Chaos Theory
...e, the population suddenly fluctuated between two values. As the rate grew, two populations became four, four became eight and soon, chaos and unpredictability erupted. It became impossible to predict populations. But once again, in the mess of unpredictability emerged order. May found certain windows where suddenly, populations WERE predictable. A certain number: 4.669 predicted exactly when the graph would bifurcate (double) again. Benoit Mandelbrot is perhaps the most recognizable figure in chaos theory for his outstanding discoveries. Mandelbrot is best known for his work on fractals. He was inspired by the works of Gaston Maurice Julia. In 1918, at age 25, Julia published a masterpiece paper dealing with the repeating iterations of functions (“The Chaos Theory” http://www.duke.edu n.p.). In the 1940’s Mandelbrot was introduced to Julia’s work, but it did not impress Mandelbrot much (“Chaos Theory” http:// www.imho.com n.p.). Only in the 70’s with his work on fractals, did Mandelbrot return to the paper with interest. He showed that Julia’s equations delivered some of the most beautiful fractals. In fact, Mandelbrot’s own fractal is the pinnacle of his mathematical achievement and has awarded him with many honors. One topic not quite as deep, but related to chaos theory is emergent behavior. This was first proposed by Evelyn Fox Keller who showed that a complex activity by slime mold cells (which had originally been thought to have been the result of a pacemaker who would lead the cells) was nothing more than a simple bottom-up consequence of the situation. There were no leaders or pacemakers, just at the right temperature and humidity, the chemicals released by the cells would be enough to spontaneously trigger an event (Johnson 18-19). Today, as mathematicians delve further into the realms of chaos theory, they uncover an astonishing and endless world. It has spawned many fresh, exciting ideas and has implications that challenge some fundamental roots of science. Chaos theory covers many topics. It covers sensitive complex systems (what Lorenz discovered), but it also covers the opposite: drawing order from randomness. Another stunning branch of chaos is the concept of emergent behavior and self-organizing systems in which a group of incapable, low-level objects collectively form a higher-level smarter system spontaneously. Yet another aspect of chaos is the production of beautiful fractals. Why can’t weathermen predict accurately predict the weather two weeks in advance—even with powerful computers and precise weather measurements? Why can’t stock brokers—spending hours pouring over data and analysis—react in a way that will always avoid economic recession? Using chaos theory, mathematicians shown that complex systems (including stock markets and weather conditions) are so sensitive to every detail of initial conditions, that we can never predict the exact outcome of the system at any future time. Practicality limits us from ever knowing the exact condition of the system at any time. As Lorenz proved to us, any miniscule difference will make exact predictability impossible. In time, this phenomenon became known as the famed butterfly effect: “A butterfly flapping its wings causes a small change in the atmosphere. Within time, that change causes the atmosphere to diverge from what it would have done. In a few months, a tornado that would have devastated Indonesia doesn’t happen. Or one that wouldn’t have happened does” (Chaos Theory” http:// www.imho.com n.p.). Another example of chaos theory is found in this ancient folklore: “For want of a nail, the shoe was lost; For want of a shoe, the horse was lost; For want of a horse, the rider was lost; For want of a rider, the battle was lost; For want of a battle, the kingdom was lost!” (“The Chaos Theory” http://www.duke.edu n.p.) That does not mean that we cannot make educated guesses in the stock market. In fact although chaos tells us how something ordered and seemingly predictable equations may fall apart into randomness, it also tells us the opposite: that from something random and—by all appearances—unpredictable, order can be derived. This was shown by the 3-d model Lorenz got when he plotted his data. It was definitely ordered, the graph kept repeating itself over and over, despite the fact that it came from unpredictable, random data. In a less involved—but related nevertheless—topic of chaos, emergent behavior stands out as one of most fascinating subjects. As exemplified in the termite nest construction, emergent behavior is about the bottom-up intelligence of a huge mass (a swarm) of dumb, low level creatures. Emergent behavior is amazing. An ant colony (not a termite colony but governed by the same phenomenon nevertheless) will know how to optimize space, clear a cemetery for their dead, dumb trash in a certain place all with perfect precision and optimal distance. Yet incredibly, we know that there is no direction or leadership involved (Johnson 18-19). All the previous aspects of chaos seem deeply mathematical and abstract. But the this branch is vastly different. Fractals are probably by far the most well known part of chaos. Known for their beautiful shapes, fractals are images generated by powerful computers from mathematical equations. Many people, having no care for the strong mathematics behind it, have been fascinated by the various pretty pictures of fractals. The Mandelbrot set and Julia sets are among the most recognizable fractals. The equations behind fractals are nothing more than simple equations. Interestingly, the most complex and beautiful fractals are also the ones with the simplest equation. That is how it relates to chaos: simple equations generating wildly fluctuation data, but ordered in the sense that it all unites together to form a beautiful picture. But as aesthetic as they are, there are mathematics behind them. The basis behind fractals, are iterating functions: take a number, plug it into a function, take the result, plug it into the same function, repeat to infinity. That’s all. And yet the result is not only pretty, but weird as well. For example, fractals are not 1 dimensional objects, nor are they two dimensional, or three dimensional. In fact, they are IN BETWEEN. A fractal known as the Sierpenski triangle has a dimension of 1.585 (Math n.p.). The dimension of a fractal known as Cantor’s Dust has a dimension of 0.631. All this may make it seem as if fractals are abstract mathematical oddities, with nothing more important than a pretty image. But it is more. Fractals make up the world as we know it. So how does fractals—and chaos in general—relate to the world? “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lighting travel in a straight line”—Mandelbrot (“The Chaos Theory” http://www.duke.edu). In almost all school math courses, kids learn about perfect shapes: spheres, cubes, and triangles to name a few. But the real world has no such shapes. There are no perfect curves. Everything is “wrinkly” and rough. People may spend their whole lives finding more digits to pi (for a circle), but it is useless since there is no such thing as a perfect circle in nature. This is where chaos comes in. Chaos theory is embedded into almost every aspect of our lives, even though we don’t know it. Fractals especially, constitute almost everything in the world—even us. The key is self-similarity. A magnified view of fractals shows exactly the same structure on a smaller scale embedded into the overall picture. Mountains are shaped as fractals: take rocks by the mountain side and one can see the similarity in contour and structure of both. Human lungs (the inside branches) have at least 15 levels of self-similarity. River bends are echoed in sand formations beside it (Math n.p.). Blood vessels, tree branches, stock market graphs—they are all examples of fractals. Understanding fractals will surely give us an understanding of all these systems. Researchers have discovered a set of three equations that graphed a fern perfectly. This has had impacts in biology, since it implies that perhaps DNA stores simple fractal equations rather than a map of where every single cell goes. Computer art has become more realistic using chaos and fractal equations. For example, instead manually creating 3-d models of trees (a painstaking process, forming every branch and leaf at precise angles and such forth), now programmers simply insert a simple formula that creates beautiful, realistic trees (Chaos Theory” http:// www.imho.com n.p.). Stock markets are chaotic and graphs of it are actually fractals. The graphs are self similar—graphs of stock within a day are the same as graphs of months and years. In fact, applying the mathematics of chaos theory to the stock market turns up profits. There are mathematicians who invest according to the laws of the theory. While it is true that chaos theory (ironically) renders in-day trading useless because of its random unpredictable effect, it also predicts order to the stock market over the long term. That order is what the theoreticians are lo...