The Real Butterfly Effect
If a butterfly flaps its wings in China today, it may
cause a tornado in America next week. Most of you will be familiar with this
“Butterfly Effect” that is frequently used to illustrate a typical behavior of
chaotic systems: Even smallest disturbances can grow and have big consequences.
The name “Butterfly Effect” was popularized by James Gleick in his 1987 book
“Chaos” and is usually attributed to the meteorologist Edward Lorenz. But I
recently learned that this is not what Lorenz actually meant by Butterfly
Effect.
I learned this from a paper by Tim Palmer, Andreas Döring, and Gregory Seregin called “The Real Butterfly Effect” and that led me to dig up Lorenz’ original paper from 1969.
Lorenz, in this paper, does not write about butterfly wings. He instead refers to a sea gull’s wings, but then attributes that to a meteorologist whose name he can’t recall. The reference to a butterfly seems to have come from a talk that Lorenz gave in 1972, which was titled “Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?”
The title of this talk was actually suggested by the session chair, a meteorologist by name Phil Merilees. In any case, it was the butterfly that stuck instead of the sea gull. And what was the butterfly talk about? It was a summary of Lorentz 1969 paper. So what’s in that paper?
In that paper, Lorenz made a much stronger claim than that a chaotic system is sensitive to the initial conditions. The usual butterfly effect says that any small inaccuracy in the knowledge that you have about the initial state of the system will eventually blow up and make a large difference. But if you did precisely know the initial state, then you could precisely predict the outcome, and if only you had good enough data you could make predictions as far ahead as you like. It’s chaos, alright, but it’s still deterministic.
Now, in the 1969 paper, Lorenz looks at a system that has an even worse behavior. He talks about weather, so the system he considers is the Earth, but that doesn’t really matter, it could be anything. He says, let us divide up the system into pieces of equal size. In each piece we put a detector that makes a measurement of some quantity. That quantity is what you need as input to make a prediction. Say, air pressure and temperature. He further assumes that these measurements are arbitrarily accurate. Clearly unrealistic, but that’s just to make a point.
How well can you make predictions using the data from your measurements? You have data on that finite grid. But that does not mean you can generally make a good prediction on the scale of that grid, because errors will creep into your prediction from scales smaller than the grid. You expect that to happen of course because that’s chaos; the non-linearity couples all the different scales together and the error on the small scales doesn’t stay on the small scales.
But you can try to combat this error by making the grid smaller and putting in more measurement devices. For example, Lorenz says, if you have a typical grid of some thousand kilometers, you can make a prediction that’s good for, say, 5 days. After these 5 days, the errors from smaller distances screw you up. So then you go and decrease your grid length by a factor of two.
Now you have many more measurements and much more data. But, and here comes the important point: Lorenz says this may only increase the time for which you can make a good prediction by half of the original time. So now you have 5 days plus 2 and a half days. Then you can go and make your grid finer again. And again you will gain half of the time. So now you have 5 days plus 2 and half plus 1 and a quarter. And so on.
Most of you will know that if you sum up this series all the way to infinity it will converge to a finite value, in this case that’s 10 days. This means that even if you have an arbitrarily fine grid and you know the initial condition precisely, you will only be able to make predictions for a finite amount of time.
And this is the real butterfly effect. That a chaotic system may be deterministic and yet still be non-predictable beyond a finite amount of time .
This of course raises the question whether there actually is any system that has such properties. There are differential equations which have such a behavior. But whether the real butterfly effect occurs for any equation that describes nature is unclear. The Navier-Stokes equation, which Lorenz was talking about may or may not suffer from the “real” butterfly effect. No one knows. This is presently one of the big unsolved problems in mathematics.
However, the Navier-Stokes equation, and really any other equation for macroscopic systems, is strictly speaking only an approximation. On the most fundamental level it’s all particle physics and, ultimately, quantum mechanics. And the equations of quantum mechanics do not have butterfly effects because they are linear. Then again, no one would use quantum mechanics to predict the weather, so that’s a rather theoretical answer.
The brief summary is that even in a deterministic system predictions may only be possible for a finite amount of time and that is what Lorenz really meant by “Butterfly Effect.”
I learned this from a paper by Tim Palmer, Andreas Döring, and Gregory Seregin called “The Real Butterfly Effect” and that led me to dig up Lorenz’ original paper from 1969.
Lorenz, in this paper, does not write about butterfly wings. He instead refers to a sea gull’s wings, but then attributes that to a meteorologist whose name he can’t recall. The reference to a butterfly seems to have come from a talk that Lorenz gave in 1972, which was titled “Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?”
The title of this talk was actually suggested by the session chair, a meteorologist by name Phil Merilees. In any case, it was the butterfly that stuck instead of the sea gull. And what was the butterfly talk about? It was a summary of Lorentz 1969 paper. So what’s in that paper?
In that paper, Lorenz made a much stronger claim than that a chaotic system is sensitive to the initial conditions. The usual butterfly effect says that any small inaccuracy in the knowledge that you have about the initial state of the system will eventually blow up and make a large difference. But if you did precisely know the initial state, then you could precisely predict the outcome, and if only you had good enough data you could make predictions as far ahead as you like. It’s chaos, alright, but it’s still deterministic.
Now, in the 1969 paper, Lorenz looks at a system that has an even worse behavior. He talks about weather, so the system he considers is the Earth, but that doesn’t really matter, it could be anything. He says, let us divide up the system into pieces of equal size. In each piece we put a detector that makes a measurement of some quantity. That quantity is what you need as input to make a prediction. Say, air pressure and temperature. He further assumes that these measurements are arbitrarily accurate. Clearly unrealistic, but that’s just to make a point.
How well can you make predictions using the data from your measurements? You have data on that finite grid. But that does not mean you can generally make a good prediction on the scale of that grid, because errors will creep into your prediction from scales smaller than the grid. You expect that to happen of course because that’s chaos; the non-linearity couples all the different scales together and the error on the small scales doesn’t stay on the small scales.
But you can try to combat this error by making the grid smaller and putting in more measurement devices. For example, Lorenz says, if you have a typical grid of some thousand kilometers, you can make a prediction that’s good for, say, 5 days. After these 5 days, the errors from smaller distances screw you up. So then you go and decrease your grid length by a factor of two.
Now you have many more measurements and much more data. But, and here comes the important point: Lorenz says this may only increase the time for which you can make a good prediction by half of the original time. So now you have 5 days plus 2 and a half days. Then you can go and make your grid finer again. And again you will gain half of the time. So now you have 5 days plus 2 and half plus 1 and a quarter. And so on.
Most of you will know that if you sum up this series all the way to infinity it will converge to a finite value, in this case that’s 10 days. This means that even if you have an arbitrarily fine grid and you know the initial condition precisely, you will only be able to make predictions for a finite amount of time.
And this is the real butterfly effect. That a chaotic system may be deterministic and yet still be non-predictable beyond a finite amount of time .
This of course raises the question whether there actually is any system that has such properties. There are differential equations which have such a behavior. But whether the real butterfly effect occurs for any equation that describes nature is unclear. The Navier-Stokes equation, which Lorenz was talking about may or may not suffer from the “real” butterfly effect. No one knows. This is presently one of the big unsolved problems in mathematics.
However, the Navier-Stokes equation, and really any other equation for macroscopic systems, is strictly speaking only an approximation. On the most fundamental level it’s all particle physics and, ultimately, quantum mechanics. And the equations of quantum mechanics do not have butterfly effects because they are linear. Then again, no one would use quantum mechanics to predict the weather, so that’s a rather theoretical answer.
The brief summary is that even in a deterministic system predictions may only be possible for a finite amount of time and that is what Lorenz really meant by “Butterfly Effect.”
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