At MIT in 1953, Edward Lorenz was put in charge of a project devoted to statistical forecasting, exploring how the newly available digital computer could be put to use. A key issue was to know how well such numerical tools, based on linear statistical models, could predict complex weather patterns. Lorenz was skeptical of the appropriateness of such tools as most atmospheric phenomena are non-linear. He sought a set of simple nonlinear differential equations that would mimic meteorological variations, thus providing a test example for the linear statistical approach. It was necessary to find a system with non-periodic behavior (think of the large-scale turbulent eddies such as cyclones and anticyclones). After several failed attempts and some success with more complicated models, he arrived in 1961 at the following system of ordinary differential equations :
$$ \begin{cases} \dot{x} =\sigma (y-x)\\ \dot{y}= rx-y-xz\\ \dot{z}=xy-bz \end{cases} $$
where $\dot{x}=\mathrm{d} x/\mathrm{d} t$, etc. Here $x$ denotes the rate of convective overturning, $y$ the horizontal temperature difference, and z the departure from a linear vertical temperature gradient. There are three (positive) parameters : $\sigma$ is the Prandtl number (ratio of fluid viscosity to thermal conductivity), r represents a temperature difference driving the system, and b is a geometrical factor. What matters for us is that Lorenz numerically solved them and made a breakthrough : he discovered what was then called “deterministic chaos”. (In his paper, he used the values $\sigma = 10$, $r = 28$, $b = 8/3$.) The crucial insight of Lorenz was to observe that although orbits do depend on initial conditions, they accumulate on a kind of “surface” with “figure-eight” shape which is insensitive to initial conditions.
Lorenz made a rough sketch of this object and understood that two close initial conditions had the property of rapidly converging toward this “surface” and of travelling together for a while after which they start separating, at seemingly random intervals – one staying in a “wing” while the other goes to the other one – before they come close to each other back again, and so forth. Lorenz had just discovered what David Ruelle and Floris Takens later called a “strange attractor” [1]. Strange because this is not a fixed point or a closed orbit (limit cycle), and not generally a manifold. It turns out to be locally the product of a Cantor set and a piece of two-dimensional manifold.
Let us emphasize Lorenz’s lucidity and stroke of genius in not attributing what he observed to an artifact of the computer or an effect specific to his model. Instead he understood that there was an underlying general phenomenon. Indeed, one should realize that Lorenz used a Royal McBee LGP-30. As all computers of that time, it was slow, noisy and not as reliable as today’s computers. Margaret Hamilton (born in 1936) and then Ellen Fetter (born in 1940, also known as Ellen Gille after she married) were Lorenz’s programmers.
Lorenz published his results in 1963 in a journal of meteorology [2]. The manuscript was sent to Ulam for review. It took almost ten years to physicists and mathematicians to realize the importance of this work. It is only in 1972 that Lorenz presented the “butterfly effect” in the 139th meeting of the American Association for the Advancement of Science, asking : “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas ?”. At this occasion, he presented the surprising picture of the attractor which now bears his name.
Let us point out that it is only in 1998 that Warwick Tucker proved in his PhD thesis the existence of Lorenz’s attractor [3]. His demonstration relies upon a numerical integrator providing a precise control of errors in the approximation of true orbits.
We finish this section by quoting Lorenz [4] who refers to the book of Ulam [5] and to the paper [6] :
“We thus see that a computing machine may play an important role, in addition to simply grinding out numerical answers. The machine cannot prove a theorem, but it can suggest a proposition to be proven. The proposition may then be proven and established as a theorem by analytic means, but the very existence of the theorem might not have been suspected without the aid of the machine. Ulam has discussed the general problem of the computing machines as a heuristic aid to reasoning, and has presented examples from a number of different branches of mathematics.”