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26 august 2021

What is the butterfly effect?


   It is a theory that a butterfly flying in Brazil could cause a storm in Texas. According to the expression, invented by meteorologist Edward Lorenz, it is enough to modify at least one parameter of a meteorological model so that it gradually amplifies and causes, in the long run, colossal changes. This notion is no longer just time, it also applies to people and the environment.

The snowdrop effect.

For example, the use of foam baths, pesticides or flame retardants in Europe is destroying the polar bears in Greenland. These toxins released into the environment, in fact, travel thousands of miles, pollute the water and accumulate in the fats of fish and other seals, which are in turn ingested by bears. Eventually all the pollutants are concentrated and suffer from behavioral, reproductive or even growth disorders.

The butterfly effect is often used as a metaphor for everyday life or for history. But do you know where this expression "butterfly effect" comes from? In fact, it is a true scientific theory.

Edward Lorenz, a mathematician working in the meteorology department at the Massachusetts Institute of Technology, is studying the problems of predicting weather forecasts using computer systems.

On March 1, 1963, he modeled the behavior of the atmosphere. E.g. Develop a system of 3 small equations, very simple and nonlinear, ie without a simple relationship between cause and effect. Try to determine the weather, study the convection of warm air in the atmosphere and temperature in different places.

At some point, a parameter equal to 0.506 is entered into the computer. Suddenly, you realize that the result is totally different from what you got before, when you used 0.5061.

A small single decimal point of difference gives a radically different result.

He published an article on the subject. At a 1972 conference, one of his colleagues reacted by saying, "If your model is right, it means that if a seagull flutters, it completely changes the atmosphere."

Edward Lorenz thinks the idea is good, but replaces the seagull with a butterfly.

Deterministic chaos.

Edward Lorenz really showed that the solution is extremely sensitive to the initial conditions. If you add a small decimal to the tenth decimal place, the result produced by the computer will be very different. This is called deterministic chaos. The equation is quite deterministic, it predicts the future well, but there is a sensitivity to the initial conditions: as soon as one of the parameters is unknown, even a little, the behavior explodes and goes completely elsewhere.

Physicists have found that this feature occurs absolutely everywhere. Once there is a nonlinear equation, a chaotic solution often automatically appears. Of course, we can calculate the future, but the forecasts are so sensitive in the initial conditions that they are not really practical. The measurement is very approximate, and the final solution is totally undetermined.

It has become such a model that the notion of chaos theory is now taught in all universities. We return to all the old theories.

The simplest model, for example, is the responsibility of the Belgian Pierre-François Verhulst, who in 1838 published an equation for modeling population evolution in the early days of sociology. Study the births that increase the population and the deaths that decrease it. The equation is extremely simple; however, we realize that this is also chaotic.

And we see that there are, in science, a whole series of equations that exhibit chaotic behavior. It has become a scientific field in itself.

Philosophical determinism.

When this expression is used to talk about human actions, it is often in the sense of a certain determinism: if you do this, the consequences can be huge.

Philosophically, this discovery was also a shock, because it means that we are not able, even if the equation is deterministic, to determine the future: we cannot calculate it, because there is too much sensitivity in its conditions.

"So there has been a whole debate about irreversibility, about why life and death, while all the laws of physics are temporarily reversible, that is, if you change time, you somehow go back to your past, but of course no one knows how to do it. ", emphasizes Pasquale Nardone. 

How does chaos theory explain irreversibility? How can we, despite this theory of chaos, still see the emergence of structures such as hives, termite mounds, animal behavior? a very interesting debate between deterministic but chaotic models in the appearance of structures in space and time.

What supports the chaos theory is that a very small deviation in a parameter can have a large influence on the resulting situation.

An example presented by Poincaré to illustrate the notion of chaos.

Later, Henri Poincaré worked on chaotic phenomena, especially reflecting on the stability of the solar system and the problem. His work had no immediate application, due to the lack of electronic computers.

The consequences of chaos theory.

In Lorenz's example, a meteorologist would not necessarily think of considering the variations in airflow caused by the flapping of a butterfly's wings. His idea of ​​"non-infallibility of the prediction system", theorized in the form of the "butterfly effect", reminds us that there is at least a difference between what is determined and what is determinable.

Recent work has shown that modeling the atmosphere is not affected by the butterfly effect, because a minimal effect is forgotten without any visible impact on the whole.

However, it is true that small factors can have huge effects. Naturalist Stephen Jay Gould also argues that the Darwinian process is only statistical and that if we returned to Earth over 65 million years ago, flora and fauna would undoubtedly follow different paths on the African continent.

On the other hand, the butterfly effect is not inevitable and Jacques Laskar's work shows that the Earth's orbit would not have remained stable for more than a billion years without the small gravitational influence of the Moon, which would have stabilized its orbit. and thus ensured that it remained in the "habitable" area, thus preventing the orbit from deviating according to minor variations.

The results of chaos theory.

The practical limits of Newton's model are better understood and a new concept of "relative determinism" appears. The term "chaos theory" reappeared and in the early 1970s the world experienced a madness for this paradigm. So, I found two surprising results:

chaos has a kind of signature;

It itself can lead to stable phenomena. We will not be able to know its details, but we can know the final states without knowing where we will end up: it is a generalization of the notion of attractiveness raised by Poincaré.

The Santa Fé Institute was founded in 1984 to try to study the conditions under which order sometimes emerges from chaos.

Extrapolation of the butterfly effect.

Otherwise it is used as a metaphor for everyday life or for history. However, we must be careful to put these issues together. In fact, one of Blaise Pascal's thoughts is often summed up in the phrase “Cleopatra's nose, if it were shorter, the whole face of the earth would have changed.

In the case of the butterfly effect, the large variation is due to a very small change in the relative value of a mathematical variable. In the case of history, unpredictability is due to the fact that we do not know how to put it into equations. Therefore, it cannot even be said that the differences that seem intuitively insignificant give rise to a small variation of a numerical argument. It is not possible to know, until history is equated (which is not necessarily possible), whether the system of equations is truly chaotic. In any case, at this point in our knowledge, chaos theory does not provide more information about these phenomena than in proverbs.


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