The Edge of Organization: Chaos and Complexity Theories of Formal Social Systems

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Russ Marion

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    Dedication

    This book is dedicated to the people in my past—my father and grandparents—whose presence in this work is indelible; and to my wife Gail, daughter Cathy, and mother Lois, whose presence in this labor was indispensable.

    Preface

    The Edge Premise

    Modern apologists (modern being the past 250 years or so) make much of the differences between faith and science, but, ironically, both science and faith are premised upon common assumptions. Granted, assumptions about the origins of life differ, as do assumptions about biological causality (need vs. accident), the nature of life itself, and the role of humans in the grand scheme of things. Nonetheless, the ancient Hebrews and Charles Darwin agreed that life resulted from the sifting of order from chaos, that chaos is an ambiance of disorder and useless randomness from which useful order is separated. To put it another way, the religious and the scientific have a common lack of appreciation for chaos and a common appreciation for order.

    But I (and Chaologists in general) maintain that chaos is not receiving due respect in all this. I do not intend to engage the Hebrews over theology, but I do argue that our perception of chaos as random, useless dynamic is sophistry. Far from being meaningless void, chaos is the source of creativity and construction in nature and in social dynamics (the focus of this book). Many unpredictable systems are quite ordered, although that order is quite complex. Weather patterns, brain waves, insect populations, and the economy are all unpredictable systems with hidden structures. Conversely, much of what we assume to be ordered behavior is actually quite complex and, in many ways, unpredictable. Many formal social organizations, for example, may appear stable in general—market share, organizational form, and products may change little from year to year—but details, such as personnel, technology, and clientele, are quite dynamic.

    I also argue, along with Stu Kauffman (who coined the coming aphorism) and other Complexity theorists, that “order is free.” It requires no external force, no “sifting.” Not only are chaos and order not what we think them to be, not only are their roles misunderstood, but the emergence of order does not necessarily follow the paths historical science has laid out for it.

    There are two issues here: (a) our mis-definition of chaos, and (b) our misunderstanding about the emergence of order. We will address the last, first. Again, faith and science have similar premises: Order is considered the product of sifting, and sifting represents external force, or work. The one attributes sifting to the efforts of God, the other to the efforts of natural selection; but both see order as the fruit of work. Biologist Brian Goodwin explains that our culture is characterized by an almost primal or subconscious acceptance of the work ethic, the conviction that good results from effort. Sifting, whether by God or by selection, he continues, represents work done to produce order. Just as hard work draws men and women from the jaws of poverty and meaningless life, the work of a sifter rescues us from the void of chaos. This is powerful metaphor in our culture, and to argue against it is, well, work. How else could order arise if not through effort of some sort?

    Chaos Theory, or rather that branch of Chaos Theory that we will identify as Complexity Theory, responds that order emerges naturally because of unpredictable interaction—interaction is the vehicle by which this occurs and unpredictability is the stimulus that promotes novelty (but I'm getting ahead of myself). The argument proposed in this book is that interacting entities—atoms, molecules, people, organizations—tend (a) to correlate with one another because of their interaction, and (b) to catalyze aggregation. Correlation is what happens when two or more people exert interactive influence over one another; a husband and wife, for example, will gravitate together in their attitudinal structures because they discuss and live their beliefs together. Autocatalysis—the second point—begins when the behavior of one system stimulates certain behaviors in another system that in turn stimulates another and another; eventually the chain of stimulation returns to motivate, or catalyze, the original system and the cycle is reinforced. Order, then, emerges not because someone or some thing expends energy to create it; rather, order emerges from the natural, and free, consequences of interaction.

    Unpredictability is an important element of this process. This argument is captured by the U.S. historian Henry Adams, who said, “Chaos often breeds life, when order breeds habit.” Novelty does not arise from formula. The mistake of traditional science is its presumption that events are the formulaic and linear results of stimulus, that time is reversible (more on this later), that everything that is or will be was predestined at the moment of the big bang. The Greek philosopher Epicurus said that “it would have been better to remain attached to the beliefs in gods rather than being slaves to the fates of physicists …[for] the latter …brings with it inviolable necessity” (Prigogine, 1996, p. 10). A flipped coin can produce only heads and tails; an action can produce only an equal and opposite reaction; the output of a linear equation is determined by, and functionally related to, its input variables; an uninspired man or woman produces only the mundane and predictable. The unpredictable, the free will, the chaotic creates novelty, and novelty is the author of new order. It was unpredictability that made possible the Theory of Relativity, Newton's Laws of Motion, and Heisenburg's Uncertainty Principle; it is unpredictability that underlies inventions and mobs and rumors and informal groups and organization. Correlation and autocatalysis build, but unpredictability is that twist of events that inspires creation and renewal. Without it, there are no happenstance meetings, no flashes of creativity, no Schumpeter-ian gales of creative destruction; there is only predictable routine with everything in its place following its predetermined course. Chaos is the spice of order.

    The second issue raised in the early paragraphs of this preface had to do with the nature of chaos and order. We usually assume that chaos means random disorder and that order means predictable stability. Both notions are true, but they are less often true than we think. Order is not always ordered and symmetrical (particularly in nature and in society), and chaos need not be random.

    Unpredictability comes in two forms: the random and the Chaotic (Chaos is capitalized here because of its specialized meaning). Random behavior exists when an entity, given choices, is likely to perform any one with equal probability. A molecule in fluid, given the choice of moving right or left, is just as likely to move one way as the other and there is no knowing ahead of time which it will do. Chaotic behavior, by contrast, is more structured, stable, and deterministic; nonetheless it, like random motion, is still unpredictable.

    Chaos is descriptive of systems rather than entities. Systems, of course, are composed of entities, but the corporate behavior of these entities is organized by correlation and autocatalysis. Consequently the dynamics of Chaotic structures, like weather systems, fluid turbulence, families, mobs, and organizations, are, to varying extents, patterned and stable; even so, their trajectories over time are unpredictable, again to varying extents.

    It is the unpredictable nature of Chaotic systems that inspires the order of which we spoke earlier. The isolated, random behavior of an individual, we will argue, is not up to this task. Individuals (such as Newton, Einstein, Martin Luther King, and Henry Ford) may be credited with the emergence of new order, but their achievements are possible only within the context of the correlated, autocatalytic dynamics of a system of actors. An individual may be the symbol of change; he or she may even be the catalysis about which change dynamics collapse. The individual within a Chaotic system, however, is influenced and delimited by correlation with the whole. One could suppose that the completely idiosyncratic behavior of an individual could catch on and influence the whole, but new order more usually involves the evolution of a system rather than the peccadilloes of an eccentric.

    We need to narrow our perspective of Chaos a bit in order to describe social systems properly. Chaotic order per se is actually too violent, too changing to describe much that goes on among living beings. Complexity theorists, who are conceptually related to Chaos theorists, argue that life tunes Chaos's intensity down a bit to a transition band between Chaos and predictable stability called the Edge of Chaos. Dynamics in this band are still Chaotic but they also possess characteristics of order. Full-blown Chaotic systems have little memory; living systems must be able to map their past. Chaotic systems flit a bit too readily from novelty to novelty; living systems need to consolidate gains. Predictable, stable systems, by contrast, possess none of the panache needed to create new order or even to respond adaptively to creative environments. Complex systems lie between these poles, at the Edge of Chaos, and they have both panache and stability sufficient to serve life.

    Chaos and Complexity theories are themselves, ipso facto, Chaotic and Complex systems, and as such are case studies in the emergence of new order (in this case, the new order is a way to describe reality). This emerging order “catalyzed” my interest in applying its ideas to formal social systems. I was helped along the way by a network of similarly interested academicians. The American Educational Research Association has a special interest group dedicated to such study; The Society for Chaos Theory in Psychology and Life Sciences has an international conference and a journal devoted to the subject; and numerous writers, such as Stu Kauffman, William Doll, Brian Arthur, Jeff Goldstein, Brian Goodwin, and Claire Gilmore, are exploring the explanatory power of these theories in the social sciences. This book, then, is part of a growing network of academics, resources, and ideas devoted to understanding the Chaotic and Complex nature of human social and psychological structures, and the study of formal social systems is its special niche in that network.

    I would like to give special thanks to two members of this emerging network of Chaologist and Complexity theorists. Raymond A. Eve, Professor of Sociology at the University of Texas at Arlington, and Sara Horsfall, a recent Ph.D. graduate in sociology from Texas A&M, provided valuable insights about this manuscript, insights that had a significant impact on the overall structure and direction of the book. I would also like to thank Kristin Bergstad for her editorial work. She has found errors that I never would have found, and I am indebted to her for that. Finally, I appreciate the efforts of Harry Briggs and Anna Howland, Sage editors who decided to give this book ago.

  • Notes

    1. Boolean algebra, developed by mathematician George Boole about 1850, allows one algebraically to manipulate logical statements. The answer to such a statement is either true or false. “If A AND B, then C” is a Boolean expression that states that C is true only if both A and B are true. “If A OR B, then C” states that C is true if either A or B is true. Boolean logic is most commonly used in computer programming.

    2. We define a specialist as one who performs a specific task; such a person requires only enough training to perform that task. The term specialist might be used differently by other theorists. Max Weber, for example, defines it much as we do. Richard Hall (1991), however, defines specialist as a highly trained individual with a broad range of skills—which is our definition of a generalist.

    3. This works for Chaotic systems—Complex systems are somewhat more stable and predictable than are Chaotic ones, however. The Lyapunov exponent has been applied with some success to Complex social systems such as stock markets; more research is needed to determine its applicability to more overtly Complex systems such as those observed in formal organizations.

    4. Existing literature on Chaos theory and social organization has made much of “sensitive dependence on initial conditions,” or the “butterfly effect.” Essentially, these arguments have invoked the butterfly effect to suggest that organizational behavior is highly unpredictable. Sensitive dependence is certainly a cornerstone of Chaos theory and is undoubtedly its best known tenant (moviegoers may remember the reference to it by the mathematician in Jurassic Park when he defined Chaos theory for the other characters), but Chaos theory involves much more than this. More important, to the extent that social systems are Complex rather than Chaotic, the butterfly effect is muted, more stable. Moreover, it is compromised by the fact that Complex attractors are loosely coupled—small and relatively isolated. To wit, social theorist should be cautious about indiscriminate reference to this phenomenon.

    5. A word of caution: Plots generated by this algorithm should not be re-sized unless they can be re-rendered in postscript; otherwise, the relationships among points will be distorted. Further, the graphs generated from the stock volume data for this chapter could be reproduced accurately only with a 600 dpi (or finer) printer; detail may be lost with anything less than a 600 dpi printer.

    Annotated Bibliography of Books and Articles

    Arthur, W. B. (1989). The economy and complexity. In D. L.Stein (Ed.), Lectures in the sciences of complexity (pp. 713–740). Redwood City, CA: Addison-Wesley.

    Reading level: moderate

    Brian Arthur argues that economy is better modeled by increasing return (them that has, gets more) than by decreasing returns, which is more traditionally accepted in economic circles. Increasing returns means that it is not necessarily the best product or service or technology that will rise to dominance, rather it is the product or technology that has an initial advantage in the market that is more likely to succeed. He provides numerous examples, and presents arguments about why this is true.

    Bak, P. (1996). How nature works. New York: Copernicus.

    Reading level: easy to moderate

    This book represents a discussion of his criticality hypothesis that is geared to the general reader. Bak describes his sandpile experiments and the principles underlying them. He then applies the results to diverse phenomena, including highway traffic patterns, earthquakes, Conway's Game of Life, cognition, and economics. Presents an excellent discussion of the power law distribution.

    Bak, P., & Chen, K. (1991). Self-organized criticality. Scientific American, pp. 46–53. http://dx.doi.org/10.1038/scientificamerican0191-46

    Reading level: easy to moderate

    Per Bak argues that change is nonlinearly related to cause, and that it comes in many different scales. Demonstrates by dropping sand a grain at a time onto a scale and measuring the size of “sand slides.” Observes a fractal dimension to the distribution of slides.

    Baker, G. L., & Gollub, J. P. (1990). Chaotic dynamics: An Introduction. Cambridge, UK: Cambridge University Press.

    Reading level: moderately difficult

    Baker and Gollub introduce chaos to a fairly sophisticated but uninitiated audience. Their coverage of strange attractors, correlation dimension, spectral analysis, Poincaré maps, sensitive dependence, basins of attractions, bifurcation, phase space, logistic equations, and Lyapunov exponents is mathematical, but not oppressively so. Includes the code for a number of computer programs in the appendix.

    Beltrami, E. (1993). Mathematical models in the social and biological sciences. Boston: Jones and Bartlett.

    Reading level: moderate

    Presents a number of mathematical models of social and biological dynamics. Of particular interest is his discussion of variations on the logistic equation.

    Briggs, J., & Peat, F. D. (1989). Turbulent mirror. New York: Harper & Row.

    Reading level: easy

    An interesting book about Chaos that devotes roughly half its pages to Complexity Theory without actually calling it Complexity. The chapters in the first half of the book are numbered from largest to least (Briggs understandably invokes Lewis Carroll's imagery often); here he discusses Chaos Theory. The chapters in the last half of the book are numbered normally; here is discusses how order emerges out of interaction (Complexity Theory).

    Cronbach, L. J. (1988). Playing with chaos. Educational Researcher, 17(6), 46–49.

    Reading level: easy

    Lee Cronbach was probably the first major social scientist (outside the field of economics) to write of the potential of Chaos for understanding social phenomena. This article is primarily a review of Gleick's popular book, Chaos, but Cronbach also muses about how Chaos might be applied in social science research. He argued, however, that much of the initial work will be metaphorical because social scientists (his specific audience was educators) aren't prepared to do the type of math demanded by Chaotic analysis.

    Feigenbaum, M. (1978). Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics, 19, 25–52. http://dx.doi.org/10.1007/BF01020332

    Reading level: difficult

    The important article in which Michael Feigenbaum develops the Feigenbaum constant, or the rate at which period doubling decomposes into Chaos.

    Gilmore, C. G. (1995). A new test for chaos. In R. R.Trippi (Ed.), Chaos and nonlinear dynamics in the financial markets (pp. 383–415). Chicago: Irwin Professional.

    Reading level: moderate

    Gilmore develops a topological technique for analyzing Chaotic structure, one that is particularly useful for the relatively small data sets available in the social sciences. The algorithm is discussed, as are techniques for neutralizing random noise. Gilmore reanalyzes several previous studies and clarifies their results with the return map.

    Gleick, J. (1987). Chaos: Making a new science. New York: Viking.

    Reading level: easy to moderate

    The classic that popularized Chaos. This book introduces the reader to the Lorenz attractor, Chaos in the logistic equation and Feigenbaum's universality, Mandelbrot sets and fractal geometry, and Poincaré maps.

    Goodwin, B. (1994). How the leopard changed its spots: The evolution of complexity. New York: Scribner.

    Reading level: easy to hard

    For the most part, this is a biology text, hence the “hard” rating of reading level above. However, the introductions to several of Goodwin's chapters, and his last two chapters, are useful for social scientists. Here he discusses the reductionist versus holistic debate, cooperation versus competition, emergence, his version of the autocatalytic process, and human culture.

    Hénon, M. (1976). A two-dimensional mapping with a strange attractor. Communications in Mathematical Physics50, 69–77.

    Reading level: difficult

    The article in which Michael Hénon introduces the famous Hénon attractor, which is generated by two simultaneous difference equations on a Poincaré Map.

    Holland, J. H. (1995). Hidden order. Reading, MA: Addison-Wesley.

    Reading level: easy

    John Holland, called by some the father of the field of genetic algorithms, describes how complex adaptive systems (CAS) interact to generate order. He describes an artificial life simulation called Echo, and describes strategies for expanding our understanding of CAS. His discussion in Chapter 1 of aggregation, tags, and building blocks is particularly useful.

    Kauffman, S. A. (1993). The origins of order. New York: Oxford University Press.

    Reading level: moderately difficult

    Stu Kauffman's seminal work on Complexity theory. He argues that natural selection is incapable of creating the amount of order observable on earth, that instead order is free, the natural outcome of interaction. He proposes a role for selection that is largely secondary: Order emerges from interaction, then selection molds it. His discussions cover such things as rugged landscapes, morphology, and phase and parameter space.

    Kauffman, S. (1995). At home in the universe: The search for the laws of self-organization and complexity. New York: Oxford University Press.

    Reading level: moderate

    A simpler and more readably version of his 1993 work, The Origins of Order. Appropriate for those wanting a less technical discussion of his argument that order is free. His discussion of patches and change in the final chapters is particularly interesting for social scientists.

    Langston, C. G. (1986). Studying artificial life with cellular automata. Physica, 22D, 120–149.

    Reading level: moderate

    Chris Langston, who is perhaps the driving force behind the popularity of Complexity Theory, shows us how different forms of order emerge using cellular automata (anyone familiar with John Conway's “Game of Life” is familiar with CA). Cellular automata is a simulation that occurs on an extended chess board. Cells on the grid are alive or dead depending upon the state of their neighbors. Langston controlled the probability that cells would live or die; he found that when the probability of death was high, pockets of repetitive or unchanging stability emerged. When the probability of death was low, Chaotic instability swirled across the board. At a middle point, however, loosely connected pockets of dynamic activity—Complexity—appeared.

    Lauwerier, H. (1991). Fractals: Endlessly repeated geometric figures. Princeton, NJ: Princeton University Press.

    Reading level: moderately easy

    An excellent, easy to understand introduction to the mathematics of fractal geometry. Perfect for the novice. Includes computer program codes in the appendix.

    Levy, S. (1992). Artificial life: The quest for new creation. New York: Random House.

    Reading level: easy

    Describes the quest to simulate life with a bottoms-up logic, as opposed to the top-down logic of artificial intelligence. Bottoms-up logic specifies adaptive rules for individual units of a system, then turns those units loose to interact with one another. Computerized simulations demonstrate that such strategy leads to spontaneous order and enables the system to deal with environmental problems with surprising effectiveness.

    Lewin, R. (1992). Complexity: Life at the edge of chaos. New York: Macmillan.

    Reading level: easy

    One of a handful of books that popularized Complexity Theory in the early 1990s, this is probably still the best introduction to the subject. Levy introduces his topics through the eyes of the key researchers of the subject, including Chris Langston, Stu Kauffman, Heinz Pagals, James Farmer, John Holland, and Craig Reynolds.

    Lloyd, A. L. (1995, June). Computing bouts of the Prisoner's Dilemma. Scientific American, pp. 110–115. http://dx.doi.org/10.1038/scientificamerican0695-110

    Reading level: not rated

    Lloyd provides computer code for competing bouts of the simulation, Prisoner's Dilemma.

    Lovelock, J. (1988). The ages of Gaia: A biography of our living earth. New York: Norton.

    Reading level: easy

    Lovelock, who is considered heretical by many biologists, argues that life, rather than existing because environmental conditions are favorable, actually manipulates that environment to make it favorable. In a computer simulation, he demonstrates that a world devoid of life becomes increasingly hot. When colored flowers are introduced, however, the temperature stabilizes at a livable level.

    Mandelbrot, B. B. (1983). The fractal geometry of nature. New York: Freeman.

    Reading level: moderately difficult

    Benoit Mandelbrot's classic book provides what may be the first major advance in geometry since Euclid. He argues that nature is not composed of stylized lines and circles; rather nature is convoluted, broken, fractal. Discusses fractal dimension, measures of the length of borders between countries, fractal landscapes and artwork, and percolation. Beautifully illustrated; the pictures alone are worth the cost of the book. Mandelbrot alerts the reader to sections that are particularly difficult, and has written in such a way that these sections can be skipped over without loss.

    Marion, R. (1992). Chaos, topology, and social organization. Journal of School Leadership, 2(2), 144–177.

    Reading level: easy to moderate

    Discusses the logistic equation within the context of information theory. Demonstrates Ruelle's procedure for generating Poincaré maps from univariate data, demonstrates the stability of chaotic systems exposed to random noise, and discusses possible applications of Chaos in the social sciences.

    Martien, P., Pope, S. C., et al. (1985). The chaotic behavior of the leaky faucet. Physics Letters, 110A(7, 8), 399–404. http://dx.doi.org/10.1016/0375-9601%2885%2990065-9

    Reading level: easy

    A classic experiment reproduced in numerous science labs for high school and college students. Martien and colleagues measured the time lapse between drips of water from a leaky faucet and plotted the intervals. A strange attractor emerged.

    May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261, 459–467. http://dx.doi.org/10.1038/261459a0

    Reading level: moderately difficult

    Robert May's classic article on Chaos in the logistic equation. Analyzes what happens to the output of this difference equation as lambda (or growth rate) approaches 3.57.

    Nielsen, F., & Hannan, M. T. (1977, June). The expansion of national educational systems: Tests of a population ecology model. American Sociological Review, 42, 479–490. http://dx.doi.org/10.2307/2094752

    Reading level: moderately difficult

    Nielsen and Hannan demonstrate that the logistic equation models the growth of public education during the first half of the 20th century.

    Nowak, M. A., May, R. M., et al. (1995, June). The arithmetics of mutual help. Scientific American, pp. 76–81. http://dx.doi.org/10.1038/scientificamerican0695-76

    Reading level: easy

    Reports on simulation experiments with Prisoner's Dilemma in which many different “actors” play against each other simultaneously and repetitively, and each “remembers” the behavior of its opponents in the previous round. His results show a pervasive tendency toward cooperation.

    Peterson, I. (1988). The mathematical tourist. New York: Freeman.

    Reading level: easy

    This book discusses a number of mathematical topics, but two chapters, one on Chaos and one on fractal geometry, are of interest. Both discuss core issues in a language that is easily grasped by beginners.

    Poundstone, W. (1992). Prisoner's dilemma. Garden City, NY: Doubleday.

    Reading level: easy

    Poundstone's book is as much a biography of noted mathematician John von Neumann as a treatise on the simulation, Prisoner's Dilemma. He traces the invention of Prisoner's Dilemma, discusses its many forms, presents the basic tenets of game theory, and discusses how game theory was used by nuclear weapons cold war strategists during the 1950s.

    Prigogine, I. (1997). The end of certainty. New York: Free Press.

    Reading level: moderate to difficult

    Prigogine presents compelling evidence that traditional scientific notions of time-reversibility are destroyed by Chaos Theory. He attributes this to two observations: unmeasurable initial conditions and sensitive dependence upon initial condition, and unpredictable resonances among interacting units.

    Prusinkiewicz, P., & Lindenmayer, A. (1990). The algorithmic beauty of plants. New York: Springer. http://dx.doi.org/10.1007/978-1-4613-8476-2

    Reading level: moderate

    Because of the beautiful illustrations, this is a coffee table book—despite the serious nature of its topic. The book presents genetic algorithms (actual computer code) for the generation of fractal images.

    Reynolds, C. (1987). Flocks, herds, and schools: A distributed behavioral model. Computer Graphics, 21, 25, 32. http://dx.doi.org/10.1145/37402.37406

    Reading level: moderate

    In this article we learn about “Boids,” a bottoms-up, computerized simulation of bird flocking. “Bottoms-up” refers to localized rules rather than global coordination; each entity responds according to simple rule sets guiding its behavior. The discussion illustrates basic principles of Complexity Theory.

    Schroeder, M. (1991). Fractals, chaos, power laws. New York: Freeman.

    Reading level: difficult

    Good introduction to some of the basic mathematics of Chaos, including correlation dimension, Hausdorff dimension, fractals and time series data, universality, and Markov process. Applications to such events as stock markets, percolation, and population distribution are available. Includes a chapter on cellular automata.

    Sterman, J. D. (1988). Deterministic chaos in models of human behavior: Methodological issues and experimental results. System Dynamics Review, 4(1–2), 148–178. http://dx.doi.org/10.1002/sdr.4260040109

    Reading level: moderately easy

    Sterman modified two business simulations, one a stock management game and the other a beer distribution game, to collect data on decision points. When the data was plotted, images similar to known strange attractors emerged.

    Stewart, I. (1989). Does God play dice: The mathematics of chaos. Cambridge, MA: Basil Blackwell.

    Reading level: easy to moderate

    This book makes Chaos accessible to the mathematically impaired. Conceptually develops key concepts, such as phase space, topology, fractals, Poincaré maps, differential calculus, and difference equations.

    Trippi, R. H. (Ed.). (1995). Chaos and nonlinear dynamics in the financial markets. Burr Ridge, IL: Irwin.

    Reading level: moderately difficult, but varies

    A collection of articles on Chaos in financial systems. Focuses particularly on efforts to measure such Chaos.

    Tufillaro, N. B., Abbott, T., et al. (1992). An experimental approach to nonlinear dynamics and chaos. Redwood City, CA: Addison-Wesley.

    Reading level: difficult

    Discusses Chaos relative to a program in a software package that accompanies the book. Code for several other programs is available in the appendix.

    Waldrop, M. M. (1992). Complexity: The emerging science at the edge of order and chaos. New York: Simon & Schuster.

    Reading level: easy

    Waldrop's book is as much a discussion of the Santa Fe Institute for the Study of Complexity as it is a treatise on Complexity. In addition to the usual topics in Complexity Theory, this book introduces the reader to efforts in economics, particularly the interesting work on increasing returns by Brian Arthur.

    Zaslavsky, G. M., Sagdeev, R. Z., Usikov, D. A., & Chernikov, A. A. (1991). Weak chaos and quasi-regular patterns (A. R.Sagdeeva, Trans.). Cambridge, UK: Cambridge University Press. http://dx.doi.org/10.1017/CBO9780511599996

    Reading level: difficult

    Discusses the phase transition between stability and Chaos as a band of dynamics in which islands of stability float within seas of Chaos—and vice versa. Illustrates with a interesting application of a difference equation for forced pendulum motion.

    References

    Preface
    Arthur, W. B. (1989). The economy and complexity. In D. L.Stein (Ed.), Lectures in the sciences of complexity (pp. 713–740). Redwood City, CA: Addison-Wesley.
    Gilmore, C. G. (1995). A new test for chaos. In R. R.Trippi (Ed.), Chaos and nonlinear dynamics in the financial markets (pp. 383–415). Chicago: Irwin Professional.
    Goldstein, J. A. (1995). The Tower of Babel in nonlinear dynamics: Toward a clarification of terms. In R.Robertson & A.Combs (Eds.), Chaos in psychology and the life sciences: Proceedings of the Society for Chaos Theory in Psychology and the Life Sciences (pp. 39–47). Mahwah, NJ: Lawrence Erlbaum.
    Goodwin, B. (1994). How the leopard changed its spots: The evolution of complexity. New York: Scribner.
    Kauffman, S. (1995). At home in the universe: The search for the laws of self-organization and complexity. New York: Oxford University Press.
    Prigogine, I. (1996). The end of certainty. New York: Free Press.
    The Edge of Organization
    Arthur, W. B. (1989). The economy and complexity. In D. L.Stein (Ed.), Lectures in the sciences of complexity (pp. 713–740). Redwood City, CA: Addison-Wesley.
    Bak, P. (1996). How nature works. New York: Copernicus.
    Briggs, J., & Peat, F. D. (1989). Turbulent mirror. New York: Harper & Row.
    Cartwright, T. J. (1991, Winter). Planning and chaos theory. Journal of the American Planning Association, 57(1), 44–56. http://dx.doi.org/10.1080/01944369108975471
    Cronbach, L. J. (1988, August-September). Playing with chaos. Educational Researcher, 17(6), 46–49.
    Einstein, A. (1988). The meaning of relativity (
    5th ed.
    ). Princeton, NJ: Princeton University Press.
    Geller, H. A., & Johnston, A. P. (1990, January-March). Policy as linear and nonlinear science. Journal of Educational Policy, 5(1), 49–65. http://dx.doi.org/10.1080/0268093900050104
    Gell-Mann, M. (1994). The quark and the jaguar. New York: Freeman.
    Gilmore, C. G. (1995). A new test for chaos. In R. R.Trippi (Ed.), Chaos and nonlinear dynamics in the financial markets (pp. 383–415). Chicago: Irwin Professional.
    Gleick, J. (1987). Chaos: Making a new science. New York: Viking.
    Goldstein, J. A. (1995). The Tower of Babel in nonlinear dynamics: Toward a clarification of terms. In R.Robertson & A.Combs (Eds.), Chaos in psychology and the life sciences: Proceedings of the Society for Chaos Theory in Psychology and the Life Sciences (pp. 39–47). Mahwah, NJ: Lawrence Erlbaum.
    Goodwin, B. (1994). How the leopard changed its spots: The evolution of complexity. New York: Scribner.
    Guastello, S. J., Dooley, K. J., & Goldstein, J. A. (1995). Chaos, organizational theory, and organizational development. In F. D.Abraham & A. R.Gilgen (Eds.), Chaos theory in psychology (pp. 267–278). West-port, CT: Praeger.
    Holland, J. H. (1995). Hidden order. Reading, MA: Addison-Wesley.
    Kauffman, S. (1995). At home in the universe: The search for the laws of self-organization and complexity. New York: Oxford University Press.
    Langston, C. G. (1986). Studying artificial life with cellular automata. Physica, 22D, 120–149.
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    Name Index

    About the Author

    RUSS MARION grew up in North Carolina where he received four degrees in education and educational administration from the University of North Carolina at Chapel Hill. He has been a public school teacher and a school principal; he helped develop an administrative software package for public schools in North Carolina; and he currently is an associate professor of Educational Leadership at Clemson University in South Carolina. Marion's current research interests include nonlinear social dynamics (as witnessed by this book), school finances and student achievement, and mathematical modeling of social dynamics. He has published articles on social chaos in the British journal Management in Education and in the Journal of School Leadership, and has spoken on the subject at a number of scholarly conventions. He maintains a Web site dedicated to studying social chaos and complexity (http://www.hehd.clemson.edu/complex/Cmplxdex.htm) and teaches a graduate-level course at Clemson University on applications of Chaos and Complexity Theory to formal social organizations.


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