Another butterfly making tornadoes thread

[BernieM]

Butterfly effect. Much ado about (almost) nothing.

Not a lot has to be said to introduce the subject of course. But once before I addressed whether, in reality, a butterfly flapping its wings could actually create a tornado. My view essentially was that the degree of disturbance of the local air patterns by a butterfly, would actually be absorbed through a hysteresis process, and essentially just increase the overall net energy in the local system rather than causing an organization of the system which would grow to become a tornado. Of course my personal opinion is that Edward Lorenz never truly believed it was a reasonable scenario, simply that he was trying to illustrate that very insignificant forces may come to influence much larger systems in sometimes spectacular ways.

Then I wondered as to whether or not one could prove the probability that a butterfly, could in effect, ever realistically be considered the cause of an organized storm system such as a tornado, and whether one can definitively prove this one way or the other.

To do this, I suppose one would have to figure out how many butterflies there were in the world, as well as moths and other small flying insects of approximate similar size, and how many times they flapped their wings in a day (are the flapping of a moth's wings at night the same value as a butterfly in the day due to a difference in air density when it's cooler during night time?) Then one would have to figure out what the average number of daily tornadoes there are on Earth (including over the ocean where nobody might see them.)

This would give a probability of a butterfly being the cause of a tornado, expressed, for example, as 1 tornado per 1 quadrillion wing flaps!

But obviously if a butterfly could be the cause, other things that disturb air currents would also have to be considered as a potential cause. And herein lies the question.

A 747 jet is flying from L.A. to Hong Kong and it takes 12 hours of flight. How much air was displaced (turbulence produced) and how many butterfly wing flaps would that be equivalent to? But further, and perhaps more important, is whether it is a single molecule that could cause an organization and tilting of an atmospheric condition to favor creation of a tornado, or whether it is the net effect of the whole disturbance.

So does a jet flying from L.A. to Hong Kong have the same probability of creating a tornado as a butterfly flapping its wings, or does it have the flight's total air disturbed, divided by the amount of air disturbed by a butterfly flapping its wings chances of creating a tornado? Is a jet assigned an equal value of probability of a single butterfly wing flap to create a tornado, or is it equal to billions of butterfly wing flaps?

I believe it depends on the model you have for what is the cause of the tornado. Whether it is the act of a single air molecule that can influence an atmospheric condition in the end or not. In a coffee cup, all molecules move randomly so all forces are generally considered to average out to a net zero directional momentum. But there is the one in an incalculable chance that all the molecules will jump in the same direction and the coffee jumps out of the cup! But even in that cup in which everything averages out to a net directional energy of zero, there will be that one molecule moving in one direction that doesn't pair with an opposing force of another molecule moving the opposite direction, giving that cup just 1 molecule of force bias in one particular direction! If it were in the air, would that be a one-molecule case that causes the tornado? Or do we have to wait until all the molecules in a small region per chance just happen to all decide to move the same way?

And finally, is there a requirement for any force to overcome a hysteresis or resistance in the system for it to be considered sufficient to eventually turn into a tornado? That a single molecule of air nor a butterfly could do it, but perhaps a jet could.

I suppose if these questions could be answered (on a theoretical basis anyhow, as I wasn't really figuring anyone would actually go out and count butterflies in the world and count how many times they flap their wings,) one could actually determine if it's possible or not that such a small force actually grows into a large phenomenon. Most of the things I have seen that respond to small fluctuations and create amplifying effects, are inherently unstable systems with some sort of feedback mechanism in them where the feedback amplifies the fluctuation. Easy to do in a computer program or math equation, but I am not convinced it's so easy to do in the physics of the real world.

All seems a lot of work to prove one way or the other, though it might yield some interesting results if solved. Tornado bombs anyone? Or maybe there is some sort of shortcut to solving it. Wave function of the atmosphere or something.

 

[anorlunda]

Your missing the point of chaos and weather prediction. The butterfly is metaphoric speech. But the relevant point of chaos is this.

A system's evolution is chaotic when an arbitrarily small change in initial conditions can result in wildly diverging outcomes. Arbitrarily small goes all the way down to a single atom or a single photon, that could change the outcome of a macro scale chaotic system.

https://en.wikipedia.org/wiki/Chaos_theory

 

[Nugatory]

It won't take much googling to satisfy yourself that a butterfly causing ("causing" isn't really quite the right word - causation is a misleading way of thinking about extreme sensitivity to initial conditions) a hurricane is something of an urban legend. I don't know that I've ever seen a rigorous analysis showing that it's bogus though.

 

[BernieM]

Your missing the point of chaos and weather prediction. The butterfly is metaphoric speech. But the relevant point of chaos is this.

A system's evolution is chaotic when an arbitrarily small change in initial conditions can result in wildly diverging outcomes. Arbitrarily small goes all the way down to a single atom or a single photon, that could change the outcome of a macro scale chaotic system.

https://en.wikipedia.org/wiki/Chaos_theory

Yes, metaphor. I stated that I did not believe Lorenz actually meant that he thought butterflies caused or could really cause a tornado. It was an illustration he was trying to make of how some systems can amplify effects. And I am in agreement with the part about the chaos theory and know how weather prediction small values in accuracy of calculations came to give us the current weather prediction system. But what it posits is a theory that the atmosphere is a feedback system unstable enough that very small changes are capable of creating huge results. If so then one could say that a large event like a jet flying might then create the red spot of Jupiter scale storms, having early introduced far more instability, resulting in far more chaos being generated much faster. Where is the limiting mechanism? What is it that allows a butterfly flapping its wing (or the equivalent in chaos theory) to amplify itself to become a tornado, yet prevents a jet from making the red spot of Jupiter? Total energy in the system? Hysteresis? Atmosphere not dense enough or large enough?

What I was saying is that this happens in an unstable system which feedback feeds further instability. But what about a stable one? And just how unstable or stable is our atmospheric system?

If however one can do some rough calculations on some of what I indicated, one might see that the odds of a butterfly (or very small effect) causing a huge effect, would have a similar certainty in the same way as they prove subatomic particles existence based on statistics. If the probabilities of it happening are too small to be realistic in any stretch of the imagination then one could say that it was not possible and lay the whole theory to rest. Or prove that it is possible and confirm it.

 

[jbriggs444]

resulting in far more chaos being generated much faster

It's not about generating chaos. Its about scenarios with similar initial conditions diverging greatly. It is similarly probable that the butterfly's wings will cancel a tornado that would have appeared over Kansas City rather than resulting in one appearing over Dubuque.

 

[anorlunda]

But such a calculation would not portray what happens with weather. There are an arbitrarily large number of "wing flapping" events, not only initially but also during evolution of the weather.

Chaos and instability are not synonyms. Local weather can be stable or unstable, but the evolution of weather is always chaotic in the sense that no matter how precise your knowledge of initial and subsequent external influences (wing flaps), weather forecasts can not be accurate for more than a few days.

Have you ever tried dynamic modeling of chaotic systems?

 

[BernieM]

I have written some programs a long time back that played with random number and a self modifying algorithm, that modified itself on the value of the calculation using the random number. Never got the effect I was looking for though which was the equivalent of a freely running oscillator whose output would limit the range of frequencies to a reasonable range, that while the values were well within limits the values of the oscillator could vary greatly and as the oscillator approached an assigned arbitrary min/max would limit the range of the oscillator. It was an attempt at artificial intelligence (without going into a lot of detail of how it worked.) But it was essentially chaos theory with limits imposed on it.

 

[anorlunda]

I have written some programs a long time back that played with random number and a self modifying algorithm, that modified itself on the value of the calculation using the random number. Never got the effect I was looking for though which was the equivalent of a freely running oscillator whose output would limit the range of frequencies to a reasonable range, that while the values were well within limits the values of the oscillator could vary greatly and as the oscillator approached an assigned arbitrary min/max would limit the range of the oscillator. It was an attempt at artificial intelligence (without going into a lot of detail of how it worked.) But it was essentially chaos theory with limits imposed on it.

That sounds like a true instability bounded by nonlinearities. We call that a limit cycle. I repeat, unstable is not the same thing as chaos.

 

[BernieM]

It was random. What can I say. I just limited the chaos for my purposes. If I had left it without limits and didn't make it self regulating it would have been truly random and as far as I am aware that's chaos, no?

 

[anorlunda]

No, chaos is not random. Coincidentally, see this post from a post just a few hours ago.

https://www.physicsforums.com/posts/5868629/

 

[davenn]

No, chaos is not random.

ya beat me to responding as such

 

[BernieM]

I can make a random number generator that will by all tests output truly random values, yet in a small area of that sequence, can find multiple areas of pattern in it. Or I can find long sequences which have pattern in it. The program that draws those lines use random number generation to do it, no? All I need to do is find a special case ( a certain point in a long sequence where there is order within the disorder) or add a rule (order) to the random number generation that dictates the line drawing, to govern how the line will be drawn. A rule is the introduction of order. This confuses the issue by allowing random values change the a rule. Nonetheless there is yet rule. Make a program without a rule in it that will generate the same result. It will happen but probably not in the length of time that this universe exists. Rolling 5 die, it's very rare to get 5 of the same kind. More frequent to get 4 of the same kind. Rolling many rolls the rolls test true as being random, yet there are times you can get 3 rolls of 5 die to be the same value consecutively. So is the random number generator flawed? Dice loaded? No. It's the fact that order is a small subset within disorder or chaos, it's just a special case of it. Or would you disagree with this?

 

[davenn]

I can make a random number generator that will by all tests output truly random values, yet in a small area of that sequence, can find multiple areas of pattern in it. Or I can find long sequences which have pattern in it. ...big snip... So is the random number generator flawed? Dice loaded? No. It's the fact that order is a small subset within disorder or chaos, it's just a special case of it. Or would you disagree with this?

did you actually bother to read the wiki link that Anorlunda linked to in post #2 ?

a chaotic system has a certain amount of predictability about it. Randomness doesn't
Again a chaotic system does not equal a random system

BUT a chaotic system will devolve into a seemingly random system after a period of time
That time period depends on the dynamics of the system. Once a chaotic system becomes apparently random, THEN it looses its predictability

 

[BernieM]

"Chaos theory is a branch of mathematics focused on the behavior of dynamical systems that are highly sensitive to initial conditions. "

"Definition of Chaos:
PHYSICS
behavior so unpredictable as to appear random, owing to great sensitivity to small changes in conditions."

... snip...

Again a chaotic system does not equal a random system

BUT a chaotic system will devolve into a seemingly random system after a period of time
That time period depends on the dynamics of the system. Once a chaotic system becomes apparently random, THEN it looses its predictability

But that's not what the definition says. Is it wrong? Behavior so unpredictable ... contradicts what you are saying; that chaos is predictable.
... as to appear random ... to me means that there is no discernible difference between chaos or randomness then.

So at this point I don't know what we are talking about any longer. I am using the definition I learned about chaos and how to tell what's chaos and what's not. Not degrees of chaos. I learned that if there is any discernible order or predictability it is a state of order that is becoming chaotic. Or something to that effect.

 

[davenn]

"Chaos theory is a branch of mathematics focused on the behavior of dynamical systems that are highly sensitive to initial conditions. "

"Definition of Chaos:
PHYSICS
behavior so unpredictable as to appear random, owing to great sensitivity to small changes in conditions."

did you only read the first paragraph ?

But that's not what the definition says. Is it wrong? Behavior so unpredictable ... contradicts what you are saying; that chaos is predictable.
... as to appear random ... to me means that there is no discernible difference between chaos or randomness then.

huh ??!

from wiki

Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems—a response popularly referred to as the butterfly effect—rendering long-term prediction of their behavior impossible in general.[2][3] 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.[5][6] This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:[7]

Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic systems are predictable for a while and then 'appear' to become random.[3] The amount of time that the behavior of a chaotic system can be effectively predicted depends on three things: How much uncertainty can be tolerated in the forecast, how accurately its current state can be measured and a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a few days (unproven); the solar system, 50 million years. In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random.[14]

what part didn't you understand ... and what did I state that was wrong ??

 

[jbriggs444]

I can make a random number generator that will by all tests output truly random values

A deterministic random number generator which generates unbounded output using a finite algorithm which processes a finite input seed will not pass all tests of randomness. In particular, it will not pass a test that compares it to itself.

 

[BernieM]

Well start with this. Computer modeling of a system and the resulting chaos of the results due to rounding errors is not due to the actual system being modeled, but the nature of how processors and computers work. When was the last time you ever found a fractional anything? A fractional photon? A fractional electron? A fractional atom? A fractional amount of energy of any kind? So a true model of the system would not be fractional, fractions are a result of a mathematical process not sufficient to fully and correctly represent the actual system. The weather turns out only one way, though the model may show many possibilities for it to turn out. It's not that the weather is variable and full of unpredictability, it's the computer system and program, the simulation that is unpredictable, and that you can vary the output or result by changing values it processes to achieve the result is no surprise at all.

But you see what is happening here is you wish to debate whether it's chaos or not, random or not, predictable or not. What my point of the post was, isn't there a way that all the debate can be bypassed? Say that there is 174577.5 TW and a butterfly flapping its wings requires .1 mW. And that all the energy from the sun that comes in each day, produces a total of 80000 TW of molecular motion in a day (regardless if it's a molecule in the air or ocean or butterfly flapping its wings or a jet flying or a tree falling) AND that there is on average 20 tornadoes a day on the planet, so divide the total energy input by .1mW and that's the 'daily butterfly equivalent' = 8x10²⁰ DBE's per 20 tornadoes. Which in my book says it's pretty impossible that any single small motion would actually be the cause of a tornado, chaos theory or not.

 

[jbriggs444]

When was the last time you ever found a fractional anything?

Pretty much every time I've ever measured a distance, a duration, a velocity, a force, an acceleration, a pressure, a voltage or a current.

any single small motion would actually be the cause of a tornado, chaos theory or not.

"The" cause of a tornado? As if every event has a single cause? In any case, it's not about how many tornados. It's about which tornados and where. It is not unreasonable that the set of tornados occurring July 4 2020 would be a different with a butterfly wing flap on July 4 2010 than without.

Of course, there is no way to run the experiment. The whole discussion avoids being empty philosophy because there are still ways to reason about the situation.

 

[BernieM]

A deterministic random number generator which generates unbounded output using a finite algorithm which processes a finite input seed will not pass all tests of randomness. In particular, it will not pass a test that compares it to itself.

I'm glad you pointed that out, but what's the purpose of pointing that out? Do you have any input as to whether or not butterflies or effects that small can cause a tornado? Or was it that you just couldn't resist pointing out the error?

Not seeing how that contributes to the thread myself unless you are trying to say I'm stupid or something like that. Was that your intent? Done with thread thanks for clearing things up for me on chaos and all that. Was a very valuable lesson.

 

[sshai45]

Your missing the point of chaos and weather prediction. The butterfly is metaphoric speech. But the relevant point of chaos is this.

A system's evolution is chaotic when an arbitrarily small change in initial conditions can result in wildly diverging outcomes. Arbitrarily small goes all the way down to a single atom or a single photon, that could change the outcome of a macro scale chaotic system.

https://en.wikipedia.org/wiki/Chaos_theory

It won't take much googling to satisfy yourself that a butterfly causing ("causing" isn't really quite the right word - causation is a misleading way of thinking about extreme sensitivity to initial conditions) a hurricane is something of an urban legend. I don't know that I've ever seen a rigorous analysis showing that it's bogus though.

It seems to me here the point is getting missed all throughout this. If an "arbitrarily small change" can indeed result in a wildly divergent outcome, that is, that the atmosphere is a truly chaotic system, then that means that if we consider a hypothetical scenario of two exact Earths (or even Universes, perhaps, given all inputs) at this point, in which in one case a butterfly flapped and one did not, at some suitable time (so many multiples of the Lyapunov characteristic time $\tau = 1/\lambda$), it could be that very well a tornado occurs (or does not occur!) in one that does not in the other. (That this is probably impossible to do is beside the point; it's a thought experiment just like Albert Einstein's ones with the trolley car approaching the speed of light. That no real trolley can do that is beside the point.) And this, I believe, is the true gist of what is being asked regarding this problem - and it comes down to "is the atmosphere truly a chaotic system", that is, does truly an arbitrarily small input 'snowball', or does it not? Not what the "probability" is of any given butterfly forming some tornado, but rather simply are butterfly-scale perturbations relevant to the long-term (many multiples of Lyapunov time constant $\tau$ on the large-scale equations that show the chaos) behavior of the system or not? And this both simultaneously avoids any talk of taking the "butterfly" thing to a too-literal extent and gets at the real physical gist of what is going on behind it. And if it is not truly chaotic, what is the lower bound at which something will have an effect?

As I seem to remember this is very similar to a thread I saw a long time ago on another forum involving a question very much like this, and one poster there was insisting that it is in fact not possible because the butterfly is "too small compared to the scale of the attractor", which I was not sure what that meant and still am not sure - once on a chaotic attractor the whole point is there is no "too small". Arbitrarily small perturbation is sufficient to trigger divergence and yes that would mean right down to the level of a butterfly. However it's possible he was right but was not doing very well at expressing some other concept - namely that there is in fact some sort of "thresholding" effect (and someone else here mentioned some kind of "hysteresis" effect - I'd be curious to hear more about that/see some more sources) below which there is no effect. That is, that the equations that suggest an arbitrarily small perturbation will diverge in terms of the global-scale weather pattern are not valid at all scales, or at least at scales down to those approximately the level of a butterfly if not a reasonable "comfort margin" smaller. And I think that's 100% legit question to ask. Thus, what might be the answer?

 

[BernieM]

SSHAI45, thank you for expressing it better than I did. I got a bit upset that people were going off on tangents about whether I could prove some system is random or not.

At the same time as I was trying to edit my post to correct errors I missed, the PC froze (not viruses I use Linux, it was new software issues,) after I had just recently loaded a new version of Ultimate Linux, so I got pretty frustrated at that point and just said the hell with it.

The issue I have with the chaos vs not chaos arguments was that the definition of chaos is a state of disorder that is indistinguishable from a random system. Order and disorder are dichotomous. Order is absolute by definition and so is chaos. Between the extremes are relative states of disorder (chaos is not a thing, just as cold is not a thing or dark is not a thing, they are simply a lack of heat or a lack of light, chaos then is a lack of order.)

So chaos by definition is when an ordered system no longer is ordered, all order has vanished. Now there are states of 'almost chaotic' in which some order is still observable, and perhaps for physics or practical purposes, one can consider them chaotic states (minimally ordered states,) but that does not mean that they meet the real test to be complete disorder, it's just someone redefining the threshold of what 'absence of order' means.

My question was, that if the God particle can be proven to exist because of statistical tests alone, 5 orders of magnitude being sufficient for proof to be accepted as real, or 7 or whatever arbitrary threshold science decides to set, as to whether one can accept the reality of a particle's existence or not, then why can't the same thing be done to prove whether the butterfly effect is a real effect in the real world in our atmosphere, or if it just exists in theory, that there are other forces at work perhaps, that counter an actual butterfly effect from happening. And if so, how can that be done. One of the issues I raised regarding figuring this out might be whether all molecules motion would be considered equal or if it required a force of X magnitude to have a chance of propagating the proposed effect. I look at it like this: If there are say 10^20 events a day that would classify as a butterfly effect once it was determined what that test would be, so that any other motions could also be taken into account, and there are only 20 tornadoes a day created, then isn't that the same as saying that something like a butterfly has a one in 10^19 chance of being the cause of a tornado? And if I told you that the odds of a particle being real in the data was only 1 in 10^19 power, would you tell me that that particle was real or not real, just a freak of statistics or method or error somewhere in the system?

My point in the whole butterfly effect thing is that a computer model is a linear algorithm generally, and the atmosphere is not linear. Progression of events is parallel in the atmosphere. A quantum computer would be far better for modeling weather systems because it can run simultaneous calculations of all possible outcomes. In a mathematical simulation created to illustrate chaos and order, such as the line being drawn, it's obvious why it works like that, it was designed to work like that. A single value or handful of values dictate the path of the line. Is our atmosphere like that? I argue that its not. What is more likely? That a random event creates an orderly event that is self magnifying, or that the random event creates a random effect without order in it?. Of course it is possible (though small) that the random event can create an orderly system, (I guess any kind of system possible within the parameters provided,) but to the same degree that a random event can create an orderly effect, there is an equal chance that it can also create an order destroying effect. One butterfly starts a tornado and the butterfly next to it cancels it with a tornado destroying wingflap. So the net result is, simply, like the cup of coffee, all the forces equal out, and the coffee stays in the cup, it doesn't jump out. Or a large scale weather system does not form. I am saying that tornadoes are a result of order, not disorder, of huge scale, not small chaotic systems, growing into large ones, via chaos theory. That is what I was referring to when I used the term hysteresis. A delayed reaction in the system that absorbs large variations at small scales. Though they can be generated, that there are other forces that neutralize them before they get too far along in development.

If my question is stupid, just tell me so. And tell me why it's stupid. I'll fix it. Don't rat pack me and nit pick every detail of the post. If it is not a stupid post, then please give a reasonable answer or opinion without getting off on tangents. If you don't want to engage me because it's beneath you, then don't. If you need clarification, ask and I will do my best to clarify it. I am perfectly aware of what I do not know, that's why I am here asking questions. And the last thing I need is for someone make me look like an idiot in a public forum with potentially millions of viewers. Would you appreciate that? I bet not. Then why do you expect me to suffer it?

 

[sophiecentaur]

a computer model is a linear algorithm generally,

"generally" they are far from linear. As soon as there is even just a polynomial involved the model is not linear. The well known Mandlebrot set images are computer generated.

The problem with concepts like the "Butterfly Effect" being used to describe Chaos is that practical situations can only be observed Post Hoc and you cannot say whether or not a specific butterfly had an effect; it is just a popular idea that is more misleading than helpful. Practical experiments are not well behaved and, in the end, it's noise (random) that determines the actual 'input values'. That can only be done mathematically where the input values are discrete and known.
Real situations can be a mixture of Chaotic and non-Chaotic and Random. In particular, the weather can be chaotic or non chaotic over significant lengths of time.