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By Jeff Chandler "I was betting my entire career that it was not a passing fad." - Roderick Jensen
What is chaos theory? Chaos usually pertains to nonlinear dynamical systems, which are created from nonlinear equations. More specifically, chaos is the term used to describe the behavior of one such system that is thought to be inherently unpredictable. A dynamical system could be any number of things. It could be any chemical, biological or physical process that evolves over time; the swinging of a pendulum; fluctuations in the stock market; changes in temperature; or even turbulence in fluid. Scientists create nonlinear mathematical models in order to study the chaotic behavior of these systems. Their ultimate goal is to predict the seemingly unpredictable future behavior of such systems. First, scientists set up mathematical models of the system they are studying. Then, solving the equations with a series of initial values, they can obtain solutions to these nonlinear equations. Many of these equations are either extremely difficult, or impossible to solve by hand, so utilizing powerful computers is crucial. However, at each stage of the solution process, these computers make approximations by rounding numbers. The slightest error or approximation can eventually lead to a completely inaccurate answer, rendering the solution meaningless. This is what we know to be chaos. Seemingly simple nonlinear equations do not have simple properties. Roderick Jensen, a professor at Wesleyan University was first turned on to chaos theory as a graduate student at Princeton University in 1977 when his advisor handed him a copy of a paper just recently published in Nature by Robert May. May's paper applied the ideas of chaos to population biology, and was actually the first to use the term "chaos" as it is technically used today. Prior to May's article, people already knew that randomness could occur in chemical, biological, physical systems etc. The revolutionary idea in this article, which May outlines in his conclusion, is that evolution of populations (in this case) can be apparently random, even when the underlying equations have nothing random about them. A system might exhibit chaotic behavior purely because there are so many factors affecting that system. In May's population study, climate, food, health, predators, habitat etc. could all factor into the randomness of the population changes. As with most chaotic systems, population growth exhibits a "sensitive dependence on initial conditions." If one were to change the initial level of the population, all other successive populations would change. Also, if two initial population levels were extremely close, their later population levels could become completely different. Robert May initially studied the population sizes of seasonally breeding insects such as orchard pests in which their generations do not overlap. May measured the percentage of some possible maximum population at each generation and compared that with the generation before it. As a model for this equation, May used the nonlinear, or "density dependent" equation Xt+1=F(Xt). However, there are numerous situations other than population regulation, which this equation could apply to. May used the more specific form of the previous equation: Xt+1=a(Xt)(1-Xt) as a model for the population sizes of successive generations of insects. [Xt] represents the population of the generation before generation [Xt+1]. [a] represents the steepness of the curve mathematically, as well as representing the outside influences (food, climate) on population size. May discovered that at lower values of "a," predictable answers could be found, but as the initial values of "a" grew higher, answers became seemingly completely random. This was the first indication of the existence of chaos. Chaos has numerous consequences in mathematics and the physical sciences. First, no matter how precisely one is able to measure the physical answers to an equation, one may never be able to predict with one hundred percent certainty the ensuing answer or motion. Furthermore, it is useless to attempt to calculate specific answers because of the rounding errors that computers may make. Instead, it is better to try for the totality of all possible solutions, since a specific solution may not be obtainable, but the totality of those solutions might be. Totality solutions can be obtained by finding limits to the unpredictability of an answer, rather than the answer itself. For example it may be impossible to tell whether the stock market will rise or fall tomorrow, but we can accurately say that it will not crash all the way down to zero in one day. After finishing his Ph.D. in Astrophysics (plasma physics), Professor Jensen went to Yale University where he was hired as a research associate. At Yale Jensen was introduced to highly excited electrons, which were orbiting the nuclei of hydrogen atoms in a strange fashion. He recognized their behavior to be chaotic, "The electron could be thought of as a planet orbiting the sun, and if you jiggle it hard enough it could become chaotic. Which means that instead of nice elliptical orbits, even slightly wobbling elliptical orbits, the orbits would start to wander around and get longer and bigger and ionize, the planet would be ejected from the solar system." This started Professor Jensen on quantum chaos, because the atoms were actually quantum mechanical, and yet classical chaos could predict when they would ionize. So, quantum chaos was born soon after Jensen became a Professor at Yale, when he published a short article on the possibility of classical chaos in quantum mechanical systems. Chaos had been around for approximately five years by this point, and quantum chaos had been discussed, but this was the first indication of an actual experiment that could study the effect of classical chaos on quantum mechanics. Since coming for a tenured position at Wesleyan University in 1994, Jensen has continued his work in quantum chaos, but has also become interested in chaos in biological systems and economics. Jensen published a paper along with David Sellover on nonlinear dynamics in coupled economies. In the article he outlines how difficult it is to deduce whose economy is driving whose. Jensen was actually able to prove that with two rising and falling economies like the U.S. and Japan, just because the U.S. economy rises first and then Japan follows, doesnít mean that the U.S. economy is driving Japan's. Professor Jensen believed that classical and quantum chaos theory were merely fun to play with, until he realized that chaos could be useful in treating patients with severe epilepsy. Looking at electroencephalogram brainwave measurements for patients having epileptic seizures, he was able to dispel the common medical belief that chaos was bad, and order was good. During a seizure, brainwaves are very normal, whereas chaos occurs naturally all the time in the electroencephalogram measurements. So, Jensen says, "Chaos is healthy, itís the regularity thatís unhealthy." Professor Jensen has recently become interested in what he would like to call "computational molecular biology," however the name has already been taken by molecular biologists who use computers to make models of big molecules. Jensen defines this as the study of how biological molecules process information, and act as a kind of specialized computer. He admits that chaos may not actually occur here, but it is more a process of feedback dynamics, which is close to nonlinear dynamics (chaos). Jensen has been so turned on by this subject recently because there are now techniques to study large scale gene expression with what are called gene chips. With these gene chips, one is able to monitor every gene in your body over time as well as the combination of genes that might cause a particular cell to become cancerous. So, Jensen foresees working with these gene chips for the next decade at least. Jensen believes there is a large future for chaos theory. In fact, in the early years of chaos theory, James Gleick asked Jensen whether or not he should spend his time writing a book on chaos. He replied, "I [have] much more at stake in believing that it [is] more than a passing fad, because I [am] betting my entire career that it [is] not a passing fad." Gleickís book, Chaos: The Making of a New Science eventually became a national bestseller. As Jensen says, chaos is a sexy term for nonlinear
dynamics, and nonlinear dynamics is a new mathematics with
an enormous future and difficult solutions. Recently
computers have become faster and more powerful than before,
enabling scientists to obtain these difficult
solutions. So, there is much more in store for the
study and application of chaos, as well as for Jensen to
work on. He states, "I am always looking for
chaos." Classical Chaos. Roderick Jensen in American Scientist, Vol. 75, pages 168-181; March-April, 1987 Simple Mathematical Models with Very Complicated Dynamics, Robert M. May in Nature, Vol. 261, pages 459-467; June 10, 1976 The Ubiquity of Chaos. Saul Krasner, American Association for the Advancement of Science, Washington D.C. 1990 Chaos in Dynamical Systems. Edward Ott, Cambridge University Press, Cambridge 1993
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