 26 '16 at 21:14

Does the flap of a butterfly's wing in Brazil set off a tornado in Texas?

This was the whimsical question Edward Lorenz posed in his 1972 address to the 139th meeting of the American Association for the Advancement of Science. Some mistakenly think the answer to that question is "yes." (Otherwise, why would he have posed the question?) In doing so, they miss the point of the talk. The opening sentence of the talk immediately after the title (wherein the question was raised) starts with Lest I appear frivolous in even posing the title question, let alone suggesting it might have an affirmative answer ... Shortly later in the talk, Lorenz asks the question posed in the title in more technical terms:

More generally, I am proposing that over the years minuscule disturbances neither increase nor decrease the frequency of occurrences of various weather events such as tornados; the most they may do is to modify the sequences in which they occur. The question which really interests us is whether they can do even this—whether, for example, two particular weather situations differing by as little as the immediate influence of a single butterfly will generally after sufficient time evolve into two situations differing by as much as the presence of a tornado. In more technical language, is the behavior of the atmosphere unstable with respect to perturbations of small amplitude?

The answer to this question is probably, and in some cases, almost certainly. The atmosphere operates at many different scales, from the very fine (e.g., the flap of a butterfly wing) to the very coarse (e.g., global winds such as the trade winds). Given the right circumstances, the atmosphere can magnify perturbations at some scale level into changes at a larger scale. Feynman described turbulence as the hardest unsolved problem in classical mechanics and it remains unsolved to this day. Even the problem of non-turbulent conditions is an unsolved problem (in three dimensions), and hence the million dollar prize for making some kind of theoretical progress with regard to the Navier-Stokes equation.

Update: So is the butterfly effect real?

The answer is perhaps. But even more importantly, the question in a sense doesn't make sense. Asking this question misses the point of Lorenz's talk. The key point of Lorenz's talk, and of the ten years of work that led up to this talk, is that over a sufficiently long span of time, the weather is essentially a non-deterministic system.

In a sense, asking which tiny little perturbation ultimately caused a tornado in Texas to occur doesn't make sense. If the flap of one butterfly's wing in Brazil could indeed set off a tornado in Texas, this means the flap of the wing of another butterfly in Brazil could prevent that tornado from occurring. (Lorenz himself raised this point in his 1972 talk.) Asking which tiny little perturbation in a system in which any little bit of ambient noise can be magnified by multiple orders of magnitude doesn't quite make sense.

Atmospheric scientists use some variant of the Navier-Stokes equation to model the weather. There's a minor (tongue in cheek) problem with doing that: The Navier-Stokes equation has known non-smooth solutions. Another name for such solutions is "turbulence." Given enough time, a system governed by the Navier-Stokes equation is non-deterministic. This shouldn't be that surprising. There are other non-deterministic systems in Newtonian mechanics such as Norton's dome. Think of the weather as a system chock full of Norton's domes. (Whether smooth solutions exist to the 3D Navier-Stokes under non-turbulent conditions is an open question, worth $1000000.) Lorenz raised the issue of the non-predictability of the weather in his 1969 paper, "The predictability of a flow which possesses many scales of motion." Even if the Navier-Stokes equations are ultimately wrong and even if the weather truly is a deterministic system, it is non-deterministic for all practical purposes. In Lorenz's time, weather forecasters didn't have adequate knowledge of mesoscale activities in the atmosphere (activities on the order of a hundred kilometer or so). In our time, we still don't quite have adequate knowledge of microscale activities in the atmosphere (activities on the order of a kilometer or so). The flap of a butterfly's wing: That's multiple orders of magnitude below what meteorologists call "microscale." That represents a big problem with regard to turbulence because the magnification of ambient noise is inversely proportional to scale (raised to some positive power) in turbulent conditions. Regarding a simulation of$1.57\times10^{24}$particles My answer has engendered a chaotically large number of comments. One key comment asked about a simulation of$1.57\times10^{24}$particles. First off, good luck making a physically realistic simulation of a system comprising that many particles that can be resolved in a realistic amount of time. Secondly, that value represents a mere 0.06 cubic meters of air at standard temperature and pressure. A system of on the order of 1024 particles cannot represent the complexities that arise in a system that is many, many orders of magnitude larger than that. The Earth's atmosphere comprises on the order of 1044 molecules. A factor of 1020 is beyond "many" orders of magnitude. It truly is many, many orders of magnitude larger than a system of only 1024 particles. • Comments are not for extended discussion; this conversation has been moved to chat. – David Z Jan 30 '16 at 12:14 • About the simulation of a septillion of atoms, you can see benchmarks concerning the struggle to simulate millions or billions of them: lammps.sandia.gov/bench.html – durum Jan 30 '16 at 21:14 • Is the concept of turbulence in the NS equations really synonymous with "non-smoothness?" I thought that turbulence refers to nonlinear behavior originating from the nonlinear terms in the NS equations, and that turbulence is precisely what makes the equations hard even for smooth initial conditions. Under this definition, turbulence will generically occur, even in smooth solutions to the equations. – tparker Jan 6 '17 at 1:22 • I read a quote on this that I believe was said by von Neuman. He said that when he heard about the butterfly effect, his mind was full of images of weather control and great benefits for humanity using only the smallest of efforts. It was only much later that he realized that what the effect was really talking about was giving a well shuffled deck of cards just one more shuffle. You know you have changed your odds of winning, but you don't know whether you made them better or worse. – Cort Ammon Apr 11 '17 at 20:15 • Great answer, but I don't think the part about the Navier-Stokes equations is correct. The Millennium Prize problem is about smoothness from arbitrary initial conditions, not just "non-turbulent conditions." I don't think it's true that "the Navier-Stokes equation has known non-smooth solutions" [that emerge from smooth initial conditions], as that would solve the Millenium Prize problem. – tparker Sep 30 '17 at 18:17 This question already has an answer (by me) on Earth Science: The butterfly is a colourful illustration of Chaos Theory, and the word butterfly came from the diagram of the state space (see below). (Apparently, my claim on the origin of the word butterfly may be historically inaccurate. Could be of interest for HSM SE) A system that is chaotic is extremely sensitive on its initial value. In principle, if you know exactly how the state of the universe is now, you could calculate how it develops (but due to other reasons, it is theoretically impossible to know the state exactly — but that's not the main point here). The issue with a chaotic system is that a very small change in the initial state can cause a completely different outcome in the system (given enough time). So, suppose that we take the entire atmosphere and calculate the weather happening for the next 20 days; suppose for the moment that we actually do know every bit. Now, we repeat the calculation, but with one tiny tiny bit that is different; such as a butterfly flapping its wings. As the nature of a chaotic system is such that a very small change in the initial value can cause a very large change in the final state, the difference between these two initial systems may be that one gets a tornado, and the other doesn't. Is this to say that the butterfly flapping its wings results in a tornado? No, not really. It's just a matter of saying, but not really accurate. Many systems are chaotic: • Try to drop a leave from a tree; it will never fall the same way twice. • Hang a pendulum below another pendulum and track its motion: (Figure from Wikipedia) Figure of a double pendulum. Compare with this youtube video. • Or try to help your boyfriend in what must be one of the loveliest illustrations of Chaos Theory ever. Suppose you are running to catch the bus. You keep sight of a butterfly, which delays you by a split second. This split second causes you to miss the bus, which later crashes into a ravine, killing everyone on board. Later in life, you go on to be a major political dictator starting World War III (Note: this is not the plot of the linked movie, but my own morbid reinterpretation). Tell me, did this butterfly cause World War III? Not really. (Figure from Wikipedia) • That diagram is not why it is called the "butterfly effect." That diagram did not exist in 1972 when Lorenz gave his talk, let alone in the 1960s when a detractor derided Lorenz by saying "if that's the case, the flap of a butterfly's wing in Brazil could set off a tornado in Texas!" – David Hammen Jan 28 '16 at 6:21 • Not this diagram specifically. When I learned about chaos theory I learned that the word butterfly was inspired by the phase diagram. Is that wrong? – gerrit Jan 28 '16 at 13:56 • The image you posted is not from the 1960s / early 1970s, which is the timeframe of the term "butterfly effect." From what I've read, that phrase and "big bang" had rather similar origins. The phrase "big bang" was originally a derogation by Fred Hoyle, who espoused a steady-state universe. The "butterfly effect" originated from a detractor in some previous talk by Lorenz who said something along the lines of "if that's true then even the flap of a sea gull's wing in Brazil can cause a tornado in Texas!" A butterfly is even smaller and prettier than a sea gull. – David Hammen Jan 28 '16 at 21:18 • @DavidHammen Perhaps I was mistaught. I admit I did not study the history in detail. Perhaps a good question for History of Science and Mathematics. At any rate, I added a note that I am not sure about the origin of the phrase anymore. – gerrit Jan 28 '16 at 22:10 • @descheleschilder Chaos theory implies that adjacent states in the state space at time$t$can correspond to states at time$t + \Delta t\$ that are arbitrarily far apart. This situation is not unique to chaos theory, but applies to (other) unstable equilibria as well. Balance a ball on the summit of a hill: the flap of the butterfly can make the difference which way it will roll. The ball might trigger an avalanche, etc.. In summary: where situations are unstable, a tiny difference in the initial state can amplify considerably. This also applies to chaos theory. – gerrit Jan 29 '16 at 18:50

The butterfly effect is a popularization of chaos theory.

This diagram is part of the narrative of chaos theory for the hoi polloi.

A plot of Lorenz attractor for values r = 28, σ = 10, b = 8/3

It does look like a butterfly after all :; ( tongue in cheek) .

Let us set up the chaos backround first, from the Wiki article:

Chaos theory is the field of study in mathematics that studies the behavior and condition of dynamical systems that are highly sensitive to initial conditions—a response popularly referred to as the butterfly effect.1 Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general.2 This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved.4 In other words, the deterministic nature of these systems does not make them predictable.4[5] This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:[6]

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

.................

Sensitivity to initial conditions is popularly known as the "butterfly effect", so-called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D.C., entitled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.

The expression is figurative, and does not have to do with energy conservation but with initial conditions in chaotic systems, from dynamical chaos.

As climate is chaotic but the dynamics entering the equations are not really well determined, the butterfly effect can just be considered a popularized fiction for dynamical chaos and not taken literally.

• What are the quantities on the two axes on the diagram (and units, if any)? – Peter Mortensen Feb 1 '16 at 18:38
• @PeterMortensen Variables in a differential equaiton, x, y ,z . Have a look here en.wikipedia.org/wiki/Lorenz_system – anna v Feb 1 '16 at 18:47

Suppose I go outside, flap my hands in the air and then go about my business as usual. A few months later I see on the news that some town in the US has been devastated by a tornado. Is the counterfactual statement that says that had I not flapped my hands, the town would not have been hit by the tornado, a rigorously correct statement? Let's assume for argument's sake that atmospheric models predict that the perturbation caused by me flapping my hands was indeed large enough to have led to totally different weather patterns (otherwise the question would be about larger perturbations or longer time scale).

In a purely classical setting, you can't escape the conclusion, except for the fact that you had no choice to do something else than what you in fact did, as there can be no counterfactuals in a purely deterministic universe. But of course, we know that the universe is not deterministic in this sense, the laws of physics are based on quantum mechanics.

In a quantum mechanical treatment, quantum fluctuations will eventually grow to become large on a macroscopic scale, this has the effect of randomizing the weather situation. This then erases the effect of the hand flapping on a sufficiently long time scale; whether or not the town would be hit is then a matter of chance, the hand flapping then amounts to shuffling the deck that was going to be shuffled anyway.

One can then ask if there exists an intermediary time scale on which the hand flapping would have caused large enough changes in the atmosphere to have affected the formation of a tornado, but such that quantum fluctuations have yet to grow large enough to affect the weather. But I think that even this question is not a good question to ask, because of the physics involved in the decision whether or not to flap my hands. As pointed out in this article, the seemingly random decisions we make are quantum mechanical in origin.

This means that whatever I, other people, butterflies, birds etc. end up doing is ultimately due to small quantum effects that have been amplified. If I then condition on me having flapped my hands or not having flapped my hands, the actions by other organisms are still fundamentally random due to quantum mechanics. The counterfactual statement is thus invalid.

• Not only your answer is straight to the point, it's also the funniest. – Marc.2377 Jan 27 '16 at 3:28
• but presumably in a quantum setting, the flapping your hands affects the probabilities of various outcomes? – innisfree Jan 27 '16 at 5:09
• @innisfree Yes, so the question is if in the sector of the multiverse where I flapped my hands the probability if that town being hit is much larger than in the sector where I didn't flap my hands. My argument is then that the probabilities are in fact similar because in both branches consist of sub branches where someone else did something or didn't do something similar with both alternatives really existing. – Count Iblis Jan 28 '16 at 18:01

The effect is real in the sense that the movement of the butterfly may have a huge effect on the weather in a far away place.

There is however no way to control this. Don't imagine some mad scientist holding the world for ransom with a cage full of butterflies. It is better to think of it as an illustration of the theory of chaos. The idea is that chaotic systems such as the weather are hypersensitive to the smallest details (such as a butterfly flapping it's wings).

Don't forget that many parameters actually enter the system. In a truly chaotic system there is no way to disentangle the effects that are caused by butterflies (small perturbations) from the ones caused by large scale perturbations (such as the gulf stream, for example). We say that the butterfly is causing the hurricane because if we had killed it, there would be no storm. However there are many other parameters that are causing the hurricane in the same way. It is as much likely that killing one butterfly and three bees will bring the hurricane back.

Finally, chaos is something that needs time to develop. On short time scales, events that are small and far away have negligible effects.

• Can't be controlled? What about the curious case of Edgar Witherspoon?? – Nacht Jan 26 '16 at 22:27
• It can't be predicted, but it can be controlled. See e.g. Ott's book on dynamical systems, or search controlling chaos. – Jorge Leitao Jan 28 '16 at 22:41
• "There is however no way to control this." But, but ... Randall can't be wrong! – dmckee Jan 24 '18 at 22:53

Obviously the effect needs time to propagate (probably the speed of sound), but the effect is quite real. Imagine a incalculably large pool table where half of table has balls, if you shot the cue ball so it clipped one of the balls on the half with balls, the effect could propagate to the end of the table, but if you missed that half nothing would happen. The difference in that could be 0.00000000001 degrees in your shot, much like the flapping of a butterfly's wings.

• yeah, but keep in mind in the example with the pool table, your shot's force is comparable to the force necessary to move another pool ball, whereas in the case of the butterfly, the flap of a butterfly's wings is incalculably less than that of a hurricane. – DevilApple227 Jan 26 '16 at 15:25
• @DevilApple227: That's the wrong comparison. The question is not how much force it takes to move a pool ball --- it's how much force it takes to divert your hand's position by 0.00000000001 degrees. – WillO Jan 26 '16 at 15:46
• @DevilApple227 The "butterfly" in the pool example is not a shot force. It's the force that caused the ball to deviate at 0.00000000001 degrees. The shot force in the original butterfly example would be a sun that pumps the atmosphere with energy. But it can produce simply warm day instead of hurricane. – OON Jan 26 '16 at 16:01
• @Sam I think it would be a good idea to add WillO and OON's clarifications to the question (reworded slightly if necessary!) – wizzwizz4 Jan 26 '16 at 19:03
• @descheleschilder, wow, you completely failed to understand the analogy, thanks for posting exactly the wrong response. – Sam Feb 1 '16 at 14:58

Could a butterfly flapping its wings cause a hurricane? Yes. However, only if the required conditions (which are very precise) existed elsewhere. There would be a critical state of the atmosphere after which a hurricane will form, and a small injection of energy from a butterfly flapping its wings could cause this threshold to be crossed.

Is it likely that a butterfly flapping its wings has ever caused a hurricane? Probably not. It is very unlikely for the required atmospheric state to exist at that exact critical threshold, at exactly the right moment, and exactly the right place with respect to the butterfly. Given infinite time though, of course, this would happen. (Or an infinite number of butterflies and infinite amount of atmosphere...)

Could we ever be sure of any of this? Probably not. Measuring (in the real world, not in simulation) the effect of a butterfly flapping its wings is such a sensitive demand that this would be almost impossible, and certainly impossible with current technology. In fact, the magnitude of wind caused by a butterfly is so small in comparison to that of a hurricane, that the very act of measuring the wind speed from a butterfly's wings may result in a significant disturbance of that wind, after which the effect of the original wind would no longer be traceable. (This reminds me of the Uncertainty Principle.)

• I'm no chaos theorist, but I think saying that the butterfly wings can (in the common-sense manner) cause a hurricane, even in a theoretical model, is rather dishonest. In a chaotic system, a smallest change in the initial conditions can, in time, yield significant changes. However, the initial change can only be called a "cause" of the latter significant changes in very relative terms, which do not correspond to common-sense notion of cause and effect. If you keep dripping water into a bowl on the edge of a table until it tips, the last drop does not cause it any more than the hundredth. – tomasz Jan 27 '16 at 22:06
• Yes I agree with that. I suppose when I mean that the atmosphere must be in a critical state, I mean that it has the potential to become a hurricane if the butterfly were to initiate that wind. – Karnivaurus Jan 28 '16 at 0:01
• I agree that there´s not one cause leading the weathersystem from one state to the next. You should consider the system (parallel), as a whole. I don´t agree that a storm can be in a state af unbalanced equilibrium, wich only needs the flapping of a pair of wings (very close the so-called unbalanced equilibrium, and not 500 km. away) to send it to one of the both sides of the equilibrium. The only way the weathersystem displays chaos is if you let the relevant variables of the WHOLE weathersystem vary VERY little. I don´t agree with what Karnivaurus says: I mean ... to initiate that wind. – descheleschilder Feb 1 '16 at 11:57
• It's not possible to exist in a state of unbalanced equilibrium for a sustained time, but it is possible to exist in that state momentarily. If this coincides with the butterfly flapping wings, then this is when a hurricane can develop. And when I talk about the atmosphere being in a critical state, I am not talking about there being "very slightly less wind" than is necessary for a hurricane, which only needs a little wind to become a hurricane -- when I refer to "conditions", I mean "initial state + other factors" which, together with a butterfly flapping its wings, would cause a hurricane. – Karnivaurus Feb 1 '16 at 16:08

Weather models have significant positive feedback loops built in because without them the return to average is too fast for accurate prediction. This is due to the grossly inadequate spatial density of weather collection stations, and results in weather models being subject to the butterfly effect.

However, the actual circumstances in which significant positive feedback loops exist in the real world are much less frequent than in the models; basically only in large rotating thunder cells (causing tornados) and in cyclonic cells over quite warm oceans (spawning hurricanes etc.) Everywhere else the feedback cycles must be negative or all butterfly wings would spawn hurricanes. The damping effect of these dominant negative feedback cycles almost everywhere is what makes our climate mostly predictable, even wen the weather isn't.

Therefore, emphatically no; the butterfly effect doesn't exist in nature; It is artificially introduced into our weather models, through positive feedback loops, in order to partially compensate for the insufficient spatial and temporal density of starting data measurements; but at the expense of instability in the models.

Climate is what you expect; weather is what you get. - anonymous

The engineer weighs in.

let's look at the disturbance created by a butterfly's wings. Air has viscosity, so the butterfly wingflap signal will have a characteristic time over which it will fall below the noise floor of air movements having a length scale of order ~one wingspan. this will be on order ~seconds, and will propagate a distance on order ~tens of centimeters before it is lost in background noise.

We now compare this to the wake vortices being shed by a wind-blown tree which is ~3 orders of magnitude bigger than the butterfly. for a tree 50 meters tall, we would be hard-pressed to detect a disturbance above the noise floor of our detection apparatus ~10,000 meters downwind of it.

So if we analyze this situation in terms of signal propagation in a dissipative medium which possesses a noise floor, it is possible to set limits on the length scale of signal propagation- which argues strongly against the butterfly effect.

• Exactly. The public confusion is due to Jurassic Park. Chaos Theory just says the distance (in phase space) between 2 parcels of air (molecules?) that flow over the wings will diverge exponentially: so what?--that doesn't make tornados. Now self-organized-criticallity is a different story. Does that one small step you take trigger the giant avalanche that kills you?--maybe. – JEB Jan 25 '18 at 2:26
• The example you gave about the one small step in the snow reminds me of the straw that broke the camel's back. You can compare the weather system with a waterfall. If you change a very little "piece" of the water upwards (even very small compared with the change in phase space that the different flapping of a butterfly's wings cause, because the weather system is so much bigger than the waterfall), this will have no noticeable effect on the water hitting the bottom. The changing water (in its entirety) on top, in the course of time, will affect the water reaching the ground. – descheleschilder Jan 25 '18 at 9:52
• @JEB, you're mistaken. Chaos Theory says that the trajectories of the whole system in its phase space will diverge, not of tiny parts of it (air molecules, in your example). – stafusa Apr 28 '18 at 19:02

Yes - but only as far as one is willing to believe the mathematical model fits reality.

In the mathematics we can demonstrate the butterfly effect; the sensitivity of particular nonlinear dynamic system models to initial conditions. And we can contrive certain experiments of systems that seem to behave in the context of a butterfly effect.

But even for real physical systems we consider relatively simple, we can never say the model exactly fits the system no matter how hard we try. So there is always remaining uncertainty. If we get good predictions from the model then we say the model is useful, but we can never say the model is the system.

• AFAIK there is a considerable amount of experimental verification of chaos in natural phenomena. Why would you not believe that chaos theory does not fit reality? The hypothesis "as far as one is willing to believe" is true for any scientific statement. – Jorge Leitao Jan 28 '16 at 22:51
• @J.C.Leitão Of course, I believe in chaotic behaviour of systems. But it depends on how the system is made up. Like I said, the weather is a system that consists of many particles. Only if you change a tiny bit the constituents of the system, it will develop after some time into something completely different as would be the case if you didn´t. Besides, the system can not have its own energy source (from within). So in the case of weather you have to change the velocity of all particles in the atmosphere and not just a few surrounding a butterfly. Or think of a river that flows down the hill. – descheleschilder May 26 '16 at 22:53
• The only way you can know for sure if the weather is sensible to a little change of the velocity of the air molecules is to do an experiment with two exact copies of the Earth, the only difference being the change of one butterfly's flaps of the wings. I´m sure that no difference will be seen on the big scale weather, only a bit around the butterfly, but this doesn´t get amplified but will dampen out quickly so nothing of the change will reach the big scale weather. And the butterfly isn´t capable of pushing the development of a storm system over some limit behind which the storm will occur. – descheleschilder May 26 '16 at 23:06

Okay, let's consider if the weather system is chaotic. Can we do that by looking at a very little patch (let's say a few cubic centimeters made up out of air particles around the butterfly) of the weather and vary this collection of air particles by a tiny amount in phase space? The most little patch is to vary just one air particle, which will respond linearly to the little change in phase space if it were isolated, just as two isolated air particles will respond linearly if we change them by a tiny amount in phase space. Three isolated air particles don't react linearly anymore to the little changes in phase space (if they interact with each other by contact forces, somewhat like billiard balls) let alone a number of air particles in the above-mentioned patch.

The patches aren't isolated though. They are part of a much, much bigger system (the weather system). If we change just one air molecule by a tiny amount in phase space, it will initially respond linearly, but by its contact interaction with the other particles, we have a many-particle system, which is clearly non-linear. The changes in the non-varied particles will be less the further they are away from the varied particle (the tiny change in phase space gets distributed over the other particles, and because energy is conserved each particle is affected less and less, the farther away, because most collisions are not head on).

The same will happen if a very little patch (compared to the whole system of which we want to find out if it's chaotic, in this case, the whole weather system) is varied in a tiny way in phase space. Let's consider this patch as a huge pool table, with a huge amount of balls in contact with its surroundings. It's clear that the balls (particles; let's consider the matter in a classical way) show chaotic behavior. But to deduce from this that if the balls on the pool table show chaotic behavior (if isolated) the huge (weather) system is also chaotic if the pool table is not isolated anymore, but becomes part of the whole system, is a different story.

The reasoning is the same. The energy of the tiny changed patch gets the more diluted the farther away you are from the patch. So the butterfly in Brazil can't cause a tornado in Texas. I think Mandelbrot was or not aware of the actual situation, or made up this (bad) example to give an example of chaos, or just not thinking right. To show if a system exhibits chaos you need to consider the whole system and not some part of it.

In determining if a dripping tap exhibits chaos, we also don't vary a very small patch of the system. Instead, we consider the whole system. And it turns out that the rate at which the drops fall determines the system's chaotic behavior. After a certain rate the drops don't fall regularly anymore but fall in an unpredictable (i.e. chaotic) way and I can't see this has something to do with changing the motion through space phase of some particles (and this are much fewer particles, relatively seen, if we compare it with the much bigger weather system) of this system.

And besides, in weather systems, there are no critical tornados, hurricanes, typhoons, which do or don't evolve, depending on tiny changes of little patches of air. Those patches don't have the ability to change a huge developing system

• Threshold behaviors are not the same thing as chaos. Systems that are chaotic exhibit sensitive dependence in a large fraction of their phase space. The system you are describing here exhibits sensitive dependence in a vanishingly small fraction of its phase space. – dmckee Mar 7 '17 at 2:05
• Your argument explains why we cannot observe chaos in linear systems, by arguing with the principle of superposition. However, a turbulent atmosphere and every other chaotic system is a non-linear system, and a quintessential feature of non-linear systems (such as a turbulent atmosphere) is that this principle does not apply. – Wrzlprmft Apr 11 '17 at 19:50
• I changed my answer in reply to your comment. – descheleschilder Apr 12 '17 at 2:55
• Why would the double pendulum not be sensitive to changes so tiny that we cannot experimentally reproduce them, while being obviously sensitive to larger changes? – Wrzlprmft Apr 12 '17 at 8:52
• Because these tiny changes will not affect the whole pendulum and these changes will dissipate quickly into the whole pendulum. So chaotic behavior will occur only if we look at the whole pendulum and not some tiny part of it, just as we have to consider the whole weather system to see if it's chaotic (which it is). To determine if a system is chaotic we can't look at a tiny piece of it. – descheleschilder Mar 21 at 3:51

I think, after reading what´s written above, we must make a distinction between systems that are big and little, like the atmosphere and a single billiard ball.

I´s clear that a billiard ball on a table makes a journey that deviates more and more from the path it would have taken hadn´t you give a slightly different direction in velocity.

Consider now a container with a gas consisting of all the same atoms, and consider these atoms as billiard balls (I think not invoking quantum mechanics is reasonable assumption). In the case of the billiard ball, the system consisted of just óne element (of course the ball consists of a lot of tightly connected particles, but for the ball as a whole they´re not important; how often do you hear to consider atoms as billiard balls?). The trajectory of an atom is surely gonna be different if the direction of this one atom gets a slightly different direction. But this time, there are no walls of a table. It bumps into other atoms who on their turn get also a slightly less velocity direction.

The number of atoms is so big that after a time the different direction in velocity of the first atom won´t be noticeable anymore for most atoms of the gas. So the whole gas-container system won´t change in comparison to a container (of course with the same dimensions) that contains the same gas. And even if we should change the direction of the velocity of each atom, there would be no macroscopic difference concerning the two boxes. Off course, all the atoms velocities would differ, but for the system as a whole, there´s no observable differences. Same pressure, temperature pression, etc.

Now the example of the boxes with gas is rather simple (but of course you can´t keep track of the individual atoms) because the gas is contained. But now imagine a river. If I change the direction of velocity of one water molecule somewhere high in the river, the molecule, of course, changes its path, along with many others (direct or indirect). But these changes dampen out and the river ends up the same as without these little change of one molecule.

But if the velocities of the molecules of the river as a whole (all the water molecules) each get a little different velocity direction, the river sure is gonna end up different down the mountain.It ends always down the mountain, but there are many ways. The difference with the container of gas is very simple: the gas is contained, the river runs wild, without container boundaries.

The same is true for the atmosphere. Change one little part, and there will develop no different weather (maybe some little turbulences around the butterfly). Change all the molecules that make up the atmosphere (conserving the energy), and a completely different weather develops.

I read once that if you connect the butterfly to a very sensitive atomic device (by means of a radio transmitter tied to the butterfly) and if the butterfly makes a slightly different movement so that it gets close enough to the bomb so that it will detonate (gee, how far have we come?), the little change in movements of the butterfly has a very great impact on a large scale. But I think in chaos theory we must consider systems that have no energy load they can trigger, wich in fact, the butterfly also has, but in this case, it´s a very little energy that the butterfly possesses, and she is part of the surroundings, not of the weather itself (the explosion of an atom bomb would also be part of the boundary, but the amount energy involved in a butterfly is negligible). Or take the story of a man who missed his plane, because he júst didn´t run too hard to catch it. Unlike an atom in a box who júst didn´t hit another atom his trajectory amidst the surrounding, won't cause a damping of the things to come (because of his energy resources). He´s pissed of, decides to go back home (he had an important meeting in London, concerning a big money deal), and angry as he is, drives much too fast on the highway, after wich he enters town and still puts the foot on the gas pedal. He doesn't see the woman and child, who both get killed.

Obviously, the last examples are no examples of chaos because parts of the systems have their own internal energy source.

It seems obvious to me that we have to look at the very small difference in the initial conditions of the whole weather system if we want to exhibit the chaotic behavior of the weather. So, for example, we have to give every air molecule an equal difference in its velocity. A little difference in the flapping of the wings of a butterfly will very fast dissipate and after a fairly small distance, this little change in the wing's flapping won't affect the further course of the weather. So this little change in the movements of the wings can certainly not cause the emergence of a hurricane thousands of miles away. We're talking here pure about the weather and not about other changes that a tiny change in the wing's flappings of a butterfly can induce. Imagine sitting in a wind still forest. Does one really think that a puff of air I blow in the air will still be noticeable 100 meters away? And because this wind stillness maps one to one to air in motion, also in a storm my little puff of air won't change the course of the gigantic weather system. To assume this is to assume a little change in the movement of a star will subsequently cause an entirely differently developing Universe (regarding the motion of all the other stars) in the course of time as this little change had not occurred.

• And because this wind stillness maps one to one to air in motion – At this point, you are (once more, IIRC) assuming the superposition principle, which only holds for linear systems. Chaos is mathematically proven to only occur in non-linear systems (such as the weather). — Also, your argument does not seem specific to the weather but can be translated to such systems as the double pendulum, whose chaoticity can easily be demonstrated experimentally. – Wrzlprmft Dec 1 '17 at 15:12
• Maybe, but in considering chaos you have to consider the whole system in question (the whole, global weather system, the whole double pendulum, etc.), and you consider just a very, very tiny part of the whole system. – descheleschilder Dec 2 '17 at 0:56
• you consider just a very, very tiny part of the whole system – Depending on your scope of consider, yes, but what do you deduce from this. Or, to be blunt: So what? – Wrzlprmft Dec 2 '17 at 9:04
• Well, in investigating if the weather system is chaotic you have to consider the whole weather system and not just a little part of it. If you look at the little piece of the system surrounding a butterfly, you don't take into account the major part of the system, which is necessary to see if the system is chaotic (which it is). – descheleschilder Jan 29 '18 at 1:11
• @Wrzlprmft-Let's not trying to tackle this problem with math. Like the butterfly effect, in the double pendulum, we should change a very little patch of the particles (their trajectories in phase space) out of which the double pendulum exists (like changing a little patch of the air particles out of which the weather exists). In proportion to the patch in the weather system, that are very much fewer particles. – descheleschilder May 8 '18 at 22:04

If you want to know if a system, like the weather system, is chaotic, you have to take into account the whole system and not just a little part of it, like the surroundings of a butterfly.

To determine if the weather system is chaotic you have to take into account all the particles that make up the system. You can argue what particles are part of the weather system, but I think we all agree that these particles are the particles that make up the atmosphere of the earth, and maybe we can let out the higher layers of the atmosphere because they hardly have any influence on the lower parts. So to find out if the weather system is chaotic (which it of course is) we have to look at very small differences of the particles that make up the weather system (we should not let all the particles make a random little change in phase space, but let all the particles make the same little changes in phase space) at a certain time and see how the weather evolves. In this case, it is possible that a hurricane can develop where it wouldn't have evolved if we wouldn't have let the little differences taken place.

Of course, this experiment is impossible to perform on earth. Even with two exact copies of the earth. What you can do if we have two exact copies is release a butterfly on each earth. Both butterflies will be flapping their wings differently, after which you can see if a hurricane develops on one of the copies and not on the other copy. I bet you can't see any difference.

Many examples above involve criticallity (like the example of the two pipes that lead to a dam), which isn't an issue in atmospheric conditions. It's just not true that if you change a very small part of the atmosphere (like the flapping of the butterflies wings) a critical behavior develops like the straw that broke the camel's back. If someone knows a counterexample please be free to mention it.

So my conclusion is that a little change in the flapping of the butterflies wings (if the little change in the flapping causes nothing else than a little change in the air's movement surrounding the butterfly) on one side of the earth surely won't lead to a hurricane on the other side of the earth.

P.S. Concerning the sun's radiation, this radiation is evenly distributed over all the particles that make up the weather system, so no critical energies develop. Which is to say, no little change in some part of the system will cause a big release of energy like a hurricane (maybe a little puff of downwards air can break the camels back, which is a completely different situation as the weather system).

## protected by Qmechanic♦Jan 26 '16 at 16:39

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