Some scattered notes on Chaos Theory

I am just summarizing what I learned from a book on chaos theory. Tomorrow, or soon, I will try to apply what I learned to the development of literary style.

Daniel Christopher June

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I recently read the book “Chaos: Making a New Science” by James Gleick. This books is written for the layman, and readily understandable without extensive knowledge of science. Gleick tells a lot stories, filling in biographical sketches and anecdotes of the many scientists who worked to define what is now known as “chaos theory.” Nevertheless, despite the reader level the author intended, I feel I did not grasp everything he had to say, so I will only share those few points I think I do grasp, and then lay out how I think they can be used to enhance a writing style (in a later email).

Chaos theory began with “the butterfly effect.” Put less metaphorically and more scientifically, the butterfly effect shows that in complex systems, where there are many variables, such as in the weather on earth, initial conditions determine how things are going to look later on. The scientist made a toy climate on his computer to calculate how the weather would develop. After collecting the data, he tried it again a few days later. The computer had changed some of his numbers, rounding up on decimal points, making something akin to a wind breeze of 12.23400000000126 mph turn into 12.23400000013 mph. He had accidentally discovered that the variables involved in weather control are so minute and dispersed, that even if we had weather equipment at every five feet over the whole globe, we could not predict the weather next week, for those spaces between the devices would have data that effected the results.

At one point, it was shown that the sensitive equations involved with complex systems can be greatly changed, even if the change is nothing more than the gravitational pull of a raindrop five miles away from where the experiment is happening.

We now believe that weather prediction is impossible, beyond the hit or miss stuff we already have.

Note that chaos theory is a science allowed by computers. The models and paradigms of chaos theory involve complex calculations that could not be drawn out by a human being. Not that the equations are difficult --- even a highschool senior who had some knowledge of differential equations could do them ---- but the math involves endless iterations of the same equation over and over again. I will try to explain more…

The sensitive dependence on initial conditions led to some further discoveries. Lorenz, the scientist who in studying weather, accidentally discovered this behavior, put weather aside and studied this. He discovered what has become known as the “Lorenz attractor.” It’s like this: when a data set is chaotic, that means if you graph it, you can see no pattern, let alone make a formula to express it. But if you graph things a little different, make a graph that accounts for periodicy, you will see surprising patterns emerge. Though the data is nonpredictable, it is patterned, the data point loops around attraction spots. The new graphs replace “time series graphs” with “trajectories in phase space,” which are fully contained in the graph, give the full picture, do not go onwards off the graph into infinity.

In every day language, they were able to look at the data differently, and suddenly it was discovered that chaos is ordered. These new graphs allowed the view to understand the equation all at once: all the info is in one picture.

Mandelbrot is a scientist who “discovered” the fractal, or a mathematical model in which the shape of the whole is repeated in the parts. Some of these graphs are beautiful, I suggest you look them up online. We call such fractal shapes “self similar.” If a coast line is variegated on a map to look rough and irregular, imagine that rough pattern being the same on every level, down to the microscipice, infinitely. No matter at what level you magnify it, it looks the same. Strangely enough, a circle with a fractal circumference has an infinitely long circumference, though it is only two inches long! These were not invented by Mandlebrot, but scientists are always playing with mathematical toys that later prove useful. Such a one is the “Sierpinski carpet, constructed by cutting the center one-ninth of a square, then cutting out the centers of the smaller square that remain; and so on. The three-dimensional analogue is the Menber sponge, a solid looking lattice that has an infinite surface area, yet zero volume”

All this turns out to be most relavent to just about every physical system. There is no part of your body where a cell is more than two cells away from a blood vessel. Yet the vesicles take up less than 5% of your body. How could our DNA have made such a complex system through the body? With a few basic rules that repeat again and again. The lungs too are fractically structured, so that your lungs contain a bigger surface area than your typical tennis court.

This applies to how lightning bolts form when they come to earth, in fractal patterns. In fact, as the Fibonicci golden ration was discovered to be all over in nature, from sea shells to sunflowers to galaxies, so are fractal shapes all over in nature. A dynamic system is one in which the variables of at least three differential equations create an infinite behavior. It appears random and unpredictable. It has been found the explain the behavior of the twitching of the eyes of schizophrenics, the orbiting of three suns around each other in three dimensional space, in heart fribulations, etc.

Fractal energy is a engineering problem. Turbulence is the energy lost in a lot of machines. Small scale motion drains energy. When you push an oar in the water, there are pockets of turbulence in which the water makes big whirls, and in those big whirls, smaller ones, and smaller, and smaller till it gets to the atomic level. That disperses energy, so that it is wasted. This is important for airplanes, because turbulence is chaotic and can crash a plane. Turbulence is disorder at all scales. Yet these systems can be graphed with incredible accuracy using a “strange attracter” or two dimensional graph that can looks like an owl mask with lines going in looping behavior.

The author gives a detailed biography of Mitchell Feigenbaum, a scientist who added a lot to this field. He was a detached intellectual as a kid, who avoided other kids and thought a lot. He would take 4-5 hour walks to think out his ideas, listened to the music of Mahler, and identified with Goethe’s Faust, his favorite book. When he got to using computers to figure out his equations, he would spend 20 hour days for months exploring these things, looking for keys behind them. He loved Goethe because Goethe, unlike Newton, was wholistic, and did not aim to “divide the light into a spectrum.” He was inspired by other outmoded types of science and seemed to find what he need to figure out his equations. To say the least, his ideas were rejected again and again. Yet new discoveries are discovered not by specialists or committees, but individual passion.

Much of his work was taken up by Libchamber, another eccentric scientist who believed in the great Man theory of history, loved Goethe and was obsessed with old books; and unlike his fellow scientists, he was not a communist. Libchamber found ways to experiment with and verify Feigenbaums equations. It explained the growth of ameba in a dish, the growth of a crystal, of rivers. It was discovered that boundaries are the most interesting places to study. When one equation imposes on another, or when system on another, interesting things happen.

Many other mathematicians contributed, before straight physicists joined in: “Simple equations often produce ellipses, parabolas, and hyperbolas of conic sections or even the more complicated shapes produced by differential equations in phase space. But when a geometer iterates an equation instead of solving it, the equation becomes a process instead of a description, a dynamic instead of static.”

It was discovered that from very simple equations, if iterated, infinite worlds could be created, and if looked at graphically, each layer could have its own organization, patterned yet unlike all the others, so that from simplicity came infinite complexity and variation. Not all these systems needed to be self similar. Some could be infinitely unique, yet patterned and structured gracefully, based on only a few equations.

These types of equations can explain otherwise impenetrable data sets. They explain how the immune system organizes itself and how sugar is converted into energy by yeast, the patterns of growth..

Reality doesn’t follow “linear equations” so much as “dynamic equations.” When an object is in motion, it creates friction in the world. The friction is not constant but related to how fast its going, so that to measure anything accurately would be impossible. Not only are there many variables, but the variables change each other. It’s like playing a game whose rules change depending on how you place it.

“Was chance necessary for biological systems? Hubbard, [a biologist studied in this book] thought about the parallels between the Mandelbrot set and the biological encoding of information, but he bristled at any suggest that such processes might depend on probability. “There is no randomness in the Mandelbrot set,” Hubbard said. “there is no randomness in anything that I do. Neither do I think the possibility of randomness has any direct relevance to biology. In biology randomness is death, chaos is death.”

A group calling themselves “the dynamic system collective” worked together hippy style to explore chaos in different systems, though they were grad students and got no funding to do so. The created many relevant discoveries on how chaos relates to physics and biology, and had to scratch their way into getting professional acceptance.

(The author shows how strange attracters are engines of information, and he goes into information theory, talking about how chaos is ordered and what this means for the second law of thermodynamics.)

The Collective did an experiment on a leaky faucet, noting how often a water dropped. By graphing it in this manner: let X be the time since the previous drop, and Y the interval between a drop and the next drop, he made a two dimensional graph. After adding the Z coordinate as X + two drops he got a three dimensional graph that resembles a Lorenz attractor, or in other word, the chaos of a dynamic system.

“The paragon of a dynamic system and to many scientists, therefore, the touchstone of any approach to complexity is the human body.” The system was applied to the human heart. “Modeling any one piece of the hearts behavior would strain a supercomputer; modeling the whole interwoven cycle would be impossible.” Yet dynamic systems were discovered in the heart. The pressure in each valve changes the thickness and flexibility of the walls, which add different pressure based on how much is in them. Artificial valves create areas of turbulence and areas of stagnation, where blood clots get formed, which can lead to stroke. “The mathematicians found that the heart adds a whole level of complexity to the standard fluid flow problem, because any realistic model must take into account the elasticity of the heart walls themselves. Instead of flowing over a rigid surface, like air over an airplane wing, blood changes the surface dynamically and nonlinearly.

Mental disorders and sleep irregularity have been shown to be dynamic systems. (He talks about mode locking of the moon and other systems, so that its face always is near earth, and satellites tend to spin in whole-number ratios of their orbital period 1 to 1 or 2 to 1. Etc, and says that “biological equilibrium is death.”)

He summarizes his work saying that simple systems give rise to complex behavior. Complex systems give rise to simple behavoir. And that the laws of complexity hold universally, caring not at all for the details of a systems constituent atoms.”

Daniel

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