Translation of part of an article by Etienne Ghys dating from around 2009, I suppose.

Since the quotes from Edward Lorenz are here a translation into English from a French translation, there may be a noticeable lack of precision compared to the original quotes. If you have the real ones, please post them in the comments.

Meteorology studies a phenomenon of inextricable complexity: the movement of the atmosphere. The equation that governs this movement has been known for a long time: it is the Navier-Stokes equation. But knowing how to write an equation does not mean knowing how to solve it! Let’s think a little about the amount of information needed to describe the atmosphere: we need to know the temperature, wind speed, atmospheric pressure, humidity, etc., not only in a given place but also in all places on the globe! Having an exact knowledge of this data is simply impossible: it requires an infinite amount of data, most of which is inaccessible.

Edward Lorenz was a meteorological theorist, with a mathematical background, who recently passed away. In 1962, he had the idea of ​​caricaturing the Navier-Stokes equation, simplifying it to the extreme, to make it “as if” the atmosphere depended only on three parameters, whereas it would require an infinite number! Simplifying a complicated problem in the hope that it will keep the essence of the phenomenon studied: this is a mathematician’s activity. And in his “atrophied atmosphere” reduced to its three coordinates, E. Lorenz can run his computer and calculate the numerical solutions that are supposed to describe the movement. Imagine Lorenz’s computer, with its small capacity, in 1962! It was then that he found “experimentally” that the slightest change in “his toy atmosphere”, for example adding 0.0000001 to one of the three coordinates, causes a considerable change in the atmospheric movement after a relatively short time. This is the phenomenon of “sensitive dependence on initial conditions”, the paradigm of chaos theory.

Look at the picture. It represents a trajectory of the simplified Lorenz equation, in three-dimensional space. These curves spin like crazy, sometimes to the left, sometimes to the right, and it seems impossible to predict whether a turn to the right will be followed by another turn to the right or to the left. And yet, for a given initial condition, for a given atmosphere, there is a well-defined future; determinism is not called into question. However, two points close in three-dimensional space, so close that they may not be distinguished in the figure, define trajectories that will start out close but may end up separating significantly: one to the left and the other to the right! Thus, if we know a point with a certain uncertainty, however small, the prediction of the future becomes illusory.

In 1973, Lorenz gave a lecture with a magnificent title that perfectly sums up this idea. […]:

“Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”

The butterfly effect was born!

Why did we have to wait for Lorenz to make this concept public?

Lorenz was not the first to understand this limitation in determinism. Henri Poincaré and Jacques Hadamard, at the beginning of the twentieth century, had understood this in a slightly different context: the movement of celestial bodies may be sensitive to the initial conditions… Lorenz knew this well, and his article cites his sources extensively.

Perhaps Hadamard and Poincaré did not know how to “find the words” and they were content to write incomprehensible mathematical articles? This interpretation does not hold. Both made efforts to popularise their ideas. Poincaré wrote, for example, in Science and Hypothesis, a popular work with a circulation of hundreds of thousands of copies:

“A tenth of a degree more or less at any point, the cyclone breaks here and not there, and it spreads its ravages on lands that it would have spared.”

The butterfly is not there, but the cyclone is! Hadamard and Poincaré were probably too ahead of their time, and society was not ready for this profound change in the concept of determinism. The physics of the early twentieth century represents the triumph of the science of determinism, inherited from Newton and Laplace. Everything is calculated, everything is predicted, and for what we cannot predict, we are confident that it is only a matter of time and that physics or mathematics will be able to answer. That was without the quantum and relativistic revolutions, which shook many preconceived ideas … In 1973, public opinion was more open to these new ideas, and without engaging in historico-sociologico-philosophical discussions, the idea that the slightest butterfly, and why not my humble person, could have an influence on the overall course of the world around me, was much better perceived in 1973 than in 1900.

The effectiveness of a butterfly in Texas?

But let’s not forget that Lorenz drew his conclusions from examining an almost absurd simplification of the “real equation” that governs atmospheric movement. Does the butterfly effect have an impact on meteorology? Lorenz does not commit himself on this point. His aim is to explain that a natural phenomenon, such as meteorology, could be sensitive to initial conditions and that this could have consequences for the impossibility of medium-term weather forecasting. Let’s give Lorenz credit for popularizing this simple idea that if the future is determined by the past, it may not be in such a naive way as previously thought. Even if the butterfly in Brazil proves to be powerless, there are many other areas of science where this idea can be applied. We have talked about planets, but some people do not hesitate to talk about history, politics, finance, etc.

Thanks to Poincaré, Hadamard, and Lorenz, our understanding of determinism has changed. We know that the present determines the future, but we also know that an imperfect knowledge of the present, as is almost always the case, makes determining the future illusory. It took a century for this simple but fundamental idea to be assimilated, unfortunately often imperfectly, by the public and even by scientists.

Lorenz’s message

Here are two quotes from Lorenz, simplified slightly. The first has been understood:

“If the flapping of a butterfly’s wings can cause a hurricane, the same is true for all the other wing beats of the same butterfly, but also for the wing beats of millions of other butterflies, not to mention the influence of the activities of countless other more powerful creatures, such as humans, for example!”

The second, however, went unnoticed:

“I propose that over the years, small disturbances do not change the frequency of events such as hurricanes: the only thing they can do is change the order in which these events occur.”

In short, even if meteorologists cannot predict the weather in Lyon in a month’s time, it should be possible to predict averages and frequencies of meteorological events with a high degree of accuracy in a given location over a long period of time. Of course, this type of prediction is more modest, but it is often just as useful. Lorenz’s second idea reframes the role of the forecaster.

Today

Of course, these ideas go far beyond the specific case of meteorology. A scientific theory cannot be based on a negative principle such as the impossibility of predicting the future; it must propose a method for overcoming this difficulty. Today, the mathematical theory that addresses these issues is called dynamical systems theory, and when it takes the perspective of Lorenz’s second quote, it is called ergodic theory: the aim is then to understand not specific trajectories, but frequencies and averages. This is a fascinating and flourishing mathematical theory, especially since the 1970s!

A battle is raging among weather specialists and Navier-Stokes equation experts over whether the butterfly in Brazil influences Texas. The question is whether Lorenz’s approximations are justified in the case of the atmosphere. To answer this question, mathematicians need to better understand dynamic systems that depend on a large number of dimensions (and even an infinite number of dimensions). A recent article is entitled “The butterfly effect no longer exists!” but other authors criticize the assumptions made in it. Of course, physicists and meteorologists must also be consulted: no mathematical theory can be applied to a concrete situation if it cannot be verified that the assumptions are satisfied in practice. So, does the butterfly effect exist “in reality”? Let’s leave the mathematicians to work with their physicist colleagues. They may soon have an answer for us. For example, the fact that the Lorenz equation, simplified to only three dimensions, actually satisfies Lorenz’s second quote is a very recent purely mathematical result: mathematicians say that the Lorenz equation has a “physical measure.” And this difficult mathematical result is not, a priori, related to the question of whether the Lorenz equation accounts for the movement of the atmosphere.

Even if the butterfly effect did not exist in the atmosphere, it would still remain a rich and powerful mathematical idea. The theory of dynamical systems is not limited to describing the atmosphere. As is often the case in mathematics, an example has become the seed of a theory whose ambition is to understand a much broader field than initially thought, and to establish connections with other areas that seemed far removed. The concept of chaos, which emerged a century ago for reasons related to celestial mechanics, has been enriched by the example of turbulence in the atmosphere and has invaded a large part of mathematics, including even number theory, which seems so “static” and immutable… See, for example, an illustration in this article.

Celestial mechanics, meteorology, and number theory have therefore recently been united in common methods. Poincaré warned us: “To do mathematics is to give the same name to different things.” Today, chaos means many things, far more than Poincaré or Lorenz could have imagined.

On this slide we show the three-dimensional unsteady form of the Navier-Stokes Equations. These equations describe how the velocity, pressure, temperature, and density of a moving fluid are related. The equations were derived independently by G.G. Stokes, in England, and M. Navier, in France, in the early 1800’s. The equations are extensions of the Euler Equations and include the effects of viscosity on the flow. These equations are very complex, yet undergraduate engineering students are taught how to derive them in a process very similar to the derivation that we present on the conservation of momentum web page.

The equations are a set of coupled differential equations and could, in theory, be solved for a given flow problem by using methods from calculus. But, in practice, these equations are too difficult to solve analytically. In the past, engineers made further approximations and simplifications to the equation set until they had a group of equations that they could solve. Recently, high speed computers have been used to solve approximations to the equations using a variety of techniques like finite difference, finite volume, finite element, and spectral methods. This area of study is called Computational Fluid Dynamics or CFD.

The Navier-Stokes equations consists of a time-dependent continuity equation for conservation of mass, three time-dependent conservation of momentum equations and a time-dependent conservation of energy equation. There are four independent variables in the problem, the x, y, and z spatial coordinates of some domain, and the time t. There are six dependent variables; the pressure p, density r, and temperature T (which is contained in the energy equation through the total energy Et) and three components of the velocity vector; the u component is in the x direction, the v component is in the y direction, and the w component is in the z direction, All of the dependent variables are functions of all four independent variables. The differential equations are therefore partial differential equations and not the ordinary differential equations that you study in a beginning calculus class.

https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html

A Yale-led study warns that global climate change may have a devastating effect on many butterfly populations worldwide, turning their species-rich, mountain habitats from refuges into traps.

Think of it as the “butterfly effect” — the idea that something as small as the flapping of a butterfly’s wings can eventually lead to a major event such as a hurricane — in reverse.

https://news.yale.edu/2025/03/31/new-warnings-butterfly-effect-reverse