Vatilla: You mention:
” the defining characteristic of chaos, i.e. the sensitive dependence on initial conditions, “
Perhaps I can add a bit of mathematical exposition. The quoted phrase refers to hyperbolic divergence of nearby orbits of the dynamical system as they pass into the future. Consider, e.g., the original Lorenz Attractor, the archetype of a “strange attractor”, a fractal, chaotic attracting set in the dynamical system. An “attractor” in a dynamical system (e.g., the solution space of a set of ordinary or partial differential equations. Again, e.g., the three-dimensional solution space to the 3D system of three partial differential equations in three variables which Lorenz simplified from the Navier Stokes equation for weather < PAUSE >
How are the Navier-Stokes equations used in weather forecasting?
Response from: Robert Naumann, retired scientist NASA
The Navier Stokes equations can be integrated by modern high speed computers and allows the known pressure, temperature, and wind velocity at every point on the Earth to be predicted at some time in the future. The problem is that we cannot precisely specify the pressure, temperature, and velocity at every point on the globe as the initial conditions and the NS equations are non-linear partial differential equations which are very sensitive to initial conditions (a small change in initial conditions can cause a rapid divergence in the solutions) so it is not possible to accurately predict the weather for more than a few days in advance.
< RESTART >. the 3D system of three partial differential equations in three variables which Lorenz simplified from the Navier Stokes equation for weather was integrated numerically by the new M.I.T. supercomputer, which was quite archaic by modern standards.
There were no graphics, only a dot printer (not even a dot matrix printer, you remember those?)
Many scientists believed that supercomputers held the key to avoiding similar weather catastrophes. Introduced in the '60s, these powerful, room-sized computers finally offered sufficient processing power to take a set of initial atmospheric conditions, crunch the numbers and spit out an accurate forecast.
A researcher at MIT, Edward Lorenz, had one of these early computers running in his office. Into this clumsy machine, Lorenz entered a streamlined computational model consisting of 12 meteorological calculations. The equations analyzed basic variables -- temperature, pressure, wind speed -- and spit out a simulated weather forecast. To "see" this weather, Lorenz would select one variable and then have the computer print out how that variable changed over time. In a bit of artistic flair, he directed the computer to print a certain number of blank spaces followed by the letter "a" in addition to simple numerical results. This produced a graphical representation of the variable being studied -- the letter "a" would meander across the page, just as capricious as the weather it was simulating.
One day in 1961, a particular output sequence caught Lorenz's eye. He decided to repeat the calculation, but to save time, he started from the middle of the run. Using the previous printout, he selected numbers halfway through the series to be his initial conditions. He entered these values, restarted the calculation and went away for some coffee. When he returned, he was astonished to find that the second run hadn't produced identical results as the first. The output pattern should have been the same, but the second graph diverged dramatically from the first after just a short time. Lorenz thought at first that his computer, notoriously finicky, wasn't working properly. Then he discovered the problem: The numbers he had entered from the printout only contained three digits, while the computer's memory allowed for six digits. This small discrepancy -- entering 0.506 versus 0.506127 -- was enough to introduce enormous unpredictability into the system.
Lorenz discovered with weather what Poincaré had discovered with interacting celestial bodies: certain complex systems exhibit sensitive dependence on initial conditions. Alter those conditions even slightly, and you'll produce wildly different results. Weather forecasting, Lorenz realized, was a futile effort at best because no one could ever quantify atmospheric conditions with certainty. To help people understand this concept, he invoked the idea of an animal flapping its wings, which would create a small area of turbulence, which would then be magnified over time and distance into catastrophic meteorological changes. At first, Lorenz favored the wings of a seagull. But in 1972, while preparing for a conference presentation, a colleague suggested he change his title to something a tad more poetic: "Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?" The image captivated the public, and soon the Butterfly Effectbecame the standing metaphor for the challenges of weather forecasting and for chaos itself.
Lorenz might have been satisfied with the results of his computer experiment, but he suspected he might be standing on the threshold of something bigger -- something profound. His now-famous "dishpan experiments" opened up the door to this wild and wonderful world we know today as chaos.
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