Does the butterfly effect apply in reality?

[BernieM]

Everyone has heard that a butterfly flapping it's wings in one place may ultimately cause a tornado to form in another place. I really have a problem with this.

It has also been said that if all the molecules in a cup of coffee were to move the same direction at the same time that the coffee could 'jump' out of the cup. However possible it may be, it is astronomically improbable. But who knows maybe someone has actually seen it happen. After all it IS possible. Of course the reason it is astronomically improbable is that all the atoms are moving chaotically at random in different directions, creating no 'net' effect of atoms moving in the same direction at the same time that would cause the coffee in the cup to jump out.

Now back to the butterfly effect problem I have:

If we look at the coffee cup full of atoms, each atom acting in a 'butterfly' fashion, causing other atoms to move in the same direction adding to the net effect intiated by the 'butterfly', it could cause the cup to empty itself. But when viewing it's effect in relation to the other atoms which are also acting in a butterfly fashion in the cup, the individual butterfly effect is fairly effectively neutralized, thereby nullifying any reasonable possibility that the coffee cup will empty itself out.

I believe that in the real world, that if there were any 'tornado' effect caused by the 'butterfly effect', that it would be very localized and minimal. In the case of the coffee cup, it could make a local group of atoms nearest to it tend to move in the same direction, and thus have a miniscule 'local tornado-like' effect, but that effect overall is lost in the overall effect of randomity in the cup.

To me it appears that the only practical place that the butterfly effect could truly cause a tornado, or make a cup empty itself out, would be in a computer simulation where all other 'counter-butterfly' effects could be eliminated, or in a astronomically huge system where the likeliehood of such an event would be certain, such as all the gaseous matter in the universe.

Anyone have any thoughts on this?

 

[Studiot]

You misunderstand the 'butterfly effect' which is the result of an ill conditioned system.

Try a Google search for ill conditioning.

 

[Archosaur]

Also "chaos theory"

 

[BernieM]

Ill conditioning has to do with numerical analysis. This would apply well in a computer simulation, such as the MIT (I think it was MIT) professor in the early days of weather prediction, who would calculate local weather patterns that would take his computer a day to generate. Going to lunch one day he reduced the precision of the calculation by some number of decimal points which allowed the simulation to be accomplished much quicker, an hour or so, and when he returned from lunch, the weather prediction was dramatically different from the 24 hour simulation, with the same input data.

My question was does the butterfly effect apply in the real world of cause and effect in our universe and if so, where.

 

[schip666!]

That phrase was coined to describe the early weather modeling work of Edward Lorenz. It is a rather poetic description of a fairly common phenomenon more rigorously named "sensitive dependence on {initial} conditions", which is an underlying feature of so-called "chaos theory" or "non-linear dynamical systems". Wiki has a pretty good article on it: http://en.wikipedia.org/wiki/Butterfly_effect although it would take a little digging through the references to get the idea. Their "chaos theory" article has a better set of refs.

Not being a math-guy I'd never heard of "ill conditioning". The first article I found kept saying it is a measure of "linear systems", whereas sensitive dependence is a feature of non-linear systems. It is defined as the relationship between changes of parameter and result in a system of equations, where "ill" means a large result for a small change. This does sound like the diverging trajectories of a "chaotic" system, so was I being misled by the linearity of that first article?

 

[K^2]

It applies to everything. You can have quantum amplification working as a butterfly effect to bring up true uncertainty of quantum process to a macroscopic level. That's kind of a big part of MWI.

 

[Studiot]

First what the butterfly effect is not. (this is what you seem to be assuming)

The BE is not a deterministic chain of cause and effect as set out in the famous statement

"For want of a nail a Kingdom was lost"

The story here is that
When going into battle the king's horse had a loose or missing nail in its shoe.
The shoe became loose
The horse strumbled
The King fell off
And lost the battle and therefore his Kingdom.

Nor is the BE a domino effect or stacked tins of beanz effect where the stability of a series of dominos or tins depends upon one domino or tin.

Schip is quite correct in noting that the output from the equations governing certain physical systems can be very sensitive to intial conditions, such that the precision available in solving these equations is less than the possible differences in ouput.

Thus we have come to numerical maths and ill conditioned systems.

Incidentally there are other forms of Chaos and also of dynamic instability that also lead to unpredictable output.

It is important to note one difference between the BE and other forms of Chaos. The BE is balanced in that whilst one flap of a wing may generate a tornado, another may prevent one so on balance the BE will not change the overall number of tornado's.

Lorenz made this point in his original (1972) paper and repeats it in his book where he devotes a chapter to the BE.

K^2 is also correct in observing that (repeated) amplification is a key characteristic of the BE.

 

[pallidin]

In my mind, the "butterfly effect" has no more substantive physical causality for tornadoes than my walking on the ground precipitates earthquakes.

 

[DaveC426913]

You guys need to understand it before dismissing it.

The butterly's wing does not cause the tornado; it creates a tiny change in the intital conditions.

Do this: balance a pencil carefully on its tip. Watch which way it falls.

Now do it again. Does it fall the same way?


Even if you use a robot to ensure perfect balancing, the system is so unstable that the tiny imperfections in the lead tip will rapidly multiply the imbalance as the pencil moves off true.

It is not that a single imperfection "causes" the pencil to fall one way, just like the butterfly deos not "cause" the tornado.

It's that, trying to repeat the experiment is impossible because the tiniest, tiniest molecular change early enough in the process leads to a competely different outcome.

 

[BernieM]

It's that, trying to repeat the experiment is impossible because the tiniest, tiniest molecular change early enough in the process leads to a competely different outcome.

There are two outcomes here, not just one. The direction of fall and the fall itself. The direction of fall as you point out is very easily influenced, but no matter which direction you influence it to fall, the larger force of gravity acting on the pencil is not influenced and the pencil still falls with the same amount of force. But in no case could the effect cause the pencil to fall up no matter how early in the process it is introduced.

As I see it, there is something in process, and the buttefly effect may only influence components of the in-process event that are of a corresponding order of magnitude or less.
Is that a fair assessment?

 

[Studiot]

Now do it again. Does it fall the same way?

I am not convinced that this experiment leads to any form of chaos.
I think that if repeated many times and some measure of where the pencil falls ( say the radial angle from some arbitrary zero ) was recorded then there would be an even distribution around the falling circle.

The BE comes from the Lorenz equations which were developed to predict certain state variables in fluid mechanics.

It was known that the states occur at substantial jumps in the values of the variables and that sudden unpredictable transitions occurred.

Lorenz was the first to observe that these jumps are not random, taken over a large number of trials and the BE was emerged from its chrysalis.

 

[schip666!]

Lets try another example... a pendulum...I love pendulums, especially of the multiple variety, but let's just use a simple single pendulum.

In common thought there is one equilibrium point where the pendulum bob is hanging down at its lowest point and is not moving. This is a very stable state space configuration since it takes a fair amount of energy to change the the two state variables: position and velocity.

There is actually a second equilibrium point where the bob is precisely balanced UP at its highest point. This is a very unstable configuration because it can be disturbed by pretty much anything, AND the disturbance can cause the pendulum to fall either way. When it falls the sign of the resulting state variables can be diametrically opposed based on a fundamentally un-measurable input. This is Sensitive Dependence on Initial Conditions, the state space trajectories of two (almost) identical inputs are as far apart as they can get. If this "experiment" is repeated very precisely you should get an even distribution of left/right falls, even though you can't measure the initial force or predict which way a particular event will occur.

This gets even more fun when you have a coupled system of pendulums or other oscillators. There are regions in the state-space that are stable and/or have predictable trajectories, and there are regions which have divergent and so-called chaotic trajectories (strange attractors in the poetical parlance). When you look to the real world, these sorts of systems are probably much more prevalent than the simple ones that scientists have always modeled by linearizing portions of differential equations. I think Newton was aware of the "three body problem" which was finally (accidentally) characterized by Poincare, but the results using two bodies were just too useful to ignore. So it took another 200 years to stumble on chaos.

 

[DaveC426913]

I am not convinced that this experiment leads to any form of chaos.

Yes, it isn't about chaos; its about only a single aspect of chaos: sensitivity to initial conditions.

I'm trying to explain why a tiny flapping of a butterfly wing versus not flapping can ultimately result in a tornado versus no tornado.
I'm comparing it to a tiny unevenness in the surface of the pencil lead (and the paper) can result in the lead falling left versus right.

People who misunderstand the butterfly effect seem think that the butterfly flapping kind of "forces" the tornado - direct cause and effect.

 

[Studiot]

Yes, it isn't about chaos; its about only a single aspect of chaos: sensitivity to initial conditions

I think it runs deeper than that.

All manifestations of chaos have a common characteristic - that there are multiple possible solutions to the governing non linear equations corresponding to different values for the state variables, also called phase or space variables, for some values (but not all) of some parameter.

For example The equation

$$\ddot x + k\dot x + {x^3} = A\cos \omega t$$

With initial conditions @ t=0, x=0, $$\dot x$$=0

solves to a periodic function of period 6(pi)

But starting at x=1 instead produces butterfly shaped chaotic oscillation in x v $$\dot x$$ phase space.

Note that this is not a small change in initial conditions.

The peculiarity of the BE is that the system is able to switch, unpredictably, from one solution to another in mid oscillation. this has nothing to do with initial conditions.

It is also true, however, that 'The Butterfly Effect' is used in popular parlance for all sorts of black magic and witchcraft the story teller ( sorry journalist) wishes to substitute for real science.

 

[DaveC426913]

I think it runs deeper than that.

Absolutely.

I'm trying to simplify it, so that people who dismiss it too quickly can grasp it easier.

All manifestations of chaos have a common characteristic - that there are multiple possible solutions to the governing non linear equations corresponding to different values for the state variables, also called phase or space variables, for some values (but not all) of some parameter.

For example The equation

$$\ddot x + k\dot x + {x^3} = A\cos \omega t$$

With initial conditions @ t=0, x=0, $$\dot x$$=0

solves to a periodic function of period 6(pi)

But starting at x=1 instead produces butterfly shaped chaotic oscillation in x v $$\dot x$$ phase space.

Note that this is not a small change in initial conditions.

The peculiarity of the BE is that the system is able to switch, unpredictably, from one solution to another in mid oscillation. this has nothing to do with initial conditions.

Ah, but that is a completely different butterfly reference! You're talking about the Lorenzian Butterfly, which, yes, has to do with phase shifts.

These two butteffly references in chaos theory are often confused, but they are distinct.

 

[Studiot]

These two butteffly references in chaos theory are often confused, but they are distinct.

Yes indeed that's what I've been trying to say.
Unfortunately they were both introduced by the same fella.

Phase space should not be confused with phase shifts or phase angles.

The double entendres will be the death of me yet.

 

[pallidin]

To think that a minor event can substantively effect the outcome of a major event is, indeed, possible, but absolutely absurd in this particular context.

 

[DaveC426913]

To think that a minor event can substantively effect the outcome of a major event is, indeed, possible, but absolutely absurd in this particular context.

Why?

Indeed, in my world it would be absurd if it were any other way.

 

[Studiot]

We should all bear in mind that the OP is looking for a real world example of the BE, presumably as more than a throwaway comment.

I have been trying to track one down, perhaps some of Thom's theory may provide one?

 

[JaredJames]

Can it not be something simple?

For example, you make a small choice in the morning and then it influences the outcome of your day. No direct cause and effect, but small decisions you make having profound changes later on.

I'm sure there are plenty of examples like that around.

 

[schip666!]

All manifestations of chaos have a common characteristic - that there are multiple possible solutions to the governing non linear equations corresponding to different values for the state variables, also called phase or space variables, for some values (but not all) of some parameter.

I think the "multiple solutions" thing may be misleading, at best. "Chaotic" systems are perfectly deterministic, from one starting point one always gets to the same ending point. However if one starts at an infinitesimally different point one can end up someplace completely different.

In that lovely (other) butterfly diagram you can find two trajectories that start at _almost_ the same position but end up on opposite lobes of the graph.

This is what I meant about trajectories diverging, and brings us back to the Ill conditioning tie-in. No one answered my question about whether I.C. is feature of only linear systems or not...?

 

[schip666!]

We should all bear in mind that the OP is looking for a real world example of the BE, presumably as more than a throwaway comment.

I have been trying to track one down, perhaps some of Thom's theory may provide one?

Pendulums...like I said. Or go back to Poincare's three-body work.

 

[Studiot]

"Chaotic" systems are perfectly deterministic, from one starting point one always gets to the same ending point. However if one starts at an infinitesimally different point one can end up someplace completely different.

The problem is that conventional applied maths relies heavily on calculus.
In turn calculus is founded on the concept of a neighbourhood (amongst other things) so that for any point x of a function f(x) we can say that the neighbourhood of x is sent to a neighbourhood in f(x).

The whole point of functions exhibiting chaos is that this does not happen for some x and their neighbourhoods.

 

[dkotschessaa]

Could it be said that the problem with the butterfly example is that it gives the wrong impression because it only gives one determining factor, i.e. the butterfly?

That is to say, the storm is not caused by the butterfly, but the flapping wings in conjunction with other events, which set up the conditions for the storm to occur. Had that singular event not happened, then no storm would occur. The butterfly is for illustrative purpose and probably has never happened that way. That's how I read it. Please de-bunk or...uh..re-bunk...

Maybe I should start another thread, but, is there anything being done with Chaos theory these days? Reading Jame's Gleick's book years ago was the first thing that got me thinking "math is cool" but I haven't heard much about it since, though I am out of the loop. It appears as though my university used to offer a course on it but doesn't anymore.

-DaveK

 

[schip666!]

"Complexity" is finally past the Jeff Goldblum "I'm a Chaos Theorist" phase. Melanie Mitchell's Complexity: A Guided Tour is the most recent popular treatment, written by an active researcher rather than some random science journalist. The field has "bifurcated' into a number of interesting directions, e.g., Multi-agent systems, Networks, Artificial Life, and the old favorite Dynamical Systems.

There are "Complexity Institutes" embedded in many universities now and most of them owe a debt to the Santa Fe Institute who's website is worth a peruse: http://www.santafe.edu/

 

[schip666!]

... calculus is founded on the concept of a neighbourhood (amongst other things) so that for any point x of a function f(x) we can say that the neighbourhood of x is sent to a neighbourhood in f(x)...

Yes. That's a nice succinct description, which I have not heard before. I wonder if means the concept of limits needs to be re-examined in dynamical systems? I should go look at all those books that I don't understand and see if there's any mention of this...

thx

 

[BernieM]

So far, I think I understand now why my 'coffee cup' example of the butterfly effect isn't really the butterfly effect at all but rather chaos theory. So big gain for me there. I can also see how the butterfly effect can apply in systems, but I have yet one unresolved question, which is, (and I hope that I can express this well enough to be understood ...): Can the butterfly effect of magnitude x manifest in a system many magnitudes greater later in time, in real world complex systems (such as fluid dynamics), where there are OTHER forces of varying magnitudes ALSO at work?

 

[schip666!]

... Can the butterfly effect of magnitude x manifest in a system many magnitudes greater later in time, in real world complex systems (such as fluid dynamics), where there are OTHER forces of varying magnitudes ALSO at work?

Yup, that's the actual exact point...

How did the Grand Canyon form? A few water molecules bumped into a grain of sand and found that dribbling to the right was ever so slightly easier than going left. In doing so they pushed the sand ever so slightly further to the left, such that the next water molecules had an easier time dribbling to the right. Und So Weiter.

We like to call these things "Frozen Accidents".

 

[BernieM]

I understand your water butterfly effect and how ultimately it causes a grand canyon to form. But to me this is similar to the lead pencil analogy in that the butterfly effect is cumulative within a process that is occurring and changes the direction that the process may take, however, like in the case of the lead pencil, the effect is not capable of creating enough change in the process to create an extraordinary result such as the pencil falling UP against gravity. So is it fair to say that the butterfly effect can not exceed the maximum order of magnitude of the process that it is involved or embedded in?

 

[statix1]

I think it does

 

[schip666!]

Of course, you need energy to do stuff and butterfly wings don't produce much. The issue here is Amplification. Some one particle must give-way to start an avalanche. The where and how of that particle may inordinately influence the where and how of the end state. In other cases it may all just average out.

This is in fact one way you can distinguish "random" events from "chaotic" events. Complexity folks get all excited when they find a non-normal distribution of things. Again using avalanches, it turns out that the size of avalanches in a sand or rice pile has a power-law distribution rather than exactly Normal/Gaussian. There are more large events than one would expect -- the so-called Fat Tail -- which is an indication that there is something else going on in the underlying dynamics.

 

[DaveC426913]

I understand your water butterfly effect and how ultimately it causes a grand canyon to form. But to me this is similar to the lead pencil analogy in that the butterfly effect is cumulative within a process that is occurring and changes the direction that the process may take, however, like in the case of the lead pencil, the effect is not capable of creating enough change in the process to create an extraordinary result such as the pencil falling UP against gravity. So is it fair to say that the butterfly effect can not exceed the maximum order of magnitude of the process that it is involved or embedded in?

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

Here's another way of looking at it: take a major event and roll it backwards until no trace of it appears to exist. Now roll it back some more. Now start the process again. Will it happen the same way? How many minute details would have to be exactly the same to how many decimals of precision in order for it to happen the same way?

Would the grand canyon have formed the way it did if millions of years ago a rock fell into a stream and slowed it a tiny bit, causing faster silting here rather than there? you;d still ge the Grand Canyon, but it might go five miles to the left. Where you're currently standing, at the cliff top might now be 400 feet higher than in the other scenario. From your PoV, locally, no Canyon here.


Let's have another look at the tornado. Perhaps it would be easier to swallow if, instread of 'tornado versus no tornado' it was 'tornado here versus tornado elsewhere'. If you rolled the weather system back several weeks, would it sweep throguh the same trailer park? How finely would you have to tune the tiny weather currents that determined whcih way it went?

 

[BernieM]

Let's have another look at the tornado. Perhaps it would be easier to swallow if, instread of 'tornado versus no tornado' it was 'tornado here versus tornado elsewhere'. If you rolled the weather system back several weeks, would it sweep throguh the same trailer park? How finely would you have to tune the tiny weather currents that determined whcih way it went?

That was my point. That the cause of the tornado was sufficient energy and conditions in the storm system to create a tornado. That the butterfly effect is not capable of introducing enough energy (or concentrating enough energy by redirection of energy already present) to create a storm system from a weaker one incapable of generating a tornado. That all the butterfly effect can do is influence where the tornado may occur. That the overall energy sufficient to create a tornado was present in the system already or would have become present in the system regardless of any butterfly effect or not.

 

[D H]

Responses to a random selection of recent posts, but in a time order (so not quite a chaotic response):

I think the "multiple solutions" thing may be misleading, at best. "Chaotic" systems are perfectly deterministic, from one starting point one always gets to the same ending point.

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.

However if one starts at an infinitesimally different point one can end up someplace completely different.

In that lovely (other) butterfly diagram you can find two trajectories that start at _almost_ the same position but end up on opposite lobes of the graph.

This is what I meant about trajectories diverging, and brings us back to the Ill conditioning tie-in. No one answered my question about whether I.C. is feature of only linear systems or not...?

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.

Could it be said that the problem with the butterfly example is that it gives the wrong impression because it only gives one determining factor, i.e. the butterfly?

Perhaps. Then again perhaps it is just the way the concept is oversimplified for presentation to the lay community. The name reflects combination of frustration and whimsy on the part of Edward Lorenz when he thought he could short-cut some computations by restarting from a checkpointed file. On investigating further and finding such incredible sensitivity to initial conditions he had to wonder whether "the flap of a butterfly's wings in Brazil sets off a tornado in Texas." It is not so much that the flap of the wing sets off the tornado as that flap just happening to put the weather on a trajectory where the true causes of a tornado can come out and play whereas without the flap the weather would just happen to follow a trajectory where those true causes are attenuated.

It is essentially the difference between weak and strong chaos. Our inner solar system is almost certainly weakly chaotic, and might be strongly chaotic. Suppose you have some predictions based on the best model available and somehow have the ability to peer a billion years into future to test those predictions. First let's suppose the inner solar system is only weakly chaotic. What this means is that your billions years lookahead would still have Mercury orbiting inside Venus, Venus orbiting inside the Earth, Earth orbiting inside of Mars. Your predictions of exactly what those orbits look like and where those planets lie on that orbit be would be utterly worthless. They would be pretty much worthless just looking a few million years into the future, let alone a billion years.

Now let's suppose the inner solar system is strongly chaotic. Now when you look a billion years into the future you might not find four planets. You might not find any inner planets at all. Taking smaller steps into the future, you find that Jupiter's perturbations of Mercury's orbit makes Mercury's orbit become ever more elliptical, eventually crossing Venus' orbit. The inevitable near-collision between the two sends Mercury on an highly elliptical orbit that goes out beyond Mars and Venus on an elliptical orbit that crosses Earth's orbit. The inevitable near collisions between Mars and Mercury and between Earth and Venus give all four a gravity assist that sends each close to Jupiter's orbit, and then it is bye-bye.

The weather is almost certainly strongly chaotic.

"Complexity" is finally past the Jeff Goldblum "I'm a Chaos Theorist" phase.

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.

 

[Studiot]

How oft is someone misquoted?

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

Here is a short extract from the original paper.

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.

1) If a single flap of a butterfly’s wings can be instrumental in generating a tornado, so also can all the previous and subsequent flaps of its wings, as can the flaps of the wings of millions of other butterflies, not to mention the innumerable more creatures, including our own species.
2) If the flap of a butterfly’s wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado.

So Lorenz made no claims, but actually asked two contrary questions. He did not actually offer an answer to either in the rest of the paper.