Does the butterfly effect apply in reality?

[pallidin]

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

With all due respect I have a problem with that(not saying I'm right, though)
The butterfly effect supposes that most anything, such as a pine cone drooping from a tree, is substantively contributive to the formation of a tornado.

 

[D H]

Lorenz did not say that the flap of a butterfly's wings can cause a tornado.

Well everyone knows that you can judge a book by its cover, and the cover of this book (or this paper) specifically asked "Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?" That his answer was not "yes" was irrelevant. That his answer was essentially "no way to tell" -- well that doesn't sell news articles. The title of his article, and pretending that the answer is "yes" -- now that does sell news articles.

 

[DaveC426913]

With all due respect I have a problem with that(not saying I'm right, though)
The butterfly effect supposes that most anything, such as a pine cone drooping from a tree, is substantively contributive to the formation of a tornado.

Not sure what you're saying. Yes. A pinecone can do the same thing.

But let's make sure we understand - it's not the energy of the butterly wing or the pinecone contributes to the tornado. That's not it at all.

It's that the Gigajoules of energy that makes a tornado can be easily redirected if nudged early enough.

A house-sized boulder perched on a peak can have massive amount of energy redirected in the direction of choice with just the touch of a finger.

The key to chaotic systems is that they diverge rapidly. The boulder, once off balance by a millimeter, will rapidly gather momentum, resulting in a force far, far greater than that contained in my finger.

Contrast with the common idea of deflecting an asteroid. This does not diverge rapidly; divergent forces are balanced by convergent forces, so that a mere butterly's wing flap on the asteroid will not cause it to diverge wildly. This is not a chaotic system.

 

[pallidin]

It's that the Gigajoules of energy that makes a tornado can be easily redirected if nudged early enough.

So, the flaps of a butterfly or of the dropping of a pine-cone is physically sufficient to "redirect" a major atmospheric event?
I don't see it. This is by no means the same as a chaotic pendulum.

 

[DaveC426913]

So, the flaps of a butterfly or of the dropping of a pine-cone is physically sufficient to "redirect" a major atmospheric event?

No, it's sufficient to deflect a microscopic eddy that was waffling on the verge of going left instead of right, to go right, which can deflect a tiny puff of air, which was on the verge of going right instead of left, to go left, which can deflect a tiny breeze that was waffling on the verge of going left instead of right, to go right...

... a strong gust which was on the verge of going right to go left...

You see, all these things are highly divergent. They are always right on the edge of going a slightly different way. It multiplies exponentially.

We are not used to highly divergent systems. We are used to highly convergent systems.

We are not used landscapes where pebbles are perched precariously on top of other pebbles, ready to fall and knock over small rocks perched precariously on top of other small rocks, ready to fall and knock over large rocks perched precariously on top of other large rocks, ready to fall and knock over large boulders perched precariously on top of other large boulders.

But this is the world of chaos. And it's highly non-intuitive.

 

[schip666!]

Since your "non-random" selection of quotes consists mostly of my assertions...

Not so fast. Non-deterministic problems can arise even in good old Newtonian mechanics. As far as I can tell, these non-deterministic situations are a space of measure zero (and hence will "never happen"). However, these non-derministic situations have a common thread: They are centered on unstable equilibria or unstable steady-state solutions. Points in phase space close to these singular points might be deterministic, but the solutions are incredibly divergent due to the proximity to the non-deterministic points.

Perhaps we are not defining deterministic the same. I don't agree that there are non-deterministic problems in classical physics (nor to push the point, at all actually). I mean that -- in theory -- systems are absolutely predictable, our (mathematical) model can exactly describe the behavior, modulo the effects of electrons on Jupiter and such. There are certainly systems that are so Complicated that they are effectively un-predictable, but Complex systems are a different beast. So called chaos occurs when the effects of un-measurably small perturbations are amplified by the dynamics of the system. Back to pendulums...the coupled pendulums' equations of motion are (fairly) easily derived and are therefore deterministic in my usage, but certain regions are not solvable over the long term due to our "Butterfly Effect". And even my single pendulum in the UP equilibrium is not predictable due to those Jupiterian electrons -- it is however still subject to causality, and thus deterministic in my sense.

As far as I know, chaos is a feature of non-linear systems only, not linear systems. Given that the Navier–Stokes equations are non-linear, whether linear systems are subject to chaos is irrelevant regarding the weather. Weather is the canonical non-linear system.

I was asking about "Ill Conditioned" which, I think, studiot brought up at the beginning of this thread. I had not heard the term, and when I looked it up I found reference to it's being a feature of Systems of Linear Equations. So I wonder if it is different from the feature of non-linear systems we are interested in here. I haven't gotten an answer yet though.

That's not fair. Is cosmology past the Carl Sagan "billions and billions" phase? Is forensics science past the Sherlock Holmes / CSI / Law & Order / NCIS / ... phase? How science is portrayed to the public and how it works in reality are very different things.

Yup, that's why, even before the Goldblum-Chaos-Theorist dig that I referenced, the real chaos folks started using Non-Linear Dynamical Systems as their nom-de-plume. I think this may finally be diffusing into the general populace as well.

 

[Studiot]

perturbations

What is a pertubation in this context?
I'll swop you for a discussion of ill-conditioned.

 

[D H]

Perhaps we are not defining deterministic the same. I don't agree that there are non-deterministic problems in classical physics (nor to push the point, at all actually). I mean that -- in theory -- systems are absolutely predictable, our (mathematical) model can exactly describe the behavior, modulo the effects of electrons on Jupiter and such.

Here is a simple classical system that exhibits nondeterministic and acausal behavior.

Imagine a flat, frictionless planar surface with a uniform gravity field normal to the plane. Next attach a frictionless cone-like surface to the plane. The base of this object is tangent to the plane at the points of contact. The peak of the object is a cusp. This cusp is obviously an unstable equilibrium position. Now place an point mass at rest on the peak of the object. Because the point mass is at rest, it will stay that way forever, right? That is after all the definition of an unstable equilibrium point.

That the point mass stays at rest forever is indeed one solution to the equations of motion. There are uncountably many other solutions. The point mass can stay at the peak for some random time T but at time T starts sliding down the cone-like surface in some random direction.

Another way to look at it: Start with the point mass at rest on the plane. If you flick that point mass just right, it will slide straight up the cone-like object and come to rest right at the peak. Once started in motion, the only forces acting on the object are gravity and the normal force, both of which are conservative forces. That means the behavior should be time-reversible. Except it is not. It might come right back to where it started, or it might just hang at the peak for a while and then come down in some random direction.

One objection is that the set of phase states that lead to this weird behavior is a set of measure zero, so it isn't real. On the other hand, the set of phase states that result in the point mass coming to rest within some (small) neighborhood of the peak does have a non-zero measure. Change some aspect of the state of one of these almost-bizarre states and you will get a huge change in behavior.

Now imagine a landscape peppered with these cone-like objects: Unstable equilibria in every direction, as far as the eye can see.

 

[schip666!]

What is a pertubation in this context?
I'll swop you for a discussion of ill-conditioned.

I mean a slight change in initial conditions. Back to that nice butteryfly diagram. There are points on the map that are infinitesimally close to each other which have highly divergent trajectories. One point's orbit may stay on the same lobe, where the second jumps to the opposite lobe. My belief is that the difference between the starting points is not measurable in any practical sense. In linear, or linearized systems, this difference would not appreciably affect the trajectories.

Ill conditioned? It sounded like exactly the above, but the harping on "linear" made me suspicious...

 

[schip666!]

Here is a simple classical system that exhibits nondeterministic and acausal behavior.

Imagine a flat, frictionless planar surface with a uniform gravity field normal to the plane. Next attach a frictionless cone-like surface to the plane. The base of this object is tangent to the plane at the points of contact. The peak of the object is a cusp. This cusp is obviously an unstable equilibrium position. Now place an point mass at rest on the peak of the object. Because the point mass is at rest, it will stay that way forever, right? That is after all the definition of an unstable equilibrium point.

That the point mass stays at rest forever is indeed one solution to the equations of motion. There are uncountably many other solutions. The point mass can stay at the peak for some random time T but at time T starts sliding down the cone-like surface in some random direction.

Your cone is a three dimensional description of my UP pendulum. I would disagree that it is non-causal and non-deterministic. Any disturbing of equilibrium requires cause/energy and once cause is known it is (should be?) deterministic. Now we may not be able to detect or measure the cause, aside from watching where the balls fall, but it's there. When you have them as far as the eye can see you can only say Random and fall back on statistics...e.g., a box of molecules.

But I'm not sure that these are the best examples of Complex systems, they may be "just" Complicated in having too many or small variables to measure. Complex systems, like coupled oscillators, can be very simple in their configuration and also usually involve feedback mechanisms. I've been trying to pin down the significant features of both, but, you know, it's just a hobby...

 

[D H]

Your cone is a three dimensional description of my UP pendulum.

No, it isn't. They are similar in that both my cone and your inverted pendulum have unstable equilibrium points. Both can be started at some initial position and velocity that will make the point mass or pendulum come to rest exactly at the unstable equilibrium point. Here is where the similarity ends. It will take an infinite amount of time for the pendulum to reach the unstable equilibrium point. My point mass will reach the peak of the cone in finite time. This distinction is not a splitting of the hairs. It is a wig factory.

I would disagree that it is non-causal and non-deterministic. Any disturbing of equilibrium requires cause/energy and once cause is known it is (should be?) deterministic.

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. The cone is non-deterministic and non-causal. Any system that has an unstable equilibrium point in which an object without any external forces (other than those described in the equations of motion) can reach that unstable equilibrium point in finite time will exhibit this behavior. If the only forces acting on the object are conservative, the system should be time reversible. But it isn't. The reason is that such systems are not Lipschitz continuous in some way.

 

[Studiot]

Extract from Chambers Dictionary of Science and Technology

Ill-Conditioned

A term used in triangulation to describe triangles of such shape that the distortion resulting from errors made in measurement and plotting may be great, the criterion often used is that no angles in the triangle may be less than 30 degrees.

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.

 

[Studiot]

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.

 

[Bob S]

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

 

[D H]

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.

Yes, it excludes quantum effects. In fact, one could argue that this quantum mechanics negate the possibility of this classical non-determinism. The example at hand requires that the full state have specific values (position exactly atop the cone, velocity exactly zero).

I intentionally made the example easy to visualize. The surface is continuous but has a discontinuity in the gradient at the peak. A similar situation arises on a surface, Norton's dome, that has a second order discontinuity.

 

[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.