Sensitivity to initial conditions....


Chaos theory and the sensitive dependence on initial conditions were described in the literature in a particular case of the three-body problem by Henri Poincaré in 1890.^[4] He later proposed that such phenomena could be common, for example, in meteorology.^[5]
In 1898, Jacques Hadamard noted general divergence of trajectories in spaces of negative curvature. Pierre Duhem discussed the possible general significance of this in 1908.^[4]
The idea that one butterfly could eventually have a far-reaching ripple effect on subsequent historic events made its earliest known appearance in "A Sound of Thunder", a 1952 short story by Ray Bradbury about time travel.^[6]
In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value. The result was a completely different weather scenario.^[7]
Lorenz wrote:

"At one point I decided to repeat some of the computations in order to examine what was happening in greater detail. I stopped the computer, typed in a line of numbers that it had printed out a while earlier, and set it running again. I went down the hall for a cup of coffee and returned after about an hour, during which time the computer had simulated about two months of weather. The numbers being printed were nothing like the old ones. I immediately suspected a weak vacuum tube or some other computer trouble, which was not uncommon, but before calling for service I decided to see just where the mistake had occurred, knowing that this could speed up the servicing process. Instead of a sudden break, I found that the new values at first repeated the old ones, but soon afterward differed by one and then several units in the last decimal place, and then began to differ in the next to the last place and then in the place before that. In fact, the differences more or less steadily doubled in size every four days or so, until all resemblance with the original output disappeared somewhere in the second month. This was enough to tell me what had happened: the numbers that I had typed in were not the exact original numbers, but were the rounded-off values that had appeared in the original printout. The initial round-off errors were the culprits; they were steadily amplifying until they dominated the solution." (E. N. Lorenz, The Essence of Chaos, U. Washington Press, Seattle (1993), page 134)^[8]

In 1963 Lorenz published a theoretical study of this effect in a highly cited, seminal paper called Deterministic Nonperiodic Flow^[9][10] (the calculations were performed on a Royal McBee LGP-30 computer).^[11][12] Elsewhere he stated:

One meteorologist remarked that if the theory were correct, one flap of a sea gull's wings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls.^[12]