OT: The History of Mathematical Chaos Theory
How Chaos Theory Works
By: William Harris | Updated: Aug 20, 2020
The Birth of Determinism
The 1600s enjoyed a slow and steady illumination as a collection of visionary thinkers brought reason, form and structure to the great mysteries of the world. First came Johannes Kepler, the German astronomer who, in 1609 and 1618, described how planets moved in elliptical orbits with the sun as one focus of the ellipse. Next came Galileo Galilei, who made fundamental contributions to the scientific studies of motion, astronomy and optics throughout the early 1600s. These empirical concepts and ideas joined the inventive thinking of philosophers such as René Descartes. In 1641, Descartes published his Third Meditation, in which he discussed the principle of causality -- "nothing comes from nothing," or "every effect has a cause."
All of these ideas set the stage for Isaac Newton, whose laws of motion and gravitation shaped science for centuries to come. Newton's laws were so powerful that, if you were so inclined, you could use them to make predictions about an object far into the future, as long as you knew information about its initial conditions. For example, you could calculate precisely where the planets would be hundreds of years from the current date, making it possible to presage transits, eclipses and other astronomical phenomena. His equations were so powerful that scientists came to expect that nothing lay beyond their grasp. Everything in the universe could be determined -- calculated -- simply by plugging known values into the well-oiled mathematical machinery.
In the late 18th and early 19th centuries, a French physicist named Pierre-Simon Laplace pushed the concept of determinism into overdrive. He summarized his philosophy like this:
We ought then to regard the present state of the universe as the effect of its anterior state and as the cause of the one which is to follow. Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it -- an intelligence sufficiently vast to submit these data to analysis -- it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes.Using this notion, Laplace's colleague Urbain Jean Joseph Le Verrier correctly predicted the planet Neptune in 1846, relying not on direct observation but on mathematical inference. Englishman John Couch Adams had made the same prediction just a few months earlier [source: StarChild Team]. Other similar scientific achievements followed and fueled numerous technological advances, from steel and electricity to the telephone and telegraph, to steam engines and internal combustion.
But the structured, ordered world of Newton and Laplace was about to be challenged, albeit slowly, fitfully. The first seeds of chaos were planted by another Frenchman and with an analysis of a system that should have been a no-brainer -- the motion of planets.
This gets to a second key concept: uncertainty or scientific error. Even greenhorn Galileos accept the presence of uncertainty when making measurements, but they also assume they can reduce the uncertainty by measuring initial conditions with increasing accuracy. Much of 19th- and early-20th-century science occupied itself with improving the quality of measuring equipment, all in the pursuit of determinism.
The Myth of Measurement
Two important concepts underpin the philosophy of determinism. First, for any given system, the same initial conditions always produce the same outcome. Think of a game of billiards. Describe the initial conditions of the table, quantify the speed and trajectory of the cue ball, and you can calculate, using math alone, what the table will look like after a shot. Set up the table exactly the same way and strike the cue ball in the same spot, with the same force, and the outcome will be the same -- without exception. Of course, this is easier said than done. Duplicating a set of initial conditions is tricky because there's always going to be slight variation in how you set up the table.
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How Chaos Theory Works
By: William Harris | Updated: Aug 20, 2020
Not So Certain After All: Dynamical Instability
By the end of the 19th century, scientists were becoming a little complacent. Newton's laws had proven to be extraordinarily robust, and everyone assumed they could solve any physical problem set before them. In addition to this sturdy mathematical foundation, astronomers were adding more information about Earth and its position in the solar system and beyond. An astronomical chart of 1900 would have displayed the eight principal planets, each in an elliptical orbit around the sun, as well as numerous satellites, the larger asteroids and a handful of comets. The same chart would have provided apparent magnitudes, orbital velocities, diameters and distances from the sun. In other words, it contained all of the information necessary to exploit Newton's equations and determine a future state of the planets.
In 1885, King Oscar II of Sweden and Norway offered a prize to anyone who could prove the stability of the solar system. It may have seemed like an unnecessary quest (after all, the solar system had obviously been stable for millions of years before the 1800s), but it had titillated scientists for years and, at the very least, it provided a means to demonstrate the power of classical mechanics. Several well-known mathematicians, including Leonhard Euhler, Joseph-Louis Lagrange and even Pierre-Simon Laplace himself, had tackled the problem before King Oscar's contest. A few managed to provide proofs of solar system stability, at least in short-term models. But no one had been able to prove, definitively, that the eight planets would stay in a bounded region of space for all time.
Enter Henri Poincaré, a French mathematician already known for innovative thinking before the contest attracted his attention. Instead of focusing on all planets and the sun simultaneously, Poincaré decided to limit his analysis to a smaller, simpler system -- two massive bodies orbiting one another around their common center of gravity while a much smaller body orbits them both. This is known as the n-body problem, which uses complex math to predict the motion of a group of celestial objects that interact gravitationally. That math usually involves differential equations -- equations that give the rate of change of a system as a function of its present state. But when Poincaré tried to describe the present state of the bodies in his simplified calculus, he discovered that small imprecisions -- rounding off a planet's mass, for example -- grew over time and became magnified at an alarming rate. Even when he shrunk the uncertainties in initial conditions to smaller and smaller values, the calculations still "blew up," producing enormous uncertainties in the final predictions. He concluded it was impossible to predict the future outcome of the solar system because the system itself was far too complex, filled with too many variables that could never be measured with absolute precision.
For his work, Poincaré won the contest. But his real accomplishment was to discover something known as dynamical instability, or chaos. It largely went unnoticed for another 70 years, until a meteorologist at the Massachusetts Institute of Technology (MIT) tried to use computers to improve weather forecasting.
A Spot of Chaos
Poincaré found chaos when he tried to calculate the future of the solar system, but studying individual planets has provided additional clues. In the 1980s, astronomers turned to chaos theory to explain Jupiter's Great Red Spot. Earth-bound observers have seen the structure for more than 300 years, but no one could explain why it existed and why it didn't fall apart. Now scientists think that the spot represents "stable chaos" -- an island of structure located inside turbulence and disorder. They came to this conclusion by entering equations for fluid flow, Coriolis force and planet rotation into a computer. When the computer modeled the data, it produced a self-organizing system wedged snugly in unpredictable turmoil, like an island of tranquility in a sea of chaos. (For the record, they've also discovered more spots on Jupiter.)
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How Chaos Theory Works
By: William Harris | Updated: Aug 20, 2020
Of Weather and Wings
It seems a strange juxtaposition today: In the 1960s, NASA was successfully launching astronauts into orbit while weather forecasters were struggling to make accurate predictions. In 1962 alone, two ferocious storms caught U.S. meteorologists with their proverbial pants down. The first, known as the Ash Wednesday Storm, came ashore on March 6 and nearly washed away some mid-Atlantic cities. When the nor'easter finally withdrew, 40 people were dead, and residents from North Carolina to New York faced $200 million dollars' worth of property damage [source: Dance]. The second storm -- the "Big Blow" -- struck the opposite coast on Oct. 12, battering California, Washington, Oregon and southwest Canada with near-hurricane-force winds. The Metropolitan Life Insurance Company declared the storm, which caused $230 to $280 million in damage, the worst natural disaster of 1962 [source: Read].
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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Note: NVIDIA and their iconic CEO Jensen Huang are now planning on using a supercomputer to meet del and predict the climate. In a private communication I pointed out to Jensen that these efforts could fall prey to the “Butterfly Effect” and “chaos”, which would make the climate mathematically unpredictable. Here is our exchange:
Climate Change And Chaos Theory?
I sent Jensen the following e-mail and he responded thus. I'm not really satisfied with his response but at least he had the decency to respond.
Hi Jensen,
I am a very happy NVIDIA investor who got a Ph.D. in mathematics in the 1980’s in Chaos Theory. I find your idea of modeling Climate Change fascinating, but don’t you run into the Butterfly Effect discovered by Edmund Lorentz, a meteorology professor at Harvard, who did a toy model simulation of weather in the early 1960’s and discovered the Lorentz Attractor, a fractal archetype for chaotic behavior in a deterministic dynamical system. This exhibits so-called “sensitive dependence on initial conditions”, with exponential divergence of orbits in the solution space of the differential equations. In brief, he discovered that long-term weather prediction is mathematically impossible!
If you could respond I’d be delighted.
Your devoted fan,
Frank Sullivan
Jensen's Response:
Uncertainty is important in the simulation outcome.
Got some more insights from Jensen.
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Jensen's Response:
Long term weather prediction seems hard.
Long term climate prediction is more possible.
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Frank's Continuation:
Thanks, Jensen.
I’m not really familiar with Climate models. They may be more tractable mathematically than weather models.
Lorenz simplified a few fluid dynamics equations (called the Navier-Stokes equations) and ended up with a set of three nonlinear equations:
When he did computer simulation of the solutions to this PDE (using the ancient computers available in the early 1960’s) he discovered the Lorenz Attractor, which exhibited the Butterfly Effect.
It is possible that the solutions to Climate models do not exhibit this chaotic behavior. As you say, there is always uncertainty, but when there is Chaos present there is so much uncertainty that no prediction is possible due to “sensitive dependence on initial conditions” and exponential divergence of solution orbits in solution space.
Thanks for your response.
Frank Sullivan
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Jensen's Response:
Climate is the average weather.
We are seeking to know the likely hood of increase in extreme weather and to what degree. i.e., high confidence that there will be 3-times more days of ideal wildfire days in western U.S. in 2055 vs 2021.
Much different than sunny with a chance of drizzles on Wednesday afternoon. |
