Ever since the dawn of the Age of Reason when scientists began to unravel the mysteries of the natural world, virtually everyone has come to accept the notion that future events are predictable, as long as we have enough information. After all, it was this very premise that triggered technological advances that set mankind on the road to Progress and Prosperity. Hence, the Industrial Era was marked by a widespread optimistic belief that the world around us -- and indeed society itself -- would become more and more manageable. NOT QUITE!
Join us now, won't you, for this "magical mystery tour" of a new-fangled body of science that "subverts the dominant paradigm" while offering hope that humanity can yet find new ways to adapt to a world of diminishing resources without sacrificing freedom and individual responsibility.
NOTE: (Some of the following text is excerpted from my dissertation, Between Power and Poverty: A Study of Political-Economic Adaptation and the Autonomy of Emerging Nation-States, With Special Reference to Peru, by Andrew G. Clem, University of Virginia, copyright 2002. All rights reserved.)
"Chaos theory" is the popular term used to describe a novel, quite revolutionary approach to a wide range of mathematical, pure science, and applied science fields. Other people prefer the terms "complexity theory" or "dynamic systems theory." It purports to be a "new paradigm," that is, a comprehensive framework to guide and evaluate research and analysis that transcends the limitations of conventional ("normal") science. The fundamental lesson of chaos theory is that the behavior of a wide range of "dynamic systems" (e.g., the atmosphere, the solar system) is extremely sensitive to minute fluxes in initial conditions, thus making it virtually impossible to obtain accurate medium- and long-term predictions. Although this finding seems quite discouraging, in fact it opens the door to solving or coping with a wide range of heretofore vexing problems.
Most people who have heard of Chaos theory associate it with the idea that tiny disturbances can cause enormous long-term differences. In the movie Jurassic Park, for example, the actor Jeff Goldblum waxed manic about the "butterfly effect," the idea that a butterfly flapping its wings could set in motion a sequence of self-reinforcing events that would ultimately result in a hurricane on the other side of the world. While within the realm of (distant) possibility, this misleading image has had an unfortunate effect, reducing Chaos theory as conceived by most people into a generic cliche for "the world's just a big, unpredictable mess." One often hears pundits invoking chaos theory to complain about frustrations with daily life, but that really does a disservice to the task of bringing the important insights of chaos theory to a wider audience.
Most scientific and quasi-scientific disciplines are grounded in what is known as the "Newtonian mechanistic" paradigm, which developed from the 16th to the 19th centuries thanks to brave scientists such as Galileo and Isaac Newton. Before their discoveries, engineers and astronomers had relied upon the theories of Ptolemy and Archimedes, which were incredibly complex and difficult to apply in novel situations. (For example, under the earth-centric cosmology of Ptolemy, the irregular observed movements of the planets were attributed to bizarre loops in their orbits, called "epicycles.") The law of gravity and subsequently discovered laws of physics allowed men to achieve a radical simplification in understanding, and the universe came to be conceived of as a vast clock mechanism. A French scientist named La Place actually envisioned a utopian future in which all future events could be forecast ahead of time. In the 19th Century, Auguste Comte led in the creation of Positivism, a science-based social philosophy that expects steady improvements in the ability of experts to predict future natural and social phenomena -- and thus manage the world around us. "Order and progress" is the basic creed of positivist philosophers, and indeed, that very phrase is emblazoned on the flag of Brazil.
To modern people, those Newtonian scientific principles were simply common sense, but during the Medieval Era, they would have been considered blasphemy. Then in the 20th century Newton's laws were challenged by Einsteinian relativity and Heisenberg's quantum mechanics, which called into question the very notions of matter and space as absolute entities. The point is that all science is imperfect and therefore eventually is bound to be superseded by something better. Thomas Kuhn, a philosopher of science who died recently, coined the phrase "normal science" to denote the conventional approach to research. He was one of the first to explicitly acknowledge that scientific pursuits are influenced by sociological factors -- prevailing customs tend to shut out "fads" (otherwise there would be no reliable standards for determining truth), but this also imbues scientists with a "deaf ear" to potentially worthy discoveries.
Although chaos theory has roots in early 20th century mathematics and physics, it really began to emerge in the 1960s. Thanks to the advent of computer technology, mathematicians and scientists in a variety of fields discovered that simple "deterministic" forces give rise to astonishingly complex forms in nature, something that the mechanistic Newtonian paradigm cannot explain. It soon became clear that their respective controversial research projects all had strong common universal underpinnings. By the 1980s these insights began to be applied to analyzing seemingly indeterminate real-world outcomes and even sudden unanticipated catastrophes such as earthquakes, plagues, and even stock market crashes. The following sections cover some of the main aspects of chaos theory: nonlinear dynamics, fractal geometry, complexity, turbulence. The latter sections are not yet finished, and all sections of this page are subject to future correction, amplification, and clarification.
Edward Lorenz was a pioneer in using computers to forecast weather trends, but in 1960 he accidentally stumbled across an anomaly in one of his simulations when he re-entered the input data using rounded figures. To his amazement, the results soon diverged far from those of the previous simulation, even though they were based on almost exactly the same numbers. This flew in the face of everything he knew about mathematics, and he drew a broad lesson from it: in certain kinds of nonlinear equations, or systems based on simultaneous nonlinear equations, results display sensitive dependence on initial conditions. This term, known popularly as the "butterfly effect," eventually came to be one of the hallmarks of chaos science, but it killed Lorenz's dream of making long-term weather forecasts, and for several years it failed to draw widespread attention. Then in 1975 mathematician James Yorke published a paper based in part on Lorenz's work, "Period Three Implies Chaos," bringing the stunning implications of nonlinear behavior to a much broader audience.
At about the same time, Australian physicist Robert May was modeling population fluctuations, using the nonlinear "logistic equation"
y = rx(1-x) , where y equals the preceding value of x (ranging from 0 to 1) in an iterated series
The virtue of using this form of an equation (in which one quantity is multiplied by its reciprocal) in population studies is that the results are "environmentally" self-limiting. (This is in contrast to the unrealistic exponential equations used by British economist Thomas Malthus, who fretted that the Earth's human population would grow without limit.) This logistic equation underwent period doubling (meaning that the results shifted from a constant stream to two cycles, four cycles, etc.) as the r parameter was increased, and to May's surprise, it yielded chaotic results when the parameter r (ranging from 1 to 4) crossed a certain threshhold -- about 3.58. This was clear evidence of deterministic stochastic (random) behavior, a seeming oxymoron! (See the "Try this at home" exercise below the following graphs.) The following charts display results for the parameter values that yield, respectively, two-period, eight-period, and N-period (chaotic) cycles:
A careful look at this last chart will reveal repeating patterns, hinting at a subtle order within this chaos, but every apparent complex cycle is slightly different than the preceding one. That raises the tantalizing question of whether the similarities are part of some larger-scale pattern. In fact, as one can see by repeating the calculation many times, they ARE. The graph below plots the calculated results of over 150,000 iterations (!) of the logistic equation, with parameters ranging in value from 2.5 to 4.0, based on an arbitrarily selected initial x value of 0.6.
Several features of this graph are worth mentioning. First, it illustrates the phenomenon of "bifurcation," which is what happens when a smooth stream of output data suddenly splits into two paths. This happens when the value of r passes the critical values of approximately 3.0, 3.4, and 3.56. Second, you will notice small bands of white space in the midst of the cluttered chaotic region on the right side of this graph. In fact, these are "windows" of order that emerge when the outer-edge "shadows" cast by each cascading branch converge with the "shadows" of other branches. The broadest such "window" is where a stable three-cycle pattern briefly prevails, and then cascades into a 6-cycle, 12-cycle, etc. How the heck do you get from a 2, 4, 8, etc. series to 3??? Well, that was what James Yorke's 1975 paper was all about. Third, you will notice that the width of the interval of each cycle is successively smaller. In fact, as physicist Mitchell Feigenbaum discovered, the intervals diminish at a constant rate -- which he calculated to be 4.6692016090 -- for ANY such system, regardless of the specific input values! This newly-discovered irrational constant number, like pi (3.1416...) or the natural logarithm e (2.71828...) was clear evidence that chaotic behavior in a wide variety of situations was an aspect of nature that was universal in scope. Feigenbaum's proof that chaos was universal brought the various strands of research into nonlinearity into a more or less coherent whole, marking the true emergence of "chaos theory."
You can experiment with the logistic equation to see for yourself how adjusting the r parameter leads to a transition from order to chaos. Begin with a blank spreadsheet, and enter the r parameter (which can range from 1 to 4, but the results are only interesting when it is 2.5 or more) in cell A2. Then enter the x parameter (which can range from 0.01 to 0.99) in cell B2, and finally enter the logistic equation as indicated in cell C2:
|1||r||x||0||=C1+1||[FILL RIGHT] ...|
|2||2.5||0.9||=$A$2*B2*(1-B2)||[FILL RIGHT]||[FILL RIGHT] ...|
|3||=A2+.01||=B2||=C2||=D2||[FILL RIGHT] ...|
|4||[FILL DOWN] ...||[FILL DOWN] ...||[FILL DOWN] ...||[FILL DOWN] ...||[FILL RIGHT] ...|
(The dollar signs are the standard spreadsheet notation for absolute cell references.) Next, enter the incremental equation "=C1+1" in cell D2, select that cell, and drag all the way to the right side of the spreadsheet, release the mouse button, and choose the "Fill right" command from the appropriate menu. (NOTE: The specific commands may vary slightly from one spreadsheet program to another.) Likewise, select cell C2 (which contains the logistic equation), drag all the way to the right, and choose the "Fill right" command as before. The calculated results will appear in Row 2. In a spreadsheet that is 100 columns wide the right-side column will be labeled "CV."
To see the graphic results of your calculations, select the entire row of cells from C3 to GR3, and choose the "Make chart" command from the appropriate menu. By altering the value in cell A3 you will see some fascinating effects, especially as it crosses from 2.9 to 3.0, then around the 3.45 threshold, then around the 3.56 threshold, etc. What is happening is that the equation results are "bifurcating" (splitting into pairs) at higher and higher cyles -- 1, 2, 4, 8, 16, etc. After 16 cycles it gets so complex that it is effectively chaotic, even though the results are derived from a strictly deterministic algorithm! If you followed the directions correctly, your results should look like the charts found above in this section.
If you are really curious and have a fast computer with plenty of memory, you can recreate the massive Logistic equation summary map shown above. Enter the incremental equation "=A2+.01" in cell A3, then drag the entire third row of cells (A3..CV3, or whatever) down at least 200 rows, and choose the "Fill down" command. It may take several seconds or more for the calculations to be peformed, depending on your hardware and the number of columns and rows in your spreadsheet. Finally, select all cells from B3 to CV500, and choose the "Make chart" command, choosing the x-y graph option. The resulting chart may appear very crude, so you may have to enlarge it and/or fiddle around with various chart options to see some of the fascinating details. (In a spreadsheet with 100 columns and 200 rows there will be a total of 20,000 cells. If your computer is extremely fast, you can a spreadsheet with up to 300 columns and 1000 rows, in which case there would be 300,000 cells.)
A fractal is a geometric shape with patterns that repeat themselves without limit at smaller and smaller scales. The word "fractal" was coined by Benoit Mandelbrot, a Polish-born mathematician who discovered that in any given stream of data (such as the voice signals on a telephone line), the relative frequency of errors is constant regardless of the interval that is measured, long or short. In other words, stochastic disturbances did not conform to a bell-shaped curve whose breadth depended on the accuracy of the measuring device, but rather showed the same degree of irregularity the closer the obsever looked. Based on this insight, he became a pioneer in devising superior error-checking and data-compression algorithms that ultimately made possible reliable long-distance digital communication -- and thus the Internet itself! The infinite self-similarity of fractals give them the bizarre quality of having a fractional dimension -- that is, more than two dimensions (like an flat shape in plane geometry) but less than three (like a solid object in the real world). How weird is THAT?
The crude stick image to the right is the "seed" of a linear fractal. By rolling the mouse over the links below you can see how the principle of self-similarity gives rise to fascinatingly complex figures. Nevertheless, the repetitive quality of linear fractals is self-limiting. NON-linear fractals, in contrast, yield amazing patterns of infinite complexity. This is where the real fun begins, as you'll see below.
During World War I, French mathematician Gaston Julia began to draw graphs of equations based on complex mathematics (see next paragraph). Following up on Julia's work while doing research for IBM in 1979, Benoit Mandelbrot caught the first glimpse of the extremely complex figure shown on the right. He created what became known as the "Mandelbrot set" as a mere catalog to derive all possible "Julia sets," but it soon won recognition as a historic breakthrough in mathematics.
As improved computer technology became available during the 1980s, the incredibly rich depth and intricate texture of this abstract, totally unreal shape were revealed to human eyes. The Mandelbrot Set is plotted by a fairly simple procedure on a modified version of the standard x,y Cartesian plane. This allows us to pinpoint complex numbers, which are composed of real numbers plus imaginary numbers, which are multiples of "i" -- the square root of negative one, which is of course impossible but which nevertheless serves real purposes. OK? One example of a complex number is 3 + 4i; the real and imaginary parts are like apples and oranges and must always be kept separate during addition and subtraction, though they can be multiplied. (By definition, of course, i squared equals -1.) Now, if you pick any such complex number and replace the real component (x) with x square minus y squared plus x, and then replace the imaginary component (y) with 2yx + y, and then repeat the algorithm several times, the result will either decline toward zero or increase toward infinity. In the former case, the complex number x,y is said to be within the Mandelbrot Set, indicated by the black interior area. Otherwise, it is outside the Mandelbrot set, and the colors on the chart below represent the number of iterations before the result rises to infinity. The closer you get to the extremely complex edge of this curve, the harder it is to calculate whether a given x,y coordinate point is insider or outside, requiring high-speed, sophisticated computers -- such as my iMac.
Even a casual observer will note that the patterns in the above shape seem to repeat themselves, ad infinitum. Upon closer inspection, however, it will be seen that each iteration of a given pattern is slightly different than the adjacent one. In fact, there is NO limit to these modified iterations, which means that the Mandelbrot set is infinitely complex! With a program such as "MandelMovie" (TM), one can "zoom in" to a particular part of the curve and explore the nooks and crannies to one's heart's content. This next image demonstrates this by zooming in on a particular part of the Mandelbrot set, on the lower right side. (You might be able to pinpoint its location by matching the color patterns.) This close-up reveals, lo and behold, a tiny version of the original Mandelbrot set tilted at an angle. Like the billions and billions of galaxies up in the heavens (apologies to Carl Sagan's spirit), there are billions and billions of variations of the Mandelbrot set as one looks closer and closer.
One fascinating pattern that was discovered is that the ratio between the width of each iteration of the main "cardoid" shape of the Mandelbrot set precisely matches the ratio between the intervals of the successive bifurcation points (3.0, 3.45, 3.55, etc.) in the logistic equation summary map, which was shown in the previous section. This is the Feigenbaum constant (4.6692016090), which was discussed in the previous section. Since the calculation algorithms used to derive the Logistic equation summary map and the Mandelbrot set have absolutely nothing to do with each other, this is strong evidence of universal principle underlying mathematical chaos, and likewise has been discovered to be a fundamental aspect of the natural world.
Even more amazing and varied nonlinear fractals can be created by calculating the Julia set for any given x,y coordinate in the Mandelbrot set. Julia sets corresponding to points within the Mandelbrot set (i.e., the black region) consist of shapes whose parts are fully interconnected, while those outside the Mandelbrot set consist of infinitely fine, though patterned scattered "dusts," and those near the complex edge of the Mandelbrot set are tantalizingly intricate shapes and it is very hard to determine whether or not their parts are interconnected. To see many more detailed Mandelbrot set and Julia set images, go to hypertextbook.com.
What about the real world? Well, the very same phenomena of period-doubling, chaos, and fractal geometry have been observed in a variety of experimental settings, and some have already been usefully applied to solving real-world engingeering problems. In 1971 physicists David Ruelle and Floris Takens published a paper on fluid dynamics, claiming that turbulent flows tended to conform to a strange attractor, a form of fractal existing in an abstract realm known as phase space. This was the first time that turbulence was interpreted as the mechanism by which a dynamic system conserved internal order by exporting entropy to the outside environment. In 1977 Albert Libchaber devised and carried out the first set of experiments that confirmed the existence of period-doubling behavior leading to chaos in liquid helium convection currents. About the same time a group of physicists in Santa Cruz led by Robert Shaw began conducting experiments on various mundane processes (such as waterwheels) and applied the tool of phase diagrams to distill the flow of information from them. Other researchers studied heartbeat data and concluded that healthy hearts do not beat in precise intervals like a clock but in fact embody a degree of dynamic irregularity, in a fractal pattern.
The fact that simple deterministic formulas can give rise to extremely complex results hints at a converse conjecture: that seemingly chaotic data gathered in the real world can be analyzed to derive simple input conditions. Subject to certain conditions, time-series data can be transformed in such a way that their underlying self-correcting tendencies can be highlighted. The original data are plotted on the x axis of a graph, while the time-lagged data are plotted on the y axis, to allow quick comparison. In sum, the finding that many forms of complex real-life behavior conform to strange attractors offers the best hope that chaos theory can be put to good use in guiding scientific research.
Once you learn how easy it is to disentangle order from disorder in seemingly random data, you'll gain a greater appreciation for the many ways that chaos theory can be applied.
Closely related to chaos theory and complexity is the field of research into the spontaneous emergence of order from chaos as manifested by lasers, crystals, or (possibly) market equilibrium. Laboratory experiments have confirmed the finding of mathematical models that the micro-level chaos in nonlinear dynamic systems embodies remarkable latent self-organizing tendencies under which behavior converges toward an infinitely complex "fractal" pattern known as a "strange attractor." In 1977 Ilya Prigogine won the Nobel Prize in Chemistry for demonstrating the existence of far-from-equilibrium dissipative structures that conserve internal order by exporting entropy. He was among the boldest champions of the new paradigm, and may have lost favor and/or credibility with some people by going too far too fast. Stuart Kauffman demonstrated how matrices filled with cellular automata (abstract mathematical entities with pre-programmed behavior) can organize themselves into recognizable patters, without any outside guidance. This led to attempts to programm computers with "artificial intelligence" and sparked a controversial round of speculation about the origin of human consciousness.
The applicability of this phenomenon is a matter of sharp dispute. For example, in his book The Cosmic Blueprint, Paul Davies reflected on Lovelock's notions of Gaia, noting that Earth's temperature has remained relatively stable over the millenia even as the sun's output of energy has climbed by as much as 30 percent. Thus, global warming may a real but transitory phenomenon of self-regulation, not an inexorable trend of doom. (See blog post of Sept. 6, 2006.)
In the mid-1980s Murray Gell-Mann and other chaos-oriented scientists founded the Santa Fe Institute, which began holding conferences and sponsoring new research projects. More and more scientists realized that complex rhythms were a necessary foundation of any vital, adaptive, self-correcting system. Chaos theory was already moving from the realm of pure numbers and inert matter toward the messy and controversial domains of biology and sociology. Economist Brian Arthur (1989) found that when production functions in an industry display the property of increasing returns, small contingencies may have decisive long-term evolutionary implications, and the industry may become "locked into" suboptimal technologies such as the VHS video tape format. In particular, some researchers have used it to model the adaptive learning behavior of agents in complex systems, echoing the research programs of Karl Deutsch and Freidrich von Hayek. As Mitchell Waldrop (1992) put it, successful evolution depends on flexibly channelled bottom-up organization, as agents are inexorably induced to move along the precarious edge of chaos in the direction of greater complexity.
If such incredibly complex results can be obtained from fairly simple inputs, many of our suppositions about the way the real world works need reconsidering. Alvin Saperstein (1984, 1988) was one of the first social scientists to apply chaos theory to understand nuclear arms races and the outbreak of war. Furthermore, Kiel and Elliott (1997) suggest that the implications of chaos make the hypothetic-deductive scientific procedure of theory development rather problematic. Beyerchen (1992) applied these insights to interpret the classic Prussian military theorist, Karl von Clausewitz, suggesting that the "fog of war" was a metaphor for the chaos that emerges in dynamic systems of mutually escalating intensity. Applying Arthur's insights of increasing returns to historical analysis, Poirot (1993) argued that the transition from feudal to capitalist property relations in England could not be specifically traced to market expansion, class conflict, or demographic flux alone, but rather to a "contingent concatenation" of these processes embedded in a complex social network of self-reinforcing mechanisms. One implication of such "path dependence" is that a nation-state may likewise become locked into a polluted industrial establishment or a backward, dependent economy. That may be considered as a macro-scale manifestation of the suboptimizing behavior studied at the micro-level by John Nash (subject of the movie A Beautiful Mind, starring Russell Crowe) and other game theorists.
One repeatedly finds works in the "mainstream" literature that bear out the relevance of chaos to political analysis, at least implicitly. For example, Jervis (1997) has written at length about the practical implications of chaos and complexity. Decades before chaos theory made plain the full significance thereof, philosopher Karl Popper articulated the "accountability principle," which holds that predictability depends on attaining the requisite degree of precision in the initial conditions. (Even small measurement errors in social science are therefore likely to be fatal.) In Popper (1982) endorsed Prigogine's research as supporting his own philosophical preference for an open society and an open universe, concluding, "the creativeness of life does not contradict the laws of physics." In a similar vein, diplomatic historian (and noted bird watcher!) Louis Halle (1977) applied insights from quantum physics and other discoveries of modern science in his masterful interpretation of world history, Out of Chaos. In it, he points to the arts of architecture, music, and literature as attempts to establish a link between human beings' sordid, ignorant, finite, chaotic existence, and their aspiration toward a sublime, infinite, ordered transcendence.
Much of the research in chaos is still tentative, and skeptics abound; see Horgan (1995). Nevertheless, there is solid mathematics behind it (see Devaney, 1989). In fact, if you consider the fact that Mandelbrot's work on fractals at IBM resulted in vastly improved error-correction and data compression algorithms, you could say that the Internet itself would not be possible without chaos theory! From the perspective of traditionalist scholars, there is nothing new in the suggestion that historical patterns embody a complex, dynamic network of unforeseeable contingencies; "We already knew that." It is indeed important to heed their admonition not to privilege the scientific method over the substance of the material under study. At the very least, the universal reality of chaos highlights the potential dangers of -- and the need to avoid -- committing the quantitative fallacy, the false notion that a factor's importance is a function of its susceptibility to measurement. From an ontological standpoint, Prigogine's research into self-organizing open dynamic systems provides a solid analytical basis for Popper's, Hayek's, and Deutsch's theories about the evolution of social order and justifies conceptualizing nation-states as cybernetic adaptive organisms. Finally, the findings about fractal geometry and complexity suggest that the appropriate model of central tendency in nonlinear situations is not the bell-shaped curve but the "strange attractor." In other words, seemingly random period-to-period fluctuations in policy behavior may in fact represent a gradual, evolutionary process toward the national interest, one that mixes random chance, purposeful choice, and forced necessity.
In any case, paradigm shifts do not happen overnight, and some of the best and brightest innovators such as Galileo suffer major career setbacks. As more and more mainstream social scientists find uses for it, we can expect chaos theory to gradually gain a serious standing in public policy debates. Presidents and members of Congress may at last stop the silly pretense that they can reliably forecast the effect of tax or spending changes on the economy several years into the future. Paradoxically, this new-fangled way of thinking may lead to a more circumspect attitude about the possibilities for comprehensive centralized social planning schemes -- such as Mrs. Clinton's health care proposal of 1994, or various kinds of global crusades against poverty or evil -- and encourage a return to common-sense approaches to problems at the local grass-roots level.
This introduction to chaos theory just scratches the surface of a vast body of research, and will probably raise more questions about the subject than it answers. For further information, please consult the following scholarly and popular works. Books whose authors are indicated by bold face type below are considered the "canonical works" in the field, or are the best-known popular works. The "popular science" books by Davies and Gleick are good place to start for those new to the subject.
W. Brian Arthur, "Competing Technologies, Increasing Returns, and Lock-in by Historical Events," Economic Journal 99 (March 1989)
Brian Barry, Long Wave Rhythms in Economic Development and Political Behavior (Baltimore: Johns Hopkins University 1991)
Alan Beyerchen, "Clausewitz, Nonlinearity, and the Unpredictability of War" International Security 17: 3 (Winter 1992/93)
Paul Davies, The Cosmic Blueprint: New Discoveries in Nature's Creative Ability to Order the Universe (New York: Touchstone Books, 1989)
Robert L. Devaney, An Introduction to Chaotic Dynamical Systems 2nd ed. (New York: Addison Wesley, 1989)
James Gleick, Chaos: Making a New Science (New York: Viking, 1987) -- the best single general introduction to the topic.
Louis Halle, Out of Chaos (Boston: Houghton-Mifflin, 1977)
John Horgan, "From Complexity to Perplexity" Scientific American (June 1995)
Robert Jervis, System Effects: Complexity in Political and Social Life (Princeton: Princeton University Press, 1997).
L. Douglas Kiel and Euel Elliott, Chaos Theory in the Social Sciences: Foundations and Applications (Ann Arbor: University of Michigan Press, 1997)
Thomas S. Kuhn, The Structure of Scientific Revolutions (Chicago: University of Chicago, 1970)
Benoit Mandelbrot, The Fractal Geometry of Nature (New York: W.H. Freeman, 1983)
C. S. Poirot, Jr., "Institutions and Economic Evolution," Journal of Economic Issues 27: 3 (Sept. 1993)
Karl Popper, The Open Universe: An Argument for Indeterminism (Totowa, N.J.: Rowman and Littlefield, 1982 )
Ilya Prigogine and Isabelle Stengers, Order Out of Chaos (New York: Bantam, 1984)
James Rosenau, Turbulence in World Politics Princeton (Princeton, NJ: Princeton University Press, 1990)
Alvin M. Saperstein, "Chaos--a model for the outbreak of war" Nature Vol. 309 (May 24, 1984)
Alvin M. Saperstein, "SDI a model for chaos" Bulletin of the Atomic Scientists (Oct. 1988)
Manfred Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (New York: W. H. Freeman, 1991)
Ian Stewart, Does God Play Dice? The Mathematics of Chaos (New York: Basil Blackwell, 1989)
Bernt Wahl, Exploring Fractals on the Macintosh (Reading, MA: Addison-Wesley, 1995) -- includes disk with FractaSketch and MandelMovie software.
M. Mitchell Waldrop, Complexity: The Emerging Science at the Edge of Order and Chaos (New York: Simon and Schuster, 1992)
James Yorke, "Period Three Implies Chaos" American Mathematical Monthly 82 (1975)
Bernt Wahl, the Web site of one of the authors listed above. He was formerly involved with "MandelMovie," which was published by Dynamic Software.
In early January 2003 the Washington Post had a feature story in the Style section on Dr. James Yorke, a pioneer of chaos theory who teaches at the University of Maryland in College Park. It was OK, but had too many of the typical chaos cliches.
Stephen Wolfram, A New Kind of Science, . (Link to a book review page on Amazon.com)
Randomness and Prime Twin Proof by Martin C. Winer