# The Butterfly Effect - Real ?

## Main Question or Discussion Point

The Butterfly Effect,

"The butterfly effect is a phrase that encapsulates the more technical notion of sensitive dependence on initial conditions in chaos theory. Small variations of the initial condition of a nonlinear dyna
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mical system may produce large variations in the long term behavior of the system."

http://en.wikipedia.org/wiki/Butterfly_effect

I was always under the belief that this Effect resulted from weather forcast supercomputers, where they changed the input values by the "amount of the beating of a butterfly's wing", and after running the simulation on computer simulation created a cyclone, and the other did not.

Does anyone really think this occures in the real world, or is this just indicating the limitations of complex systems (weather) with computers.

I would think information theory and sampling rate theory would indicate that to be able to accurately simulate a complex system you would need i very high percentage of sample points. Possibly EVERY point in the system needs to be sampled.

If you start a simulation with a very small sample data set, (mabey a few thousand measurements of temp, wind speed, pressure and so on). that you will always get very different specific results, and only and 'estimation' of what the system is doing. ?

so is the Butterfly effect a real world effect, or just an indication of the limitations in measuring complex systems. (such as long term weather forcasting).

(why is a Quantum leap, considered big ?? )

Related Other Physics Topics News on Phys.org
Concept of "Butterfly effect" encapsulates true reality of non-linear dynamical systems. It is not result of limitations of computers modelling complex systems.

DaveC426913
Gold Member
Of course it is real.

Stand a pencil on its end.
Record the direction in which it falls over.
Repeat 20 times.

Fill your sink with water and pull the plug.
Record the direction it drains in.
Repeat 20 times.

Sensitive dependence on initial conditions.

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DaveC426913
Gold Member
I think I see what he's saying.

"Sensitive dependence on initial conditions." If we had an arbitrarily powerful computer so that we could know to an arbitrarily fine degree what those initial conditions are (such as right down to the butterfly's wings' vorteces), our computer should be able to accurately predict the weather, a week - and a thousand miles - away.

But there's two things to consider here:
1] Whether an initial set of conditions will accurately lead to a result.
2] Whether an initial set of conditions will consistently lead to a result.

Just because we know to an arbitrarily high degree what those initial conditions are, still doesn't mean we can predict. There are external factors - such as all the butterflies in England and the weather currents across the Pacific - that will give us a completely different result each time we run the simulation, even if our starting dataset is identical every time we run it.

You need more than simply an arbitrarily fine degree of initial data; you need a closed system, AND you need to know about every data point in that closed system.

It would still be very sensitive to initial conditions - you could still kill one butterfly and throw off the whole system - but you could predict the various outcomes.

DaveC426913
Gold Member
I like the time you edited your post Dave (post #3 edited at 07:07):P
I long ago concluded that servers exist in an atemporal plane, where time is perpendicular to IP address and their clocks are set by the teethmark patterns on an acorn by a hungry squirrel.

Thanks everyone, and i like your description Dave.

But im not convinced, if it was a real world effect, then would you not expect the number to cyclones to approximately equal the number of butterfly wing beats.

Im sure there are a vast number of wing beats that dont do anything, or is what i consider to be below the noice floor, or real world effect threshhold.

I dont really know, but if i do get my pensil and drop it 20 times, i can be fairly sure it will almost always end up lying on its side, somewhere near where i let it go. quite predictable within a range.
If the pensil suddenly jumped out of the window, then i would think some odd chaotic behaviour was occusing.

your also right, i was talking about two identical super computers running weather simulations, with their input conditions change the the wing beat.

Seems weather is more of a "Bulk" type system to analyse, in that its the trend of large changes that cause things like cyclones, and major weather things.
Like air pressure over a large area, ocean surface temps, time of year and so on.

i would think any prediction of a complex system can only ever be as accurate as the sampling rate, and quantization error. Unless as you said you read every point, and down to the level of the wing beat.

BTW: would dropping a pensil move more air than a butterfly wing beat, we need to watch that. lest we cause 20 extra cyclones !! :)

what would you expect from two side by side computer simulations of weather, with identical input data?
or do simulations make assumptions when real data is not available, or do they have any randomness built in, so would the results be different even with identical input data.

Gokul43201
Staff Emeritus
Gold Member
But im not convinced, if it was a real world effect, then would you not expect the number to cyclones to approximately equal the number of butterfly wing beats.
Not at all. For one thing, the Butterfly Effect is not a popularization of a statement that predicts that every butterfly wing-beat makes a cyclone, but that it can, if it happened at the right time and place. Furthermore, there's no telling how often the butterfly over here destroys the cyclone that the butterfly over there creates.

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Sure you arn't teasing these uptight physics types, Darryl? :)

They're not saying that butterfly wingbeats cause cyclones, just that any tiny change in the beginning can result in an entirely different outcome 6 months down the track.

As for the pencil example, whether it falls to the left or right is still quite a big difference, say 30cm, given that it was decided by a truely infintesimal difference in the initial tilt of the pencil. A mathmatician would say it does not stop there since, like a butterfly wingbeat, this event is large enough to affect large scale future events such as weather.

DaveC426913
Gold Member
But im not convinced, if it was a real world effect, then would you not expect the number to cyclones to approximately equal the number of butterfly wing beats.
As B.E.M. points out, this is a complete misunderstanding of the effect.

Let the world run for one day. Cyclones occur here and there.
Rewind the world, putting it back EXACTLY where it was the day before.
Add a butterfly to the mix.
Let the world run again. The cyclones will be completely different.

It is not that butterflies cause cyclones - it is that a tiny, tiny change in initial conditions creates a huge change in final conditions.

I dont really know, but if i do get my pensil and drop it 20 times, i can be fairly sure it will almost always end up lying on its side,
That's not what you were asked to measure. The experiment requires measuring which direction it fell.

The direction the pencil falls is dependent on tiny imperfections in the surface and in the pencil's tip. They lead to big changes in the end result.

Take the pencil test one step further.
Write the words Clear Sunny Day at the top of a piece of paper and Hurricane at the bottom.
Balance the pencil in the centre of the paper. Let it fall and record the result.
Repeat 20 times.

Think of the tiny imperfections in the paper as equivalent to the butterfly.

for every action there is a reaction

because that is the law of physics, hey it could be wrong but lets run with it. when u drop said pencil or a butterfly flaps, that energy MUST go somewhere, and it will travel around and affect our weather. its impossible to predict the weather completly right because, u would need a computer the size of the world u r trying to predict.

if we get right down to atoms, thats the whole dam universe, hence u would need another universe to perdict the first uinverse, in fact it would be simpler 2 have the exact world, just 1 year earler.

and thats not going to happen. sorry, i rambled a bit

This is why, for example, you can never, ever, under any circumstance understand what a woman is thinking. The nerve endings in the cortex fire completely randomly, and so gratuitously, it is simply impossible to predict.

J77
As said, the BE is just a nice way of explaining sensitivity to initial conditions.

In real life, in a system as robust as the weather, the small perturbation of a butterfly flapping its wings would, of course, have no effect whatsoever.

DaveC426913
Gold Member
As said, the BE is just a nice way of explaining sensitivity to initial conditions.

In real life, in a system as robust as the weather, the small perturbation of a butterfly flapping its wings would, of course, have no effect whatsoever.
I don't know why you say this. The whole point of chaos theory is that the weather is dependent on tiny differences in initial conditions.

It does not mean to suggest that a butterfly's wing will develop into a hurricane it is meant to suggest that you cannot simply rewind conditions and expect the same result.

Do you guys know any programming? Do you know about random number generation? This is going to be a bit simplistic, but go with me.

You start with a seed (a large number - sometimes it's the date in seconds since activation). Say the computer starts with a seed of 4619837461238740001. The computer will use this in a long calculation to generate a decimal number between zero and one, say it generates the decimal .421. That's your random number, which you use in calculations.

Next time you run the exact same program, the system generates another seed, but now it is four seconds later. This time it spits out the number 4619837461238740005. Note that this number is almost identical to the first. Yet the computer uses a process that generates a number that is WAY different than the first - it might generate the decimal .992.

In non-chaotic systems, you expect that starting seeds very near each other would produce result values near each other, whereas seed different from each other would produce results very different.

In but chaotic systems, that tiny, tiny change produces a result that is totally unlike any other.

So yes, tiny perturbations - because there are BILLIONS of them - means you can't expect the same result. It's not simply one butterfly wing beat, it's an iterative process - working its way up the casue-effect ladder.

Look at it the other way: The beat of a butterfly's wing is just as likely to diffuse that hurricane.

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As said, the BE is just a nice way of explaining sensitivity to initial conditions.

In real life, in a system as robust as the weather, the small perturbation of a butterfly flapping its wings would, of course, have no effect whatsoever.

J77
You point?

I'll reply to the other post above when I've had some coffee. For now, I could rant for pages as to why this statement is wrong:
The whole point of chaos theory is that the weather is dependent on tiny differences in initial conditions.

vanesch
Staff Emeritus
Gold Member
I'll reply to the other post above when I've had some coffee. For now, I could rant for pages as to why this statement is wrong:
To me, it sounds quite right...

J77
To me, it sounds quite right...
It's right from a populist point of view but the first question I would address is, "what does one mean by 'chaos theory'?"

e2a: this could be the start of an interesting topic - I have to do some offline corrections to a paper today - hopefully, more later...

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vanesch
Staff Emeritus
Gold Member
It's right from a populist point of view but the first question I would address is, "what does one mean by 'chaos theory'?"
From a "populist" PoV

Never mind.

"Chaos theory" is the study of phenomena which are strongly sensitive on initial conditions in a rather well-defined way (Liapunov-instable). In other words, the study of phenomena for which the time evolution is known, but for which nearby solutions diverge exponentially with time.
This statement implies by itself already several hypotheses, but which are satisfied for weather systems: "deterministic" predictability on the short term (which means that there IS a definite relationship between initial conditions and the state at a later moment, which is NOT purely stochastic), and gradual loss of predictability on the longer term due to the divergence of the time evolution.

HallsofIvy
Homework Helper
The whole point of chaos theory is that the weather is dependent on tiny differences in initial conditions.
Well, the problem I would have with that is that chaos theory deals with a lot more than just the weather. It might be true that "the whole point of the application of chaos theory to weather is that the weather is dependent on tiny differences in initial conditions." Of course whether or not that is true would depend on how well the mathematical model matched reality, not on chaos theory. Chaos theory is mathematics, not meteorology!

DaveC426913
Gold Member
Yes, sorry. Sloppy writing. "The whole point of applying Chaos Theory to weather is that weather is indeed highly sensitive to initial conditions."

Note that it does not (at least, should not) say that initial conditions CAUSE final conditions. Butterfly beats do not CAUSE hurricanes, but if you rewound one Earth day, duplicating it virtually exactly, you simply could NOT duplicate the day's weather because such small imperfections as tiny air currents will throw everything off.

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This is like the Schrodinger's cat thought experiment, it's not meant to represent reality but to represent what we assume about the quantum. Now that said it differs in one way it is at heart what Chaos theory is trying to say, with a small variable the longer you progress with said variable the more random the system becomes, so in essence if you have two pendulums swinging by an equation 2nx-1 or whatever we should see predictable results, but we don't, that's chaos theory, given a simple equation we should see x but we see y, why, what are we missing?

I'm kind of loathe to put this up, as it was ignored by nearly everyone, but it demonstrates in real terms what the butterfly effect posits, so forgive me.

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Thats all very interesting, and thanks for various, and wise comments.

i can certainly see your points that it appears the weather is highly sensitive to initial conditions, but does that mean that the weather is totally deterministic, i would put forward the theory that if you started with the exact same starting conditions and "ran" the weather for a period of time,

that you would get different results each "run", even with the exact same starting conditions.

but i still think thats its more an artifact of computer simulations of weather systems that promts people to assign these variations to the BE.

what i initially said that by changing the input values of a weather simulations computer, or two, and running the system you get vastly different results.
an extension of that would be to run 1000 simulations of the weather all with a butterfly wing beat difference, AND run the real world weather, i would expect that NO TWO computer sims or the real world would produce the same results.

out of interest, how would it be possilble to "work backwards" from one future weather pattern to a vanishingly small change in input values. ??
Without using a computer and weather simulation software ?

its also been quite well established that cyclones and any weather events are the result of "bulk" changes in initial conditions, as i said before, its average sea surface temperatue, and barametric pressure, ie low pressure and high sea surface temps, time of year, sunlight intensity, time of year and so on. these are all "bulk" effects.

if weather was so deterministic as stated, that would not weather forcasting be highly accurate, and as its clearly not that accurate, how can it be that we have so much "faith" that a very small initial change will make large changes "in the real world", as indicated with computer simulations of the same.

take two super computers with the same simulations software, change the REAL initial conditions by 1 butterfly beat, and run both simulations in parallel with the REAL weather, i would expect 3 different results.

but its not possible to proove, you cant work backwards through real weather from a cyclone and narrow it cause to a minute change in input conditions.

but you're right the thread was posted for a bit of fun, but its great to see such bright, and very intelligent responses.
i think, making smart people think is a fair goal ! :D

great thanks to everyone.

as we are talking about chaos, can it be said that if something is chaotic thats its not predictable, regardless of slight changes in staring conditions, if truly chaotic, you will get different results even with identical starting conditions. making weather insentive to small initial changes to the starting conditions !!.

thanks all, very interesting..

J77
"Chaos theory" is the study of phenomena which are strongly sensitive on initial conditions in a rather well-defined way (Liapunov-instable). In other words, the study of phenomena for which the time evolution is known, but for which nearby solutions diverge exponentially with time.
I would say that sensitivity to initial conditions is characteristic of chaotic trajectories and can be quantified in terms of Lyaponov exponents.

The thing which I find hard to freely speak of is Chaos theory.

From the first, I would rather use bifurcation theory. It's much more important knowing, under variation of parameters, how systems come to exhibit apparent chaotic dynamics.
what i initially said that by changing the input values of a weather simulations computer, or two, and running the system you get vastly different results.
an extension of that would be to run 1000 simulations of the weather all with a butterfly wing beat difference, AND run the real world weather, i would expect that NO TWO computer sims or the real world would produce the same results.
Of course, this depends on the parameters of the system. However, let's assume that we're using Lorenz's toy-model for the 'weather' for parameters which exhibit chaos.

As you'll be aware, the chaos trajectories converge to a strange attractor which, quite nicely, looks like butterfly wings. Therefore, I would say that while you will always get different results from different runs, the dynamics are constrained to regions of phase-space and, thus, are not "vastly" different.
its also been quite well established that cyclones and any weather events are the result of "bulk" changes in initial conditions, as i said before, its average sea surface temperatue, and barametric pressure, ie low pressure and high sea surface temps, time of year, sunlight intensity, time of year and so on. these are all "bulk" effects.
These are more gradual changes in parameters than changes in initial conditions. Of course, the weather could be perturbed such that the trajectory shoots off into a region of phase space which exhibits the dynamics of a cyclone, before settling back down to the attractor (this is a bit hand wavy due to the enormity - infinite dimensionality - of the phase space of a spatial-temporal system and one which may be influenced by the past).

vanesch
Staff Emeritus
Gold Member
i can certainly see your points that it appears the weather is highly sensitive to initial conditions, but does that mean that the weather is totally deterministic, i would put forward the theory that if you started with the exact same starting conditions and "ran" the weather for a period of time,

that you would get different results each "run", even with the exact same starting conditions.
Yes and no. For short runs, they would be the SAME. It is for long runs that they would be different. This is the whole point. It is what distinguishes chaotic systems from stochastic systems: in stochastic systems, even for a short run, there is a finite probability of having different outcomes. Not so for chaotic systems: for short runs, they are "deterministic". This is BTW what you allude to when you say:

its also been quite well established that cyclones and any weather events are the result of "bulk" changes in initial conditions, as i said before, its average sea surface temperatue, and barametric pressure, ie low pressure and high sea surface temps, time of year, sunlight intensity, time of year and so on. these are all "bulk" effects.
What you mean here, is ON THE SHORT TERM.
So, indeed, given certain initial "bulk" (coarse grain) conditions, we can say with certainty for the next day that a cyclone will develop, or not at all. It is not that there's 20% chance for a cyclone to develop in the next 2 hours or so, as would be the case with a purely stochastic system.

However, on the long term, we don't know. Now, given that the long term is a succession of short terms, this sounds a priori illogical, but the way out is the exponential divergence of trajectories. A weather forecast for the next day only needs bulk data (all compatible trajectories go to about similar places). But for a 3-day forecast, some trajectories diverge, and we need to know more precisely whether we are on them, or on their neighbors (more detailled initial conditions needed). And for a 500-day forecast, the trajectories all go so wildly apart that we would need an accuracy of initial data which would be different with or without butterfly flapping.

but i still think thats its more an artifact of computer simulations of weather systems that promts people to assign these variations to the BE.
No, what is meant is that the trajectories which only differ by a flap of a butterfly wing, will go so wildly apart, that the weather will be totally different whether one is on the first or the second. Of course, after a while, these differences in trajectories have accumulated in bulk properties, indicating whether or not for the next day there will be a cyclone.

what i initially said that by changing the input values of a weather simulations computer, or two, and running the system you get vastly different results.
an extension of that would be to run 1000 simulations of the weather all with a butterfly wing beat difference, AND run the real world weather, i would expect that NO TWO computer sims or the real world would produce the same results.
In the long run, yes. But for short runs, they would be nearly identical. That's the whole point. It means that there is a form of determinism at work.

out of interest, how would it be possilble to "work backwards" from one future weather pattern to a vanishingly small change in input values. ??
Without using a computer and weather simulation software ?
Very often, chaotic systems are chaotic in both time directions.

if weather was so deterministic as stated, that would not weather forcasting be highly accurate,
On the short term (a few hours to a day), it is quite accurate.

as we are talking about chaos, can it be said that if something is chaotic thats its not predictable, regardless of slight changes in staring conditions, if truly chaotic, you will get different results even with identical starting conditions. making weather insentive to small initial changes to the starting conditions !!.
In principle, a chaotic system is (by definition) deterministic. That means that *identical* initial conditions always lead to *identical* time evolution. This is the difference with a stochastic system, which is random and which will give different results for identical initial conditions.

Now, probably, at a certain point, the weather system is in fact stochastic. There are phenomena which are probably truly random. For instance, the formation of a specific droplet out of over-saturated air is probably purely random.