Does the butterfly effect apply in reality?

[schip666!]

... Both staying put forever and magically starting to slide down the cone in a random direction at a random time are solutions to the equations of motion. No disturbance is needed...

I'm sorry, but I don't really understand your argument. I don't see how it would take an infinite amount of time for a pendulum to reach an unstable equlibria, nor why that would be different from your cones. But more specifically I don't get how a system at equilibrium can "magically start[ing] to slide" without any additional energy applied. Unless, for instance, one considers shifting balance due to molecular motion to be no additional energy...in which case it may come down to the scale of the equilibrium at hand.

Most likely I'm using fuzzy logic someplace. But I don't know where...

 

[schip666!]

Ill-Conditioned ...

It was in this more general sense I was using the term. Note the example given concerns nonlinear mathematics.

I expect what you have found refers to the condition number for matrices. Matrices are said to be ill conditioned if their condition number is large. This, of course, is a narrow application of the same idea to a linear system.

Huh, is Trig considered non-linear? It's just the ratios of real values, no exponentials are harmed in the making of those ratios are there? But yes, the matrix application is what started me off down the rabbit hole. Thanks.

 

[schip666!]

The occurence of either instability or chaos in some systems is scale dependant. That is you get a different answer to the same initial conditions depending upon what scale you are working at.

I think perhaps the best non deterministic examples are the self coplouring automata. Squares can be coloured black or white according to scale.

Ok here we go again...

I don't know what "self coplouring automata" -- even presuming that you meant coloring (or colouring in your funny Brit-lish usage) -- are. All 2D Cellular Automata that I know are deterministic, some are just not predictable without actually iterating them directly. By "scale" do you mean the size of the rule's precursor set?

 

[schip666!]

Does this discussion exclude the effect of the Heisenberg uncertainty principle on determining the maximum length of time a perfectly balanced pencil or UP pendulum will remain in that state? This is certainly non-deterministic.

Bob S

Quantum effects are usually considered to be much smaller than the "distrubances" leading to chaotic behaviors. They could be contributors, but are not considered to be necessary conditions.

 

[D H]

I'm sorry, but I don't really understand your argument. I don't see how it would take an infinite amount of time for a pendulum to reach an unstable equlibria, nor why that would be different from your cones.

It's right there in the math. That the period of an ideal pendulum (non-inverted) is well-known to be

\tau = r\sqrt{\frac l g} K \left(\sin\frac{\theta} 2\right)

where l is the length of the (massless) pendulum rod, θ is peak angular displacement of the pendulum, and K is the complete elliptic integral of the first kind. K(x) becomes unbounded as x approaches 1, and since sin(θ/2) approaches 1 as θ approaches π, the period becomes infinite. This means you can give the pendulum bob just the initial velocity so that it will come to rest in an inverted position, but it takes an infinite amount of time to reach that inverted position.

The opposite is true for my one. I assume you played with Hot Wheels when you were a kid, or if you didn't you at least know what they are. Imagine draping the track from the top of a dresser down to the floor. Now imagine giving a car a shove from floor level so it goes partway up the ramp and then comes back down. It does this in finite time. Now imagine doing the same with a point mass and a curve that follows the centerline of the track instead of a car and a track. Now use this curve from the floor up to this critical point to generate my cone by rotating the curve about the vertical axis that passes through the critical point. You will end up with a surface of revolution with a cusp at the critical point.

Now give the point mass the same initial velocity that sent it up to the critical point the first time around. If you do it just right, it will still go straight up the cone, it will still come to rest just at the critical point, and it will still do so in a finite amount of time.

 

[BernieM]

So far all the examples given in this thread demonstrate extremely small examples of the butterfly effect, such as an inverted pendulum, a single point at a critical point on a cone and the way a pencil falls when balanced on it's tip.

Since it was Lorenz himself who framed the buttefly effect in relation to weather systems, I think it's only fair I restate my question in relation to weather patterns and effects.

NASA identified some time back, that the 'weather engine' of the world was the Outback of Australia. That all weather patterns around the world are generated and determined by what happens there because of the huge amount of energy injected into the atmosphere there.

Is anyone here suggesting that I am to believe that a single nearly infinitesimally small influence is capable of significantly influencing a huge amount of energy distributed over a huge area which contains a huge amount of order AND chaos within it, with a nearly infinite number of potential miniscule 'butterfly effects' at work in it? That a single infinitesimal event is capable of determing the outcome of a huge dynamic system that is also interacting with other gigantic atmospheric phenomena as it moves say from Australia to Mongolia and will ultimately manifest in Mongolia in a differnt way than if that one infitesimally small influence at the beginning had not happened?

Or is it more likely that the butterfly effect may greatly influence other miniscule events in an extremely small local area, but that that effect is ultimately lost in the infinitely large sea of chaos and order it is within; in other words that it could create ultimately a micro-vortex in it's locality, but the micro-vortex would quickly be diluted in the surrounding system and quickly disappear, and never truly spawn a tornado or even a little dust devil.

Is it possible that a butterfly efffect of a much larger mangitude is needed to make manifest effects in huge systems? Lorenz mentioned the flaps of millions of butterfly wings, this would be a much larger force, but then if all the butterfly effect wing flaps were all contributing to tornado forming events it would also imply some sort of a larger order that coordinated the butterfly effects and thus would not be random, and therefore possibly be part of and caused by the storm system's order itself.

If a single miniscule effect could manifest a dramatic change in an atmospheric event even where other huge dynamic forces are at play, why is it then that the 'great spot' of Jupiter never ends up at the poles but always exists in a certain region around the equator?

I lack the education clearly that the rest of the particpants in this forum demonstrate in this thread. If anyone desires to re-state my question more eloquently and clearly, please feel free to do so.

 

[DaveC426913]

Is anyone here suggesting that I am to believe that a single nearly infinitesimally small influence is capable of significantly influencing a huge amount of energy distributed over a huge area which contains a huge amount of order AND chaos within it,

Yes.

I know it's counter-intuitive. Here's why:

Let's return to the giant boulder perched in a razor-sharp peak. It takes the slightest touch of a finger (or a butterfly's wing) to determine which way (or if) that mass falls. Tiny input, very large change in outcome. Pretend I am camped at the base of the peak, and that the boulder has a picture of a tornado painted on it. Quite literally, the flap of a butterfly's wing has made the difference betwen whether I (or possibly anyone) experiences a tornado.

What is non-intuitive is that there are systems that are metaphorically a landscape filled with giant boulders perched on razor-sharp peaks. We don't normally expect these kinds of things because gravity is a convergent system; it tends to bring divergent forces to convergent results, such as boulders to the valley floor.

Weather on the other hand is a divergent system. It does not reduce everything to its lowest potetnial energy state; weather is continually metaphorically picking up boulders and balancing them on top of razor sharp peaks. That's what's so fascinating about it.

 

[BernieM]

Your example is a microscopic event blown up to world sized proportions, and isolated from all other surrounding influences. An infinite number of razor sharp peaks exist around your one peak with an equally infinite number of butterflies ready to interact with the boulders; some having boulders balanced on top, others having already fallen, and I believe in the larger picture which way the boulder has fallen/is falling/will fall, would be purely random and evenly distributed. The event or non-event that happened to the camper below is a very localized phenomena and the fact that you personally being one of those campers who experienced a tornado, does not prove a tornado existed or will exist for all other campers below all the other peaks around.

The net effect of ground vibrations caused by boulders rolling downhill and influencing other boulders precariously perched on other peaks, movement of air currents created by the rolling boulders possibly blowing away nearby butterflies, microgravitational effects by the redistribution of the mass of the boulders as they change position (which if your boulders were perched so precariously could also cause nearby boulders to shift and fall), etc, have not been included in your model. Given all these additional interactions, how instrumental is the butterly effect and how much does it really impact the larger picture?

In a mathematical model it is easy to include or exclude anything you like and view the results without the influence of things that you don't want in the model. Actual atmospheric phenomena you can't do that.

 

[D H]

So far all the examples given in this thread demonstrate extremely small examples of the butterfly effect, such as an inverted pendulum, a single point at a critical point on a cone and the way a pencil falls when balanced on it's tip.

You've missed the point, in a couple of ways. There is a wide-spread belief that while quantum mechanics is inherently random, Newtonian mechanics is deterministic. It isn't.

The other point (never made explicitly) is that chaotic behavior can result even in systems that are seemingly simple. The weather is anything but simple. It is the quintessential chaotic system. The underlying equations that describe fluid dynamics, the Navier-Stokes equations, are highly non-linear. The weather is one unstable equilibrium position after another. Couple non-linearity and unstable equilibria and you get chaos.

NASA identified some time back, that the 'weather engine' of the world was the Outback of Australia. That all weather patterns around the world are generated and determined by what happens there because of the huge amount of energy injected into the atmosphere there.

You either misread something or read something that incorrectly reported some statement that came out of NASA. If anything, it is the oceans, and particularly the tropical parts of the Pacific Ocean, that act as the "global heat engine". El Nino and La Nina events have a strong impact on the Australian Outback. Just because weather in the Australian Outback is strongly affected by these event does not mean that the Outback causes these events.

Is anyone here suggesting that I am to believe that a single nearly infinitesimally small influence is capable of significantly influencing a huge amount of energy distributed over a huge area which contains a huge amount of order AND chaos within it, with a nearly infinite number of potential miniscule 'butterfly effects' at work in it? ...

Once again, you have either misread things, or more likely have read some lay article that completely misrepresented things. Some items to note:

• One reason students major in journalism is so they don't have to take any science class except maybe for "Physics for Poets".
• Sensationalism sells newspapers and captures television audiences. Long-winded explanations by scientists don't.

When Lorentz noticed how sensitive the weather models were to initial conditions he initially suspected something was wrong with the models, something along the line of "#@$%! This says a flap of a butterfly's wings in Brazil could cause a tornado in Texas. What's wrong?" Later he came to the realization that the models were essentially right. The weather is incredibly sensitive to initial conditions.

That is not to say that the flap of a butterfly's wings in Brazil does cause a tornado in Texas. While weather models have initial conditions, the weather doesn't. It is a continuously operating system. There is no way to say that a butterfly's wings in Brazil does cause a tornado in Texas. What can be said is that the weather is chaotic.

One consequence is that it is impossible to accurately predict the weather for more than a week or so.

 

[BernieM]

Well the actual article I read said that a space shuttle survey of the planet showed that huge thunderstorms forming in the Outback of Australia moved out over the pacific to the east and shadowed the ocean over huge areas, thereby reducing the total solar input into the ocean in that area, which is where the El Nino/El Nina forms. I have tried to find this article again but so far have not had any luck finding it.

I agree totally that the weather is sensitive to intial conditions and may dramatically change based on the 'miniscule' intial difference or condition; but what I am trying to say is that that 'miniscule initial condition' is a condition affecting a huge area and not an initial condition of a single air molecule. In that the 'miniscule initial condition' that COULD influence a large weather pattern would be something on the order of the difference of temperature of the air molecules in the model, over a significant geographical region of .000000000001 C for example as opposed to .000000000002 C. Although very small, it's magnitude is huge and thus more capable of changing the outcome of the weather pattern than the fact that a single air molecule was 1 million C instead of 10C.

 

[DaveC426913]

Your example is a microscopic event blown up to world sized proportions,

No, I've literally used a butterfly flapping. As for the size of the tornado, you can make the boulder as big as you want.

An infinite number of razor sharp peaks exist around your one peak with an equally infinite number of butterflies ready to interact with the boulders; some having boulders balanced on top, others having already fallen, and I believe in the larger picture which way the boulder has fallen/is falling/will fall, would be purely random and evenly distributed.

The event or non-event that happened to the camper below is a very localized phenomena and the fact that you personally being one of those campers who experienced a tornado, does not prove a tornado existed or will exist for all other campers below all the other peaks around.

But you can literally map the campground at the base of the mountain onto the continent. Tornado sweeps through Alabama. Butterfly beats its wings, tornado does not sweep through Alabama.

The net effect of ground vibrations caused by boulders rolling downhill and influencing other boulders precariously perched on other peaks, movement of air currents created by the rolling boulders possibly blowing away nearby butterflies, microgravitational effects by the redistribution of the mass of the boulders as they change position (which if your boulders were perched so precariously could also cause nearby boulders to shift and fall),

You are describing a classical convergent system. i.e. your assumption is that disturbing another boulder will cause it to fall. i.e. that lots of disturbances reduce the whole system to a lower entropy state. No. In a chaotic system, one boulder falling will just as likely cause another boulder to land on a peak.

etc, have not been included in your model. Given all these additional interactions, how instrumental is the butterly effect and how much does it really impact the larger picture?

What do you mean "larger picture"? All we are demonstrating is that a butterfly flapped its wings and a tornado occurred in Alabama. Roll the process back, butterfly does not flap its wings, no tornado in Alabama.

In a mathematical model it is easy to include or exclude anything you like and view the results without the influence of things that you don't want in the model. Actual atmospheric phenomena you can't do that.

Actual atmospheric phenomonea require you use the right model. Don't use a classical model.

 

[D H]

Butterfly beats its wings, tornado does not sweep through Alabama.

As Studiot not in [post=3037993]post #35[/post], that is a misrepresentation of what Lorentz said. The title of the paper, "Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?", implies that the answer is yes. The body of the paper says, in no uncertain terms, that the answer is "nobody knows". The sensationalist concept that a butterfly flapping its wings in Brazil can cause a tornado in Texas sells newspapers and magazines. That the answer is "nobody knows" sells absolutely nothing.

In a chaotic system, one boulder falling will just as likely cause another boulder to land on a peak.[/B]

No.

Actual atmospheric phenomonea require you use the right model. Don't use a classical model.

Weather models are purely classical.

 

[DaveC426913]

As Studiot not in [post=3037993]post #35[/post], that is a misrepresentation of what Lorentz said. The title of the paper, "Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?", implies that the answer is yes.

I wasn't suggesting it was otherwise. The ponit is merely that chaotic systems can have this property.

No.

Yes. In chaotic systems, small differences can result in widely divergent behaviour. Strange attractors and Julia sets show this visually.

Starting a strange attractor pendulum at point x=1,y=1 may cause it to come to rest at point A,
while starting it at point x=1,y=1.00000000000000000000000000001 may cause it to come to rest at point B, yards away.
Starting it at point x=1,y=1.00000000000000000000000000002 may cause it to come to rest at point A.

Weather models are purely classical.

What they are today is not what this is about.

 

[DaveC426913]

Here is a strange attractor pendulum plot.

If the pendulum is started over any yellow point, it will ultimately end up pointing at the yellow magnet.

ANY yellow point - even that point squished 1 micron wide between red and blue.

So:
I place the pendulum at point x,y, over a yellow point.
I look away, and while I'm looking away, a butterfly wafts by and moves the pendulum by 1/2 micron. It is now on a red point.
When turn back to my rig, and release the bob, it will oscillate and eventually come to rest on the red magnet.

That tiny, tiny difference in initial conditions will cause the pendulum to wind up centimetres from the spot where it would have.

Now, label yellow as Alabama and blue as Kansas - hundreds of miles apart.

Note that the thinness of the lines can be infinitely small - the closer you look, the more divisions you will find. Go look at a Mandelbrot (or any Julia) set to see this. This means that when you scale that Strange Attractor up by 10,000 times (from cm to kms), you still get yellow point squished between red and blue lines, all the way down to the scale at which butterfly wings can move them. This finely-detailed pattern over a wide range of scales is a hallmark of chaotic systems.

So, even tiny tiny tiny perturbations will move it enough to go from one colour to another.




Contrast this with a classical pendulum, where even the largest changes in the starting point of the pendulum make no difference where it comes to rest. This is the convergent system that we are used to.



Conclusion:

I know what you're thinking: there are uncountable butterflies; they will all cancel out.

No one is saying that a butterfly causes a tornado. What we're saying is that tiny perturbations (it could be millions, it could be merely one) cause the result to be unpredictable.

Yes, there may be millions, but unpredictable is unpredictable. With the strange pendulum, you CANNOT count on it landing on the colour you choose - you can't count on it doing so when there were a milllion butterflies, you STILL cannot count on it if you try to eliminate ALL butterflies and miss just one.

 

[schip666!]

Sorry, I can't help myself...

Now give the point mass the same initial velocity that sent it up to the critical point the first time around. If you do it just right, it will still go straight up the cone, it will still come to rest just at the critical point, and it will still do so in a finite amount of time.

OK, I think I understand your argument for "infinite time" needed to get a pendulum bob into the UP equilibrium condition, even though there may be a bit of Zeno involved. However I still don't see how it would be any different in your Hot Wheels/cone scenario, aside from the energy decay function not necessarily being a sine.

But for this thread, I'm not that interested in how it got there, but more in what happens when/how the equilibrium is disturbed. It sounded from your initial postings that you were positing that it "just collapses" without any cause, which I have a hard time swallowing. The impetus for the collapse is the butterfly effect...some tiny little "random" energy vector which is amplified by the dynamics.

 

[BernieM]

I agree with your point that where a tornado may become manifest, even whether a tornado does manifest itself or not at all, may be caused by minute effects of a butterfly flapping it's wings a month ago in a far away place.

I do disagree with the fact that the butterfly effect can somehow cause to come into existence the required energy to create the tornado, and without that energy, no matter how much a butterfly flaps its wings, it will not create sufficient energy to create a huge storm system.

However, having said that, the United States is the tornado capital of the world ... and the only place a huge migration of butterflies, monarchs, fly over us each year. I wonder if there is some connection there ... maybe millions of butterflies CAN create enough energy to make tornados =o After all Lorenz DID say millions ...

Joking aside however, the butterfly effect is basically a chaotic and random event of small magnitude, and given any huge system of chatoic events, it is likely evenly distributed with random and chaotic events, some of those events, just as likely to counter the existence of a tornado as create one, or create some other random effect on the storm system. So I think that overall there is no bias in the system beyond a very local region where each butterfly effect is observed. Anyone for a hurricane spawning over Ohio? That would also be a possiblity if the resulting location and type of atmospheric event was solely tied to a random variation induced by a butterfly flapping it's wings. So I think the fact that we have not SO FAR ever observed a hurricane spawning over land far away from the ocean, empirically proves that such dramatic weather events are tied to more powerful forces with a considerable degree of order and energy in them.

 

[schip666!]

...

That is not to say that the flap of a butterfly's wings in Brazil does cause a tornado in Texas. While weather models have initial conditions, the weather doesn't. It is a continuously operating system. There is no way to say that a butterfly's wings in Brazil does cause a tornado in Texas. What can be said is that the weather is chaotic.

This was one of my little enlightenments... One always hears Sensitive Dependence on Initial Conditions, meaning the starting point of some "experiment". However in (so called) reality one can leave out the Initial...in chaotic regimes it's always sensitive to un-measureables. Even if the equations of motion are absolutely determined, it's still unpredictable. This, for me, is the wiggle-out from classical determinism.

 

[pallidin]

The flapping of a lone butterfly's wings in NO WAY substantively effects major atmospheric events.
Period.

 

[D H]

OK, I think I understand your argument for "infinite time" needed to get a pendulum bob into the UP equilibrium condition, even though there may be a bit of Zeno involved. However I still don't see how it would be any different in your Hot Wheels/cone scenario, aside from the energy decay function not necessarily being a sine.

Zeno is not needed. Zeno did not know calculus and did not know about elliptic integrals. The period of a pendulum is easily derived from the calculus-based equations of motion (once you know about elliptic integrals, that is). The period of a pendulum is approximately constant for For small amplitudes only. As the amplitude increases, so does the period. The period becomes unbounded as the amplitude approaches pi. This is not the case for my cone, or for Norton's dome (googled that phrase), or for Painlevé's conjecture.

This was a bit of a side-track, started by me. The point was to demonstrate that Newtonian mechanics is not as predictable and deterministic as people like to think.


So, back on topic: A pendulum can make for a very good example of chaotic behavior. Just hang one pendulum from the bottom of another. Google "double pendulum". The chaos arises because the underlying differential equations are coupled. If a system as simple as this exhibits chaotic behavior, what do you think the weather is going to do?

 

[Buckleymanor]

The flapping of a lone butterfly's wings in NO WAY substantively effects major atmospheric events.
Period.

It is not possible to say that this is allways the case.
Something allways effects major atmospheric events and "sometimes" this might be a lone butterfly.
Doubt it's very often.

 

[pallidin]

A double pendulum is case specific AND isolated.

This DOES NOT occur with regards to butterfly's wing movements affecting the creation or alteration of a major atmospheric event.

 

[DaveC426913]

I do disagree with the fact that the butterfly effect can somehow cause to come into existence the required energy to create the tornado, and without that energy, no matter how much a butterfly flaps its wings, it will not create sufficient energy to create a huge storm system.

Correct. No one is claiming that.

the butterfly effect is basically a chaotic and random event of small magnitude, and given any huge system of chatoic events, it is likely evenly distributed with random and chaotic events, some of those events, just as likely to counter the existence of a tornado as create one, or create some other random effect on the storm system. So I think that overall there is no bias in the system beyond a very local region where each butterfly effect is observed.

Correct. We're not talking about biasing the world away from or towards some thing, such as tornados.

The point is, systems such as weather become unpredictable as to whether any given day will result in a tornado - because tiny perturbations affect it. And a tiny event such as the butterfly is enough of a perturbation to introduce unpredictability if introduced early enough. This is not the same thing as causing a major event.


Review the magnetic pendulum. The repositioning of the pendulum by a micron has nowhere near enough energy to cause the pendulum to move. In chaotic systems, it doesn't need to. It only has to move a micron. The pendulum goes to a different magnet; the boulder falls a different way, the tornado passes over land instead of water, and dies.

The flapping of a lone butterfly's wings in NO WAY substantively effects major atmospheric events.
Period.

This DOES NOT occur with regards to butterfly's wing movements affecting the creation or alteration of a major atmospheric event.

Sorry, Unilateral Unsubstaniated Declarations is two doors down, next to Priesthood and Status Quo. This is the Science Forum. Move along.

 

[Studiot]

Sorry, Unilateral Unsubstaniated Declarations is two doors down, next to Priesthood and Status Quo. This is the Science Forum. Move along.

As Studiot not in post #35

Not quite sure if this is a yes or no?

However I've been thinking again about what I said in post#7 about the horseshoe nail.
This is indeed a good example of effect amplification, as posted by schip. What's more the audit trail is trackable or deterministic.

It is also true that a single horseshoe nail doesn't, by itself, possesses the power to loose or gain a kingdom. And that the loss of most horseshoe nails will not result in this.

I do disagree with the fact that the butterfly effect can somehow cause to come into existence the required energy to create the tornado, and without that energy, no matter how much a butterfly flaps its wings, it will not create sufficient energy to create a huge storm system.

Of course it doesn't have enough energy, but do you meet every experience in life head on? The energy is already in the system.
Even in deterministic systems we have the principle of amplification (archimedes once said give me a long enough lever and a fulcrum and I will move the world)

Control theory and (as I have already mentioned ) catastrophe theory are both about the application of small energies to affect larger ones

I place the pendulum at point x,y, over a yellow point.
I look away, and while I'm looking away, a butterfly wafts by and moves the pendulum by 1/2 micron. It is now on a red point

That is supposing you can determine the colour of an infinitesimal point x,y.
Some Chaotic systems do not have an explicit formula for this. The only way we can establish the colour is by chosing a starting point of finite size and dividing.
Each time we divide we get a number of smaller points of each colour.

This is also what I was referring to in the cellular automata comment, but I can't locate the reference at the moment.
Perhaps DH can help?

Saying that the outcome is indeterminate is not due to a chance small deflection or disturbance, but to the fact that we don't, and can never know the final colour of the starting point.

This is the scale factor at work.

 

[D H]

Locked pending moderation.

Edit:
Unlocked.

 

[D H]

Mentor rant on:

This thread is getting as bad as some in Politics and World Affairs. To all involved: Cease and desist with the use of fallacious and non-scientific reasoning.[/color]

'Nuff said, I hope.

----------------------------------------------------------------------------------------------------------------

Time for a recap: The term "butterfly effect" arises from Edward Lorenz' 1972 talk to the 139th meeting American Association for the Advancement of Science (link: http://eapsweb.mit.edu/research/Lorenz/Butterfly_1972.pdf). The talk had the rather sensationalist title "Predicability: Does the flap of a butterfly's wing in Brazil set of a tornado in Texas?" The very first sentence in the talk:

Lest I appear frivolous in even posing the title question, let alone suggesting that it might have an affirmative answer, let me try to place it in context by offering two propositions.​

Later in the talk he clarifies the question raised in the title:

In more technical language, is the behavior of the atmosphere unstable with respect to perturbations of small amplitude?​

This is the key issue raised in the talk: What is the sensitivity, if any, of weather phenomena such as tornados to extremely small-scale disturbances such as flaps of butterfly wings?

It perhaps would have been better to title the paper using the word sensitivity rather than cause. Or perhaps not. The title did a great job of drawing attention to the topic and does give an incredible visualization of the nature of the problem. There is a very strong urge to come up with an eye-catching title or to give a memorable presentation; I certainly am guilty of feeling and occasionally succumbing to that urge. Sensationalism sells, after all.


The immediate cause of a tornados is fairly well known, enough so that warnings of the potential for severe weather are now given a day or more in advance of the event. That's quite a leap from 60 years ago, when the Weather Bureau forbade the use of the word "tornado" in weather forecasts.

The flap of a butterfly's wing in Brazil of course has absolutely nothing to do with this immediate cause. The question remains, what caused the immediate cause of some tornado in Texas? If we chase events back far enough (and we cannot do that yet), would it come down to whether a butterfly in Brazil did or did not flap its wings? We don't know, yet, and it is hard to say whether we ever will. There is no way to prove this conjecture because we can't go back in time, kill the butterfly, and see the alternate timeline that plays out. We can't simulate it either. Our weather models simply do not have that kind of small scale detail.

Another issue here is that sensitivity is not really the same as causation. Lorenz did make this distinction in the body of his talk. That is one of the downsides of a sensationalistic title. Everybody remembers the title. Very few remember or even know the details behind the sensationalistic title.

Yet another issue is scale. The cold front that triggers tornados is a medium scale event to a meteorologist. The tornado itself is a small scale event. The smallest events presently of concern to meteorologists are microscale events, things that happen over the course of a few seconds to minutes, and over the space of tens to hundreds of meters. The flap of a butterfly's wings is orders of magnitude smaller in time, space, and energy than these microscale events. Whether the weather is sensitive to sub-microscale events is an open question.

 

[dkotschessaa]

The problem seems to be treating the butterfly effect as if Lorenz was proposing a law of nature, when in fact it was only meant to raise certain questions, not necessarily answer them.

 

[BernieM]

As I see it the butterfly effect is in essence similar to or in fact the same as cumulative error in a dynamic system. The thing is that in complex systems such as weather, cumulative error is not happening in only one place at one time, rather, an ongoing continuous process at an infinite number of places in the system.

This then becomes the straw that broke the camel's back problem, in that the cause of any specific event was not caused by an individual straw, but the effect of all the straws together, and so any effect of a butterfly can never be fully the cause of any effect later in time.

 

[DaveC426913]

As I see it the butterfly effect is in essence similar to or in fact the same as cumulative error in a dynamic system. The thing is that in complex systems such as weather, cumulative error is not happening in only one place at one time, rather, an ongoing continuous process at an infinite number of places in the system.

This then becomes the straw that broke the camel's back problem, in that the cause of any specific event was not caused by an individual straw, but the effect of all the straws together, and so any effect of a butterfly can never be fully the cause of any effect later in time.

It's more than cumulative error. A car with poorly-tuned steering will drift off course. That's cumulative.

On a bridge with no railings, a car that deviates from the bridge will fall in the drink. That's wide divergence from a course. That's what we're talking about.

 

[mrspeedybob]

I cannot prove whether a butterfly flapping it's wing has ever or will ever cause a major difference in weather. I'm pretty sure no one can. What I can prove however is that a slight change in initial conditions can cause a huge difference in outcome.

On October 31, 1998 I made a decision between an orange soda and a Mountain Dew. I chose the mountain Dew. As a result of the caffeine in that beverage I wasn't as sleepy as I would have been, so when I was invited to a Halloween party I decided to go. At that party I met the woman who later became my wife. The whole course of my life was irreversibly altered by the choice between orange soda and mountain dew.

This does not mean that every choice of beverage has drastic consequences or that drinking Mountain Dew will lead you to true love. It does mean that there are so many unknown variables in the world that the final outcome of any particular decision is unknowable.

It is entirely possible that the flapping of a butterflies wing could lead to a hurricane.

Certainly. A butterfly's wings cannot result in all the atmosphere leaving the planet. But it can result in a tornado.

I beg to differ.
Suppose that tornado kills the person who would otherwise have invented a super-weapon which would have protected us from the invading aliens who come to steal out atmosphere.

It's a stretch for sure but of the billions of butterfly wing-flaps how many cause a tornado. I would submit that for a given cause there is an inverse relationship between the magnitude of a given outcome and the probability of that outcome so that no outcome is impossible, but some are highly improbable.

 

[Pythagorean]

DH:

are there any other well known systems in nature that have such discontinuities?

Also, I'm curious if there's any insight into quantum chaos from this indeterminate system?

Lastly, what do you think of this paper?
http://www.nature.com/nature/journal/v412/n6848/full/412712a0.html

 

[Claude Bile]

"Butterfly effect" is a poetic expression. One should not get too carried away with the poetic aspect of the expression to the detriment of the scientific content it is intended to convey. I can't help but think that people debating whether butterflies actually cause tornados (or some other variety of weather phenomenon) are missing the point somewhat.

Some points raised in this thread bear repeating. The "butterfly effect" was coined to describe the hyper-sensitivity of some categories of physical models with input parameters. Note that it is the models that were demonstrated to possesses this behaviour, not the physical systems themselves, and this is what made the discovery remarkable. Hyper-sensitivity to initial conditions is a fundamental property of the model itself and is NOT due to random imperfections and unpredictable elements that the real-world throws up, i.e. even when parameters are known/computed to infinite precision, the hyper-sensitivity to initial conditions persists.

The remarkable thing in my eyes is that these principles extend to a massive variety of systems, from chaotic lasers, to weather, to the motion of planets through the solar system. That is why limiting discussion to butterflies and hurricanes is somewhat limiting in my view.

Claude.

 

[willem2]

It's a stretch for sure but of the billions of butterfly wing-flaps how many cause a tornado. I would submit that for a given cause there is an inverse relationship between the magnitude of a given outcome and the probability of that outcome so that no outcome is impossible, but some are highly improbable.

All the butterfly flaps cause a tornado, they even cause many tornado's.
If you would compare 2 worlds with the butterfly flap as the only difference and you look
at the weather after a few months, The weather will be completely different. Any place that had a tornado in the first world is very likely to not have one in the second (because tornado's are rare), so the single flap causes every weather phenomenon after few months. (The butterfly may need some more time to case an El Nino)
Larger inputs do not have a larger probability to cause something, they can just cause
something sooner.

This certainly is the case for weather models which go completely out of sync after about two weeks for the smallest variation in their input. I don't see why this kind of amplification wouldn't work at smaller scales.
phenomena like turbulence and draining bath-tub vortices (wether caused by the coriolis force or not) can amplify very small causes to larger cooridinated movement of air or water)

 

[Ken Natton]

The trouble with a fast moving thread, as this one was for a while, is that you can go away to formulate some ideas of something you want to say, and come back to find that it has moved on a long way – quite apart from the basic problem of trying to track everything that has been said. However, the moderator intervention that has taken place leaves me feeling that I cannot exactly be going to derail the thread, though I’m not sure if what I have to say is of any consequence to the discussion.

While I do understand that the original post was very specifically about the butterfly effect, which is clearly a point about weather systems, and DH has given us the precise provenance of that term, as others have pointed out, the term does refer to chaos theory, and according to the account that I read, the origins of chaos theory were not in modelling weather systems, but in population modelling. Weather system modelling is just one of the other fields to which chaos theory has subsequently been applied. Political voting patterns is another – apparently Al Gore invested a great deal of effort in the study of chaos theory.

In any case, the main point that I wanted to make, for those prepared to doubt chaos theory, is that, whereas with so many important theories in physics which we have no choice but to take on trust – its not so easy to build a particle accelerator in your attic – chaos theory is something you can try out for yourself with nothing more difficult to access than a computer spread sheet. Chaos theory is, essentially, just a mathematical formula, and not a particularly complicated one, though it is an iterative one, which simply means that one of the parameters is the previous iteration’s result. In the case of populations, each iteration is of course a new generation, and one of the parameters is the current population (or more accurately, the current population as a proportion of the population capacity of the environment). The origins, as I understand it, were two populations scientists with widely different views of population modelling. One was observing populations that, from a small start, would grow steadily, until they found an equilibrium that, provided external circumstances didn’t change, could be maintained pretty much indefinitely. The other was observing populations that would oscillate wildly, following repeated exponential explosions with near total collapses. The breakthrough came when these two models were shown to both be special cases of a more fundamental model, and they key change was the addition of the iterative element in the formula.

As a control engineer, the connection with the control algorithm known as ‘PID’ is quite obvious, and any control engineer of any experience knows well enough how tiny injudicious changes to the tuning parameters can turn a previously stable control system into a violently oscillating one. So for me, it is not so much of a stretch to conceive that the flap of a butterfly’s wing could be an earlier event in a sequence of which a tornado is a later event.

 

[Studiot]

are there any other well known systems in nature that have such discontinuities?

Two real world examples, one natural, one man made.

1) Go to Switzerland in March, walk into the sowfields near the mountain sides. Start shouting and sooner or later you will trigger an avalance.

Does the ratio of the energy in your shout to the energy of the avalance compare with the ratio of the energy of a butterfly flap to the energy of a tornado?


2) Obtain a length of detcord and wire it to 40 kilos of Semtex. Supply an electrical signal from a battery.
Do this in a safe place not near Guantano Bay.

Does the ratio of the electrical energy to your det cord to the energy of the explosion compare with the ratio of the energy of a butterfly flap to the energy of a tornado?

 

[Pythagorean]

Two real world examples, one natural, one man made.

1) Go to Switzerland in March, walk into the sowfields near the mountain sides. Start shouting and sooner or later you will trigger an avalance.

Does the ratio of the energy in your shout to the energy of the avalance compare with the ratio of the energy of a butterfly flap to the energy of a tornado?


2) Obtain a length of detcord and wire it to 40 kilos of Semtex. Supply an electrical signal from a battery.
Do this in a safe place not near Guantano Bay.

Does the ratio of the electrical energy to your det cord to the energy of the explosion compare with the ratio of the energy of a butterfly flap to the energy of a tornado?

You seem to be talking about the butterfly effect in general. If so, that is not what I refer to. I was talking about DH's non-deterministic system.

 

[D H]

All the butterfly flaps cause a tornado, they even cause many tornado's.

This is wrong. First off, you are conflating sensitivity and causation. Secondly, you are stating this as as if it were fact. The fact is, we do not yet know the answer to the question raised by Lorentz way back in 1972. Meteorologists do not yet have the tools to even begin answering that question.

Regarding the imbroglio over Lorentz' question, Claude put it very nicely (emphasis mine),

"Butterfly effect" is a poetic expression. One should not get too carried away with the poetic aspect of the expression to the detriment of the scientific content it is intended to convey. I can't help but think that people debating whether butterflies actually cause tornados (or some other variety of weather phenomenon) are missing the point somewhat.

Once again, very nicely put. A lot of the moderation in this thread resulted from people taking Lorentz question far too seriously and being far too adamant in expressing their opinions.

While I do understand that the original post was very specifically about the butterfly effect, which is clearly a point about weather systems, and DH has given us the precise provenance of that term, as others have pointed out, the term does refer to chaos theory, and according to the account that I read, the origins of chaos theory were not in modelling weather systems, but in population modelling.

Robert May did most of his work in the early 1970s and published his seminal paper on the logistic map in 1976. Lorentz, however, preceded May by more than a decade. Modern chaos theory pretty much started with Lorentz.

If you dig deeper, you will find that neither Lorentz nor Mays can truly be called the "father of chaos theory". KAM theory and ergodicity theory were developed well before either Lorentz found that weather is chaotic and Mays found that populations can be.

 

[D H]

We're getting off-topic here, but it is my fault.

are there any other well known systems in nature that have such discontinuities?

Well, there's Painlevé conjecture, which has since been answered in the positive. Painlevé proved that the only singularities in the three body problem involve collisions. He raised the question whether non-collision singularities can arise in the n-body problem, with n>3. In response to this question, Von Zeipel quickly proved that singularities in the (Newtonian) n-body problem will arise only in the case of collisions or in the case of objects going off to infinity in finite time.

And yep, there are configurations of the Newtonian n-body problem that result in objects going off to infinity in finite time. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.146.2656&rep=rep1&type=pdf

 

[Ken Natton]

Robert May did most of his work in the early 1970s and published his seminal paper on the logistic map in 1976. Lorentz, however, preceded May by more than a decade. Modern chaos theory pretty much started with Lorentz.

I confess that I had forgotten, but when I returned to my source I found that the author of the essay I had read is indeed Robert May. The populations scientists with the differing opinions of what drove population change he referred to were Charles Birch and John Nicholson. May does actually give quite a detailed account of the development of the science of ecology, under which population studies falls, and contained within his essay is the following statement:

‘Modern chaos theory actually began with a set of equations relating to weather forecasting, published in 1963… The equations were the work of the great meteorologist Edward Lorenz at the Massachusetts Institute of Technology.’​

Clearly that was something that didn’t lodge itself in my memory on first reading!

May also describes how he developed his ideas in conjunction with Jim Yorke, who had himself done work on the logistics map with Tien-Yien Li.

 

[BernieM]

I think a few things have to be kept in mind when dealing with the butterfly effect in real world situations such as complex atmospheric phenomena:

In all complex dynamic systems such as the weather, any particular event will have a nearly infinite number of independent forces in play, each if you tracked back as far back as you wanted, could probably be tied to a small event such as a butterfly flapping it's wings, an acorn falling from a tree or a rock rolling down a hill, so a tornado occurring would be connected to an infinite number of butterfly effects, and it would be impossible to determine which particular butterfly effect caused the tornado. In fact I would state that the effects of all of them had to be in play for the tornado to spawn, and that no individual butterfly effect is in fact the ultimate cause of the tornado.

The guy who met his wife at a Halloween party could equally state that the reason he met her was caused by the fact that the store wasn't out of the soda that he ultimately claims casued him to meet her, that if the store had been out of the soda, he wouldn't have been presented with the choice and might possibly not have met his wife. Would he have gone to the Halloween party anyhow regardless of the soda he chose?

In a weather prediction model like Lorentz was using, variables would have been tied to things such as atmospheric pressure, wind speed, water vapor content, etc., and these variables would apply to large areas and volumes of atmosphere, not cubic centimeter resolution values, so there really isn't any representation in his model for the impact of antyhing anywhere near as small as a real life butterfly flapping his wings. That a value in his model with a difference of .00000001 may spawn a tornado but that value would be many orders of magnitude larger than the impact of a butterfly. So I would have to believe that Lorentz was never implying in reality that a butterfly would in fact have an impact in the weather.

I think the proof of the impact of the butterfly effect is limited to simple physics and mathematical models and computer models where infiinitely many things are not at play.

 

[pallidin]

So I would have to believe that Lorentz was never implying in reality that a butterfly would in fact have an impact in the weather.

That's exactly my take on this. Just my opinion...