Foundations and Applications

Chaos Theory in the Social Sciences

Chaos Theory in the Social Sciences Foundations and Applications

Edited b y

L. Douglas Kiel and Euel Elliott

Ann Arbor

T H E UNIVERSITY OF MICHIGAN PRESS

First paperback edition 1997 Copyright © by the University of Michigan 1996 All rights reserved Published in the United States of America by The University of Michigan Press Manufactured in the United States of America © Printed on acid-free paper 2004

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No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, or otherwise, without the written permission of the publisher. A CIP catalog record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Chaos theory in the social sciences : foundations and applications / edited by L. Douglas Kiel and Euel Elliott, p. cm. Includes bibliographical references (p. ). ISBN 0-472-10638-4 (he : alk. paper) 1. Social sciences—Mathematical models. 2. Chaotic behavior in systems. I. Kiel, L. Douglas, 1956- . II. Elliott, Euel W., 1951- . H61.25.C48 1995 300'. 1'51—dc20 95-35470 CIP ISBN 0-472-08472-0 (pbk. : alk. paper)

To Our Parents

Contents

Introduction Euel Elliott and L . Douglas

Kiel

Part 1. Chaotic Dynamics in Social Science Data 1. Exploring Nonlinear Dynamics with a Spreadsheet: A Graphical V i e w o f Chaos for Beginners L . Douglas Kiel and Euel Elliott 2. Probing the Underlying Structure in Dynamical Systems: A n Introduction to Spectral Analysis Michael McBurnett 3. Measuring Chaos Using the Lyapunov Exponent Thad A. Brown 4. The Prediction Test for Nonlinear Determinism Ted Jaditz 5. From Individuals to Groups: The Aggregation o f Votes and Chaotic Dynamics Diana Richards Part 2. Chaos Theory and Political Science 6. Nonlinear Politics Thad A. Brown 7. The Prediction o f Unpredictability: Applications o f the New Paradigm o f Chaos i n Dynamical Systems to the O l d Problem o f the Stability o f a System o f Hostile Nations Alvin M. Saperstein 8. Complexity in the Evolution o f Public Opinion Michael McBurnett Part 3. Chaos Theory and Economics 9. Chaos Theory and Rationality in Economics J. Barkley Rosser, Jr.

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Contents 10. L o n g Waves 1790-1990: Intermittency, Chaos, and Control Brian J. L . Berry and Heja Kim 11. Cities as Spatial Chaotic Attractors Dimitrios S. Dendrinos

Part 4. Implications for Social Systems and Social Science

215 237

Management

12. Field-Theoretic Framework for the Interpretation o f the Evolution, Instability, Structural Change, and Management o f Complex Systems Kenyon B. De Greene 13. Social Science as the Study o f Complex Systems David L . Harvey and Michael Reed

273 295

References

325

Contributors

347

Introduction Eue I Elliott and L . Douglas Kiel

The social sciences, historically, have emulated both the intellectual and methodological paradigms o f the natural sciences. From the behavioral revo lution, to applications such as cybernetics, to a predominant reliance on the certainty and stability o f the Newtonian paradigm, the social sciences have followed the lead o f the natural sciences. This trend continues as new discov eries in the natural sciences have led to a reconsideration o f the relevance o f the Newtonian paradigm to all natural phenomena. One o f these new discov eries, represented by the emerging field o f chaos theory, raises questions about the apparent certainty, linearity, and predictability that were previously seen as essential elements o f a Newtonian universe. The increasing recogni tion by natural scientists o f the uncertainty, nonlinearity, and unpredictability in the natural realm has piqued the interest o f social scientists in these new discoveries. Chaos theory represents the most recent effort by social scientists to incorporate theory and method from the natural sciences. Most importantly, chaos theory appears to provide a means for understanding and examining many o f the uncertainties, nonlinearities, and unpredictable aspects o f social systems behavior (Krasner 1990). Chaos theory is the result o f natural scientists' discoveries in the field o f nonlinear dynamics. Nonlinear dynamics is the study o f the temporal evolu tion o f nonlinear systems. Nonlinear systems reveal dynamical behavior such that the relationships between variables are unstable. Furthermore, changes in these relationships are subject to positive feedback in which changes are amplified, breaking up existing structures and behavior and creating unex pected outcomes in the generation o f new structure and behavior. These changes may result in new forms o f equilibrium; novel forms o f increasing complexity; or even temporal behavior that appears random and devoid o f order, the state o f "chaos" in which uncertainty dominates and predictabil ity breaks d o w n . Chaotic systems are often described as exhibiting lowdimensional or high-dimensional chaos. The former exhibit properties that may allow for some short-term prediction, while the latter may exhibit such

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variation as to preclude any prediction. I n all nonlinear systems, however, the relationship between cause and effect does not appear proportional and deter minate but rather vague and, at best, difficult to discern. These discoveries have given rise to a new mathematics that belies pre vious scientific commitment to prediction and certainty. Natural scientists have now applied this mathematics to numerous fields o f inquiry. A brief and partial listing o f the fields includes meteorology (Lorenz 1963), population biology ( M a y 1976), and human anatomy (West and Goldberger 1987). These studies consistently show that nonlinearity, instability, and the resulting uncer tainty are essential components in the evolutionary processes o f natural sys tems. Moreover, these inquiries have given precedence to a greater concern for the extent o f and challenges o f understanding the inherent complexity o f natural systems. The emerging paradigm o f chaos thus has profound implications for the previously dominant Newtonian view o f a mechanistic and predictable uni verse. W h i l e a Newtonian universe was founded on stability and order, chaos theory teaches that instability and disorder are not only widespread in nature, but essential to the evolution o f complexity in the universe. Thus, chaos theory, as relativity theory and quantum theory before i t , presents another strike against a singular commitment to the determinism o f a Newtonian view o f the natural realm. This understanding also suggests that the relative successes in knowledge acquisition by the natural sciences are the result o f a focus on "simple" systems that function in an orderly and consistent manner. As natural scien tists have shifted their investigative focus to more complex systems, the previous quest for certainty has given way to a greater appreciation o f uncer tainty and the enormity o f potential generated by the uncertainty o f disorder and disequilibrium. W i t h the focus o f chaos theory on nonlinearity, instability, and uncer tainty, the application o f this theory to the social sciences was perhaps a predictable eventuality. As Jay W. Forrester (1987, 104) has noted, "We live in a highly nonlinear w o r l d . " The social realm is clearly nonlinear, where instability and unpredictability are inherent, and where cause and effect are often a puzzling maze. The obvious fact that social systems are historical and temporal systems also stresses the potential value o f chaos theory to the social sciences. Social systems are typified by the changing relationships between variables. The obvious metaphorical value o f applying a theory o f chaos to the social realm has served as an impetus for the emergence o f the application o f this theory to social phenomena. Yet chaos theory is founded on the mathe matics o f nonlinear systems. Thus, social scientists, in their efforts to match the mathematical rigor o f the natural sciences, are increasingly applying this

Introduction

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mathematics to a variety o f social phenomena. Time-series analysis is essen tial to these efforts, as researchers strive to examine how nonlinear and cha otic behavior occurs and changes over time. Clearly, the fundamental gap between the clear success o f knowledge acquisition in the natural sciences versus the rather minimal successes in understanding the dynamics o f the social realm is the inherent nonlinearity, instability, and uncertainty o f social systems behavior. The seeming "chaos" of social phenomena has always been a stumbling block to knowledge acqui sition i n the social sciences. Social scientists have long argued that this rela tive knowledge gap was due to the relative complexity o f the phenomena examined by the two scientific cultures. Yet chaos theory teaches that the "gap" between the t w o sciences may have largely been artificial. As natural scientists more intensively investigate complex natural phenomena, they too must contend with the challenges that have long served to keep the social sciences in the position o f a scientific stepchild. Chaos theory seems to repre sent a promising means for a convergence o f the sciences that w i l l serve to enhance understanding o f both natural and social phenomena. Chaos theory has now been applied to a wide variety o f social phenom ena ranging across the subject matter o f the traditional social science disci plines o f economics (Grandmont 1985; Baumol and Benhabib 1989; Arthur 1990) and political science (Saperstein and Mayer-Kress 1989; Huckfeldt 1990; K i e l and Elliott 1992). Economists and political scientists have applied phaos theory w i t h considerable methodological rigor and success to the tem poral dynamics o f a variety o f phenomena in their fields. Chaos theory has also been applied to sociology. In this field, however, more than in economics and political science, such efforts have tended toward metaphorical and post modernist or poststructuralist usages (Young 1991, 1992). Thus, while this volume does not include rigorous mathematical assessments o f chaotic dy namics in the subject matter o f sociology, the applications in political science and economics should serve as foundations for the development o f such research in sociology. W h i l e no specific chapter contends solely w i t h these postmodernist and poststructuralist issues, David Harvey and Michael Reed's concluding chapter examines the relevance o f these elements to chaos re search and social science inquiry. The increasingly evident value o f chaos theory in the social sciences is thus its promise as an emerging means for enhancing both the methodological and theoretical foundations for exploring the complexity o f social phenomena. Exploring this emergent and potential value is the purpose o f this book. B y examining applications o f chaos theory to a range o f social phenomena and by providing means for exploring chaotic dynamics, the chapters in this book afford the reader a comprehensive vision o f the promise and pitfalls o f chaos theory in the social sciences.

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This book seeks to provide knowledge to both social scientists new to this area o f study and the well-informed chaos researcher. Chapters range from the mathematically and methodologically sophisticated to chapters with a strictly theoretical emphasis. The book is organized by both disciplinary area and general methodology. The disciplinary sections examine chaos theory i n political science and economics. The first section o f the book exam ines methods for exploring and examining the existence o f chaotic dynamics in time-series data, which cut across the disciplines. First, though, an initial primer on chaos and nonlinear behavior is necessary to provide the basics o f this theory and an introduction to the unique vocabulary it utilizes. The M a t h e m a t i c s and Behavior of Chaos A brief examination o f the mathematics and behavior o f chaotic systems provides a means for understanding the relevance o f this theory to the com plexity o f social phenomena. Distinguishing between linear and nonlinear equations also reveals both the relevance and the challenge o f contending with the nonlinear mathematics o f nonlinear systems. Linear equations are typified by the superposition principle. This principle, simply stated, means that two solutions o f a linear equation can be combined, or added together, to generate a new solution. This means that linear equations allow problems to be broken down into smaller pieces that may generate several separate solutions. In such linear mathematics, the individual solutions can be added back together to form a complete solution to the entire problem. The superposition principle, however, does not hold for nonlinear equa tions. A nonlinear equation cannot be broken down into bits and then refor mulated to obtain a solution. Nonlinear differential equations, and the phe nomena or problems they describe, must be seen as a totality, that is, as nondecomposable. This further means that nonlinear equations are partic ularly intractable for the analyst. The inherent nonlinearity o f many social phenomena and the intractability o f the relevant mathematics thus must ex plain, in part, the challenges social scientists face when attempting to under stand the complexity o f social dynamics. Another element o f the mathematics o f nonlinear equations is the fact that a simple deterministic equation can generate seemingly random or chaotic behavior over time. One nonlinear differential equation, the logistic map, is such an example. The logistic map is described in detail in the chapters in this edition by K i e l and Elliott (chap. 1), Diana Richards (chap. 5), and A l v i n Saperstein (chap. 7). These examples o f the logistic map also detail an essen tial element o f chaotic behavior. Chaotic behavior occurs within denned pa rameters. The logistic map shows that a simple system can create very com plex and chaotic behavior. This realization has obvious impact for the social

Introduction

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sciences. Social systems o f initial relative simplicity may result over time in very complex behavior. The varying mathematics o f linear and nonlinear systems also result in divergent temporal behavior for these types o f systems. Linear systems, char acterized by stable relationships between variables, respond to changes in their parameters, or to external "shocks," in a smooth and proportionate manner. Consequently, linear systems w i l l exhibit smooth, regular, and w e l l behaved motion. Even large waves or pulses in a linear system w i l l be dis persed over time, generally resulting in a move back to the typical behavior o f the system. Nonlinear systems may be characterized by periods o f both linear and nonlinear interactions. During some time periods behavior may reveal linear continuity. However, during other time periods relationships between vari ables may change, resulting in dramatic structural or behavioral change. Such dramatic change from one qualitative behavior to another is referred to as a "bifurcation." Nonlinear systems are consequently capable o f generating very complex behavior over time. Studies o f nonlinear systems evidence three types o f temporal behavior. Nonlinear systems may evidence behavior that (1) is stable (a mathematical equilibrium or fixed point); (2) oscillates between mathematical points in a stable, smooth, and periodic manner; or (3) is cha otic and seemingly random, devoid o f pattern (nonperiodic behavior). Chapter one presents graphical images o f these three temporal regimes. These behav iors may occur intermittently throughout the " l i f e " o f a nonlinear system. One regime may dominate for some time periods while other regimes dominate at other times. It is the potential for a variety o f behaviors that represents the dynamics o f nonlinear systems. Chaotic behavior is the behavioral regime in nonlinear systems o f great est interest. Chaotic behavior, while occurring within defined mathematical parameters, appears random and without pattern over time. Chaotic behavior does not retrace previous points during its temporal evolution. This creates the appearance o f randomness. Chaotic behavior, however, is not random behav ior, since it can be generated with a completely deterministic equation. This understanding is an essential foundation o f knowledge for chaos researchers. Even deterministic systems can generate very erratic behavior over time. Moreover, as noted above, a chaotic system may appear more or less random depending on its complexity. A system mapped by the logistic equation may allow for some predictability and is an example o f a low-dimensional chaotic system. This point raises another distinctive point regarding nonlinear systems. Nonlinear systems are historical systems in that they are determined by the interactions between the deterministic elements in a system's history and "chance" factors that may alter its evolution. In systems operating in a chaotic

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regime, this fact is referred to as sensitive dependence on initial conditions. In short, the combination o f factors that defines the initial condition o f the phe nomenon and the insertion o f chance elements during its " l i f e " may generate very divergent outcomes from systems that initially appeared quite similar. This distinguishes chaotic behavior from truly random behavior. In a genu inely random system, such a system is insensitive to its initial condition. Uncertainty is also an important element o f nonlinear systems since the outcomes o f changing variable interactions cannot be k n o w n . Thus, the com plexities o f both internal dynamics and environmental "disturbances" generate considerable uncertainty during change processes in nonlinear systems. Fur thermore, a wide and complex array o f possible outcomes is available to nonlinear systems. This is particularly true during chaotic regimes. As a result, any effort at long-term prediction in nonlinear systems is highly sus pect (Baumol and Quandt 1985). Graphical Analysis in Chaos Research The intractability o f nonlinear equations and the inherent difficulties in under standing the dynamics o f complex time series have led chaos researchers to formulate new methods for analyzing data from nonlinear phenomena. This point is w e l l stated by Hasslacher (1992, 60), w h o notes in reference to complex nonlinear systems: In these systems complexity is usually both emergent and Byzantine. This means that organized and extended structures evolve and dominate a system, and the structures themselves are so complex that, when first seen, they produce a sense o f beauty followed by a deep feeling o f unease. One instinctively realizes that the analytic tools that worked so well i n the past are going to be o f little use. Many o f these new analytical methods are graphical in nature and are based on researchers' efforts to examine the dynamical motion o f time series gener ated by social science data. Chaos researchers have thus focused on examin ing the morphology (Abraham and Shaw 1982) o f the graphics generated by these time series. For example, the chapter in this volume by Brian J. L . Berry and Heja K i m , on the dynamics o f the economic long wave, relies solely on this graphical approach to data analysis. These graphical representations are lagged mapping o f data at adjacent time periods that result in an amazing array o f geometric structures, resulting in what Abraham and Shaw (1982) label the "geometry o f behavior." These mappings reveal that nonlinear systems possess an underlying order k n o w n as an attractor, where the mathematical points describing the systems' behavior

Introduction

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create pattern and structure. These geometric formulations are used through out the chapters in the text as a means o f examining the underlying structure o f longitudinal social science data. Studies o f the attractors o f nonlinear time series reveal that each o f the three behavioral regimes emanating from nonlinear differential equations cre ates a uniquely shaped attractor. A stable equilibrium generates a point extractor, in which the data are attracted to a single point on the mapping. A stable periodic oscillation generates a circular mapping, or limit cycle, as the data revolve back and forth between consistent mathematical points. The chaotic attractor is represented by a variety o f unique shapes resulting in the labeling o f such attractors as strange attractors. These attractors are typified by the creation o f form without retracing previous mappings. It is an examination o f these attractors that serves as a graphical founda tion for the notion o f "order in chaos." Even though the numerical data describing a chaotic regime appear disorderly, their geometric representation creates unique shapes o f order. A n d since chaotic regimes function w i t h i n defined parameters, a stability also exists in chaos. We then begin to see that chaotic behavior is globally stable, but locally unstable. O r g a n i z a t i o n of t h e Book This volume represents research spanning a range o f disciplines, meth odologies, and perspectives. We have incorporated many different substantive areas in order to provide the reader w i t h as balanced a perspective as possible o f the k i n d o f social science research and writing that is currently being done. Moreover, while the book is organized into four separate sections, (1) explo ration and method, (2) political science, (3) economics, and (4) implications for social systems management and social science, we would emphasize that chaos theory is really about not only the interdisciplinary but also the m u l t i disciplinary character o f the social sciences. Thus, the reader w i l l occasion ally note references and allusions in one section to chaos research being conducted in other areas o f the social or even natural sciences. In addition to the desire on our part to incorporate a diverse array o f substantive areas in this volume, a number o f other considerations were criti cal to our thinking about this work. First, we were very concerned that the subject matter be treated in a fashion that would make the arguments and concepts as accessible and "user friendly" as possible to the professional social scientist, as well as to graduate students with an interest in nonlinear dynamics. W h i l e we recognize that some o f the chapters deal with rather complex arguments and formulations, the authors o f these chapters have done an admirable j o b in presenting the material in such a way as to allow anyone reasonably comfortable w i t h undergraduate mathematics to capture the gist o f

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the arguments. M o r e sophisticated readers, however, are not shortchanged. W h i l e accessible, we have insisted that the integrity o f the material not be jeopardized. A t the same time, it w i l l be readily apparent that there is substan tial variation among the chapters in terms o f methodological rigor. We were also concerned that the contributions represent both macro- and micro-level phenomena. Readers w i l l observe that this is particularly the case in our economics and political science sections. It is crucial, i n our view, to demonstrate that chaotic processes can occur at the level o f individuals and small groups, as well as at highly aggregated levels o f analysis. Indeed, Thad A . B r o w n , in his overview chapter for political science (chap. 6), argues that an important next step in the research agenda is to attempt a linkage o f the two perspectives using theoretical approaches drawn from chaos theory. Finally, we did not want to fall into the trap o f seeing an emerging intellectual and methodological paradigm as a singular solution to the chal lenges o f understanding the complexities o f social systems behavior. As w i t h all efforts to understand the complexity that constitutes the human and social realms, a mature and reasoned skepticism is appropriate. The final chapter o f this book, by David L . Harvey and Michael Reed, attempts to make sense o f the evolution o f chaos theory in the social sciences and its prospects for enhancing knowledge in the social sciences. Chaotic D y n a m i c s in t h e Social Sciences: Exploration a n d M e t h o d The first section o f this volume examines the dynamics o f nonlinear and chaotic systems and focuses on methodological approaches to testing for the presence o f chaos in a time series. Testing for actual "chaos" in time-series data is a particularly important issue, due not only to the technical challenge involved, but also to ensuring that chaotic time series in social science data emulate chaotic time series discovered in data from the natural sciences. W h i l e a variety o f techniques exist to test for chaos, we have concentrated our attention on those approaches that appear to be most often used by chaos researchers. W h i l e each chapter has been written in what we consider a highly accessible fashion, many readers may well prefer to start with the substantive chapters in sections t w o and three and then return later to this section. For those researchers w h o are either beginning to apply chaos theory to empirical work or otherwise interested in some o f the more technical methodological facets o f empirical work in the area, this section should be an invaluable resource. The editors o f this volume lead off this first section with a brief explora tion o f the time series o f nonlinear and chaotic systems. This chapter is highly recommended as a starting point for researchers new to chaos theory. It shows

Introduction

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how an electronic spreadsheet can be used to generate nonlinear and chaotic time series. These series can be used to create graphs and phase diagrams essential to investigating nonlinear time series. This chapter is intended to reveal for social scientists how the dynamics o f a nonlinear differential equa tion emulate much o f the temporal dynamics o f social system behavior. The remaining chapters i n this section are mathematically rigorous ap proaches to the statistical analysis o f chaotic dynamics in social science data. Michael McBurnett's chapter examines the use o f spectral analysis in investi gating the dynamics o f a time series. McBurnett begins w i t h a necessary but tractable mathematical introduction to spectral density and spectral distribu tion functions. He then examines the different types o f time series—periodic, random, and chaotic—and demonstrates with regard to the first two types the problems i n resolving the nature o f a series when "noise" is introduced into the analysis. He concludes by examining known chaotic time series as well as an " u n k n o w n " series. McBurnett's study is an excellent introduction to both the advantages and the limitations o f spectral analysis in testing for chaotic dynamics. A second approach for examining chaotic dynamics relies on the use o f Lyapunov exponents. The Lyapunov exponent measures the extent to which "small" changes in initial conditions produce divergence in a system over time. Thad A . Brown's chapter explores in detail both the advantages and disadvantages o f such an approach. The Lyapunov is shown to be linked to the information gained and lost during chaotic episodes, and hence is linked to the amount o f information available for prediction. This chapter guides the reader through the formal nature o f unfolding subspaces and the state space reconstruction needed to estimate the Lyapunov exponent. The mathematics here are downplayed, i n favor o f words and even some humor. Ted Jaditz's chapter is concerned with the development o f empirical techniques for predicting a time series exhibiting deterministic chaos. A s Jaditz notes in the introduction to chapter 4 , "The Prediction Test for Non linear Determinism," standard linear statistical models provide good fits w i t h data that are taken " i n sample," but out-of-sample predictions do much worse. For this reason, economic forecasters have been attracted to the possibility o f nonlinear determinism in economic data. Jaditz discusses the problems inher ent in determining whether or not a series truly manifests chaos and demon strates some analytical tools for improving forecasting models. Specifically, by using "near neighbor techniques," Jaditz shows how this new approach provides dramatic improvements over conventional linear prediction models typically used by economists, at least with data that are known to be chaotic. Diana Richards's contribution, "From Individuals to Groups: The Aggre gation o f Votes and Chaotic Dynamics," presents another method for testing for chaotic dynamics. Richards applies Devaney's (1989) three-part test for

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chaotic dynamics. A l t h o u g h the application o f chaos theory to the social choice problem leads to several research questions specific to social choice, the intent is to introduce chaotic dynamics to a broader social science audi ence using the case o f the generic social choice problem. The social choice problem is only one o f potentially many examples o f interaction among indi viduals or groups that is nonlinear, and therefore a potential candidate for the domain o f chaotic dynamics. Richards's chapter provides a rich understanding of the translation o f individual preferences into group outcomes and the clas sic problem o f intransitivities. The application o f chaos theory is a relatively small modeling step; it is only a small extension o f existing frameworks into the nonequilibrium realm. However, chaos theory has major implications, in terms o f complex outcomes from simple relationships, in terms o f instability and structural constraints, and in terms o f prospects for prediction for all o f the social sciences. This chapter demonstrates that chaotic dynamics are present in many social choice settings, including some cases o f Arrow-type social choice and in nearly all cases where t w o or more issues are considered simultaneously. Since the aggregation o f individual preferences into outcomes is inherently nonlinear, it is natural to expect chaos theory—the theory o f nonequilibrium nonlinear dynamics—to apply to social choice. It also therefore becomes impossible to make long-term predictions concerning group interactions. However, Richards emphasizes as well the underlying order o f chaotic pro cesses. Specifically, she suggests that a complex "fractal structure" exists, indicating, at a fundamental level, structured stability in the system. Chaos T h e o r y a n d Political Science The use o f chaos and related theoretical and methodological constructs in political science is still in its infancy. Many o f the features that have attracted economists to chaos theory also exist among political scientists. Like eco nomics, much o f political science is concerned w i t h analyzing change and exploring the evolution o f some phenomenon over time. Studies o f changes in aggregate-level electoral fortunes and trends in public opinion such as presi dential approval or attitudes toward particular issues all fit this genre. Accord ingly, such data raise the question o f whether underlying deterministic, and thus potentially chaotic, processes exist. The methodological advances in statistical analysis that have been made in recent years, advances that to a great extent have been borrowed from economics, have made some political scientists more w i l l i n g , and able, to explore the existence o f complex non linear processes. The highly formal game-theoretic and social choice work has required the application o f mathematical tools that are invaluable in chaos research.

Introduction

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The lead chapter in the political science section is Thad A . Brown's "Nonlinear Politics." B r o w n introduces the reader to the role o f chaos in understanding political phenomena. B r o w n points out that politics at every level results from the interactions o f individuals. The difficulty is that, "For mally treating interactive political behavior within massively diverse collec tives is tricky. Interactive behavior is peculiar in that it can neither be pre dicted nor analyzed by observing sets o f individuals cross-sectionally or even the time series from a given individual or group." B r o w n suggests that this characteristic, together w i t h the likely existence o f spatial and temporal phase transitions, calls into serious question traditional methodologies for investi gating chaotic phenomena. B r o w n goes on to explain that cellular automata simulations provide an innovative means o f investigating complex dynamical systems. He also discusses specific applications o f such an approach including game theory, electoral behavior, and social choice theory, all o f which are represented in this volume. Physicist A l v i n M . Saperstein was among the first natural scientists to rigorously apply chaos theory to social phenomena. His chapter, "The Predic tion o f Unpredictability: Applications o f the New Paradigm o f Chaos i n Dynamical Systems to the O l d Problem o f the Stability o f a System o f Hostile Nations," is in much the same spirit as other chapters in this section. L i k e Richards and McBurnett, Saperstein is concerned with the fundamental prob lem o f prediction. Saperstein points out that while the international system shows considerable stability and hence predictability in an overall sense, crises represent episodes o f fundamental instability, ergo unpredictability. Saperstein then points out that an important political "technology" w o u l d be to know when given national security policies w i l l produce instability. I n other words, the aim should be to "predict the unpredictable." Saperstein models several facets o f international interactions, asking fascinating (and long-standing) questions such as whether bipolar or multipolar international systems are more likely to produce conflict, and whether democratic or nondemocratic states are more likely to go to war. The conclusions illuminate w i t h great clarity some o f the most fundamental questions o f the nuclear age. Saperstein's study has the added advantage o f providing simple algorithms that can be used by anyone with a desktop computer to generate an evolution o f national sanity behavior on the part o f nations. The chapter "Complexity in the Evolution o f Public Opinion" by Michael McBurnett explores the dynamics o f public opinion in presidential nomination campaigns. Using data from the 1984 National Election Study's "rolling thunder" survey, McBurnett demonstrates the series to have properties charac teristic o f chaotic behavior. Utilizing techniques discussed in the methodology section, McBurnett shows how different analytic techniques reveal a complex nonlinear deterministic chaos pattern to public support for Democratic presi-

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dential candidates during the 1984 primary season. McBurnett's analyses and findings should be o f profound interest to all serious students o f public opin ion. I f , indeed, the solution o f public opinion can be described as exhibiting deterministic chaos, then the question is raised how one can deduce governing or predictive equations from this time series. Certainly, McBurnett's study suggests how and why drastic shifts in public opinion may occur, and the consequences for predicting the evolution o f public opinion. Chaos Theory and Economics The third section o f the book examines chaos applications in economics. A m o n g all the disciplines we cover, chaotic and, more generally, nonlinear dynamical approaches are most developed in this field. This may be at least partly explained by the mathematical rigor and statistical sophistication that have typified economics for the past several decades. However, the interest in chaos may also have resulted from an increasing dissatisfaction with orthodox equilibrium-based models o f both micro- and macro-level economics phe nomena. Relatedly, the obvious difficulties that economists have encountered in developing adequate predictive models o f behavior almost certainly have helped explain the developing interest by many economists in applications o f chaos theory. J. Barkley Rosser, Jr.'s "Chaos Theory and Rationality in Economics" provides an illuminating theoretical overview o f the implications chaos theory has for orthodox microeconomic theory. Rosser points out that standard neo classical theory makes a number o f information assumptions and that eco nomic agents, more generally, are in possession o f some basic model o f reality. The existence o f nonlinearities that are characteristic o f chaotic sys tems, however, calls into serious question such assumptions o f neoclassical economic theory. The "sensitive dependence on initial conditions," especially, means that the most seemingly trivial initial errors in economic judgment can produce totally unexpected outcomes. Rosser employs these basic characteris tics o f chaotic systems to show how they can produce a variety o f economic phenomena. Most important, he concludes w i t h a discussion o f the kind o f decision-making rule that can be employed where chaos exists. Brian J. L . Berry and Heja K i m ' s chapter is entitled " L o n g Waves 1790¬ 1990: Intermittency, Chaos, and C o n t r o l . " Berry, whose earlier research fo cused on economic and urban geography, has in recent years devoted his considerable abilities to demonstrating the existence o f economic long waves, their origins and impact. This macroeconomic study addresses two questions. First, the annual fluctuations as well as longer-run fluctuations in prices in the United States over the 1790-1990 time period, and second, the fluctuations and swings in the rate o f economic growth over the same period. Using

Introduction

13

graphical analysis, Berry and K i m demonstrate that the inflationary and stag flation cycles o f the last 200 years are characterized by a chaotic limit cycle. Their w o r k shows how chaotic processes can be contained within a larger and more extensive stable series. Berry and K i m also draw some very important policy conclusions from their analyses o f the p o s t - W o r l d War I I period relat ing to Keynesian macroeconomic management techniques. The forms o f human settlement in physical space are the subject o f Dimitrios Dendrinos's "Cities as Spatial Chaotic Attractors." Using an it erative process that places a time series o f human activity in this space, Dendrinos shows how human settlements such as cities can take the form o f periodic, quasi-periodic, or nonperiodic (or chaotic) attractors. Dendrinos indicates, for example, that chaotic patterns are the result o f laissez-faire-type market processes. This analysis distinguishes two distinct forces in locational choice: (1) those that determine the current location o f a population at particu lar points i n time w i t h i n a given space, and that are associated with the attributes o f location at a distance from where populations are currently formed, and (2) those locational forces that determine current location o f populations and are associated with the location o f prior settlement activity. Chaos Theory I m p l i c a t i o n s f o r Social S y s t e m s M a n a g e m e n t a n d Social Science The two chapters in this section serve the purpose o f examining what chaos theory means for social systems management and public policy and for re search and knowledge generation in the social sciences. These chapters attempt to provide insights to both the practical potentialities and the meth odological limitations o f chaos theory as a tool for both altering and under standing the dynamics o f social systems. These chapters raise the philosophi cal issues o f the relevance o f chaos theory to social systems and social science investigation that must be considered i f this research paradigm is to remain robust. Kenyon B . De Greene's "Field-Theoretic Framework for the Interpreta tion o f the Evolution, Instability, Structural Change, and Management o f Complex Systems" begins by pointing out that theories relating to the man agement o f complex systems have tended to lag behind changes in technology and society. He goes on to point out that an increasing gap exists between management capabilities and reality. De Greene develops a model for under standing organizational dynamics and change by employing a field-theoretic framework. The author demonstrates the history o f field theory i n the natural sciences and employs similar approaches to understanding organizational management. Like many other authors, De Greene demonstrates the linkages between macro-level phenomena, in this case the " f i e l d , " which is produced

14


by micro-level events and resulting feedback loops. De Greene's major theoretical contribution is to apply this particular theoretical approach to Kondratiev long wave behavior, showing how such waves encompass much more than just economic waves, but also institutions, technologies, and the like. David L . Harvey and Michael Reed's chapter, "Social Science as the Study o f Complex Systems," provides a capstone for this volume. The authors begin w i t h a discussion o f some important epistemological issues. A m o n g them is the question o f the relationship between the natural and social sci ences, and the role o f chaos theory as a bridge between t w o scientific and intellectual traditions. Specifically, the authors "explore the circumstances under which research strategies employing the deterministic chaos paradigm can and cannot be deployed in the human sciences." As such, Harvey and Reed provide a useful antidote to those who would uncritically apply non linear and mathematical methods and paradigms originally developed for the natural sciences. Taking what they consider a "middle course," the work o f the British philosopher Roy Bhaskar and his modified naturalist epistemology becomes critical for understanding the form a future science o f society might take. In elaborating upon this science, Harvey and Reed present a rigorous demonstration o f how chaos theory fits into various modeling strategies em ployed in the social sciences. These authors also provide a vision o f both the prospects for and limitations o f chaos theory as a means for enhancing our understanding o f the behavior o f complex social systems. Finally, this volume brings together a comprehensive bibliography o f both the chaos literature from the natural sciences and the relevant chaos literature from the social sciences. This definitive bibliography should serve as a valuable resource for all chaos researchers, regardless o f the level o f their mathematical or scientific sophistication and whether or not they are new to the field or experienced chaos researchers. Conclusion The process o f knowledge acquisition in the sciences traditionally follows a logical flow o f hypothesis development, quantification, testing, and validation or falsification. Validation and replication then generally lead to theory devel opment. Such theory aims at explaining the behavior o f systems and expedites prediction o f the future state or behavior o f the system. Such an approach to theory development is founded on assumptions o f global stability and, implic itly, o f linearity in the relationships between variables. Stability in such rela tionships allows prediction. Thus, the behavior o f nonlinear systems chal lenges traditional notions o f theory development. By inhibiting prediction, a fundamental element o f theory building is restricted. Thus, chaos researchers

Introduction

15

face the compound problem o f dealing with highly intractable data that are not easily amenable to traditional empirical analysis, as well as w h i c h , by their nature, may preclude or limit traditional hypothetic-deductive means o f theory generation. The dynamics in the relationships between variables over time in non linear systems may generate complexities that defy generalization. This diffi culty in developing such generalizations underscores the challenge o f building theories that are relevant to complex social phenomena such as government budgeting. The fundamental dynamics o f social phenomena clearly exacer bate theory building in the social realm. A t the same time, however, chaos theory suggests a much richer and interesting w o r l d for the social scientist to explore. For, as Heinz Pagels (1987, 73) has noted, "Life is nonlinear, and so is just about everything else o f interest." Indeed, as the following chapters convey, it is this richness and complexity that readers w i l l find most fas cinating.

Part 1 Chaotic Dynamics in Social Science Data

CHAPTER 1

Exploring Nonlinear Dynamics w i t h a Spreadsheet: A Graphical V i e w of Chaos for Beginners L . Douglas Kiel and Euel Elliott

The mathematical foundation o f chaos theory and the unique vernacular o f this new science can deter some researchers from exploring the dynamics o f nonlinear systems. Terms such as periodicity, sensitive dependence on initial conditions, and attractors are not the usual vernacular o f the social sciences. However, the modern microcomputer and electronic spreadsheet software provide means for the novice to chaos research to explore the mathematics o f chaos. The graphics capabilities o f spreadsheet software also provide a visual means for exploring chaotic dynamics. This is particularly important consid ering the reliance o f chaos researchers on graphical analysis. The intrac tability o f nonlinear mathematics, while often defying solution, is now ex plored via visual analysis. This chapter should thus bring to light the amazing behavior and visual imagery o f nonlinear dynamical systems and their rele vance to social science. Fortunately, the dynamics o f time-based nonlinear systems can readily be explored by researchers new to the study o f chaos theory. This exploration is accomplished here via the use o f a simple algebraic formula and the computa tional and graphical powers o f an electronic spreadsheet. The electronic spreadsheet readily allows the researcher to generate nonlinear time series and then examine these series graphically. O n l y a minimal knowledge o f spread sheets is necessary for the reader to follow the examples in this chapter. Readers are also urged to examine in greater detail the mathematical formula tion and its various dynamics presented here. By examining the chaotic and, more generally, nonlinear behavior in this chapter, the reader w i l l understand that a simple deterministic equation can generate very complex behavior over time. This has considerable value for social scientists as we learn that systems evolve from the simple to the com plex (Prigogine and Stengers 1984). This chapter also reveals the importance o f history to social systems. The initial starting point o f a social system has much to do with its eventual structure and behavior. 19

20


As noted in the introduction to this volume, nonlinear systems can take on a wide array o f behaviors over time. Scientists have, however, classified these behaviors into three distinct types o f time-based regimes. These behav ioral regimes are (1) convergence to an equilibrium or steady state; (2) peri odic behavior or a stable oscillation; and (3) chaos. The most widely used mathematical formula for exploring these three behavioral regimes is a firstorder nonlinear difference equation, labeled the logistic map. This mapping takes on the form x1 t+

—

kx,( 1

—

x) t

The variable to be examined is the value x. The parameter, or boundary value, o f the formula is a constant, k. Remember, chaotic behavior occurs w i t h i n denned parameters. The subscript t represents time and is the current value o f the variable x. The subscript / + 1 represents one time period o f the variable x f o l l o w i n g the previous x r

Mapping this formula also requires an initial starting value. The starting point, usually called the initial condition, is represented by the first value o f x„ x . Once the first value o f x, and the parameter value are determined, a simple "copy" command w i t h the spreadsheet can be used to generate the time series. The copy command serves the purpose o f recursion or feedback by using the previous value to generate the current value o f x . 0

t

A couple o f rules must be followed when using the logistic map. First, the initial condition must be a fractional value such that 0 < x < 1. Second, the parameter value or constant, k, must be greater than 0 and less than 4. Adventurous readers are welcome to explore higher values o f k. The follow ing is the general framework for inputting the logistic map into a spreadsheet. 0

1. I n cell A l , input a fractional value for x between 0 and 1. This number is the initial condition. 2. I n cell B l , input the k value or constant. Remember, this number must be greater than 0 and less than 4. 3. I n cell A 2 , input the formula ( $ B $ 1 - A 1 ) · (1 - A l ) . Cell A 2 repre sents the value x . 4. Next, copy cell A 2 down to cell A 6 1 . This affords sixty iterations o f the equation. 0

t + l

5. Then produce a line graph o f the values in cells A 1 - A 6 1 w i t h the graph function in your spreadsheet. This graph provides a visual image o f what happens as the system "evolves." 6. To change the dynamics o f the time series, simply change the values in cells A l (x) and B l (k). To make a longer time series, just copy more cells down from A 6 1 .

Exploring Nonlinear Dynamics with a Spreadsheet

21

Stable Equilibrium A fascinating aspect o f the logistic map is that each behavioral regime occurs w i t h i n defined mathematical boundaries. For example, values o f k between 0 and 3 (Stewart 1989) w i l l converge to an e q u i l i b r i u m . Table 1.1 reveals three recursions o f the equation w i t h the same initial condition (x ) o f 0.97, but 0

w i t h varying values o f the parameter constant (k). This table reveals that the iterations rapidly converge to a steady state and do not leave this state, or mathematical point, once convergence occurs. Note that as the constant ap proaches three, convergence to a stability requires more iterations. Figure 1.1 reveals the graph generated for the constant (k) o f 2.827 w i t h 100 iterations.

TABLE 1.1. N u m e r i c a l I t e r a t i o n o f Stable Equilibria f r o m t h e Logistic M a p k = 1.95

k = 2.35

k = 2.65

k = 1.95

k = 2.35

k = 2.65

0.97 0.056745 0.10437376 0.18228576 0.29066244 0.40204669 0.46879004 0.48560058 0.48709568 0.48717528 0.48717928 0.48717948 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949

0.97 0.068385 0.14971496 0.29915591 0.49270488 0.58737494 0.56955921 0.57612956 0.57388008 0.57467307 0.57439624 0.57449322 0.57445929 0.57447116 0.57446701 0.57446846 0.57446795 0.57446813 0.57446807 0.57446809 0.57446808 0.57446809 0.57446808 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809

0.97 0.077115 0.18859593 0.40552289 0.6388463 0.61141252 0.62960622 0.61798591 0.62561021 0.6206885 0.62390086 0.62181873 0.62317452 0.6222943 0.62286688 0.62249489 0.62273676 0.62257957 0.62268176 0.62261534 0.62265852 0.62263045 0.62264869 0.62263684 0.62264455 0.62263954 0.62264279 0.62264068 0.62264205 0.62264116

0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949 0.48717949

0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809 0.57446809

0.62264174 0.62264136 0.62264161 0.62264145 0.62264155 0.62264148 0.62264153 0.6226415 0.62264152 0.6226415 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151 0.62264151

22


0.90.8¬ 0.78 0.6-

I

0.5-

&

0.44 0.30.20.10

I 111 I I I 11 11 11 11 I I I I I III 11 I I M I II I I I I I ! ! I ! I I ! II I I ! I I I II II I I I I I I I I I I I I I II I I ! I I I I I 11 I 11 I I I I I I I I

25

50 Iterations

75

x =037 fe = 2.827 o

Fig. 1.1. S t a b l e e q u i l i b r i u m

Periodic Behavior A second type o f nonlinear behavior that can occur over time is periodic behavior. Periodic behavior is cyclical or oscillatory behavior that repeats an identifiable pattern. Such periodic behavior starts to occur when k > 3. This regime initiates instability into the equation as the data start to oscillate. Such a change in the qualitative behavior o f the time series is referred to as a bifurcation, or a branching to a new regime o f behavior. This can be seen in column one o f table 1.2. This first column represents a two-period cycle in which the value o f x moves back and forth between t w o values. A t approx imately the (k) value o f 3.5 (Stewart 1989) a four-period cycle occurs in which four numbers alternate in a consistent pattern. This four-period cycle is shown in column 2 o f table 1.2 and is presented graphically in figure 1.2. This process o f cycles doubling i n the number o f alternating and contin uous patterns o f values is labeled period doubling. It is this continuous bifur cation o f period doubling that eventuates in the "road to chaos" (Feigenbaum 1978). B y exploring the range o f (k) between 3.56 and 3.57, this period

100

Exploring Nonlinear Dynamics with a Spreadsheet TABLE 1.2.

N u m e r i c a l I t e r a t i o n of Periodic Behavior f r o m t h e Logistic M a p

k = 3.25 0.97 0.094575 0.27829935 0.65275867 0.73666056 0.63047328 0.75717435 0.5975494 0.78157337 0.55482842 0.80273 0.51465229 0.81180226 0.49653289 0.81246093 0.49519654 0.81242501 0.49526949 0.81242727 0.4952649 0.81242713 0.49526519 0.81242714 0.49526517 0.81242714 0.49526517 0.81242714 0.49526517 0.81242714

23

k = 3.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

97 10185 32016802 76181161 63509139 81112611 5362019 87041298 39477979 83625048 47927466 87349661 38675099 83011131 49359283 87485632 38318959 82724365 50019058 87499987 38281283 82693509 50089707 87499718 38281989 82694088 50088382 87499727 38281968

k = 3.567 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

97 1037997 33182132 79086073 58998192 86286891 4220694 87008697 4031981 85832504 43375848 87609822 3871983 8463627 46382729 88708271 35729561 81910968 52851887 88884887 35240733 81404791 53995074 88605685 36012471 8219613 52199806 89002388 34914287

k = 3.25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

49526517 81242714 49526517 81242714 49526517 81242714 49526517 81242714 49526517 81242714 49526517 81242714 49526517 81242714 49526517 81242714 49526517 81242714 49526517 81242714 49526517 81242714 49526517 81242714 49526517 81242714 49526517 81242714 49526517

k = 3.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8269407 50088422 87499726 38281968 82694071 50088421 87499726 38281968 82694071 50088421 87499726 38281968 82694071 50088421 87499726 38281968 82694071 50088421 87499726 38281968 82694071 50088421 87499726 38281968 82694071 50088421 87499726 38281968 82694071

k = 3.567 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

81057266 54769367 8836362 3667706 82843549 50697817 89157631 34481474 80584785 55808244 87971647 37744353 83817334 48382357 8908166 34693493 80817906 55297655 88173916 37194968 83326232 49558553 89168049 34452367 80552531 55878584 87942325 37823754 83886531

doubling can be more closely examined. Table 1.2, column 3 shows the result of

the value 3.567, an eight-period cycle (Stewart 1989). This process o f

period doubling continues, as k increases, to periods o f 32, 64, 128, 256, and so o n , until the onset o f chaos. Chaos Chaotic behavior occurs at the (k) value o f 3.8 to 4 . This mathematical regime represents another clear bifurcation or qualitative change i n a system's behav ior. Three divergent values o f (k) are shown in table 1.3 to show the diverse forms that chaotic regimes can take.

24


I

0.1

'I I I I I I I I I I I I I I I I I i t

f-

I I I I I I I I I I I I I I i I I I I I I I I I I I I I M

20

I I I I I I I I I I I

40

60

Iterations - * = 0 . 9 7 fr = 3.50 o

Fig. 1.2. Periodic behavior, f o u r - p e r i o d cycle

Figure 1.3 reveals the chaotic series when k = 3.98 a n d x = 0.90. What distinguishes chaos from the other regimes o f behavior is the lack o f pattern i n its longitudinal behavior. Chaotic behavior does not repeat itself and is thus labeled aperiodic. A close examination o f the decimals in the values o f (k) evidences this point. The reader w i l l also note that chaotic behavior remains w i t h i n definable parameters. W h i l e such chaotic behavior appears random, it is not. Chaos can be generated by a deterministic equation. 0

S e n s i t i v i t y t o Initial C o n d i t i o n s In nonlinear dynamical systems, operating in a chaotic regime, small distur bances can have explosive and disproportionate (nonlinear) effects. W h i l e systems operating in a steady state or periodic regime w i l l "damp" such disturbances, chaotic regimes tend to generate positive feedback and amplify such disturbances. This phenomenon o f chaotic regimes is referred to as a sensitivity to initial conditions. I n short, systems operating in chaotic regimes are very sensitive to small changes. It is this sensitivity that has generated the "butterfly metaphor" in chaos theory. Can the flapping o f a butterfly's wings in

Exploring Nonlinear Dynamics with a Spreadsheet TABLE 1.3.

25

N u m e r i c a l I t e r a t i o n of Chaotic Behavior f r o m t h e Logistic M a p

k = 3.8

k = 3.89

k = 3.98

k = 3.8

k = 3.89

k = 3.98

0.9 0.342 0.8551368 0.47073584 0.94674571 0.19158941 0.58855506 0.92020041 0.27904015 0.76447163 0.68420808 0.82105605 0.55830744 0.93708092 0.22404903 0.66063403 0.8519475 0.47930525 0.94837256 0.18605577 0.57546828 0.92835725 0.25273825 0.71767419 0.7699482 0.67308629 0.83615632 0.52059592 0.94838807

0.9 0.3501 0.88509166 0.39563016 0.93012599 0.25281746 0.73482408 0.75799626 0.71357355 0.79506285 0.63382848 0.90282986 0.34126233 0.87448115 0.42698145 0.95175965 0.17860241 0.57067696 0.95306855 0.17399539 0.55907466 0.95892462 0.15322008 0.50470295 0.97241396 0.10434944 0.36356186 0.90008623 0.34983162

0.9 0.3582 0.91497318 0.30963308 0.85076653 0.5053121 0.99488769 0.02024297 0.07893611 0.2893667 0.81842177 0.59145814 0.96170892 0.14656298 0.49782745 0.99498121 0.01987452 0.07752849 0.28464095 0.81040951 0.61151083 0.94551003 0.20505283 0.64876454 0.90691907 0.33597916 0.88792671 0.39606122 0.95200299

0.18600293 0.57534218 0.92842951 0.25250299 0.71723187 0.77067919 0.67158454 0.83812323 0.51555619 0.94908042 0.18364175 0.56968635 0.93154649 0.24231699 0.69767797 0.801509 0.60455083 0.90846267 0.31600134 0.82134908 0.55759212 0.93739596 0.22300214 0.6584343 0.85461458 0.47214431 0.94705143 0.19055107 0.85611717

0.88477839 0.39656834 0.93088436 0.25027741 0.72991427 0.76687238 0.69545082 0.82389802 0.56440038 0.95636658 0.16232792 0.52895274 0.96923916 0.11597882 0.39883289 0.93268669 0.24422287 0.71800866 0.78761696 0.65070553 0.88414971 0.39844881 0.93238381 0.24524209 0.7200328 0.78416786 0.65837716 0.87492586 0.42568503

0.18185933 0.59217033 0.96118843 0.14847484 0.50319163 0.99495946 0.01996024 0.07785607 0.28574213 0.81229239 0.60684438 0.94956543 0.19060589 0.61401563 0.94326173 0.21300576 0.66718455 0.88375632 0.40886972 0.96194719 0.1456871 0.49536021 0.99491432 0.02013807 0.07853545 0.28802318 0.81616199 0.59716556 0.95742424

China create a tornado i n Texas? I n a nonlinear system such small occurrences may have massive results as behavior alters, changes, and perhaps, explodes over time. Furthermore, systems w i t h very similar starting conditions i n their evolutions may diverge to very different systems and structure over time. This point has obvious implications for social scientists as we explore how vir tually identical systems generate unique histories. The phenomenon o f sensitivity

to initial conditions

can also be examined

using the logistic map. This is best examined by comparing t w o time series w i t h only slightly different initial conditions. By changing the initial condition by a mere one m i l l i o n t h (the last decimal place), a system w i t h the same parameters or boundary values can show very different results over time. These t w o time series i n i t i a l l y appear quite similar and actually map each

26


20

40 Iterations x = 0 . 9 0 k = 3.98 o

Fig. 1.3. Chaotic b e h a v i o r

other perfectly. But once the divergence starts, the time series continue to behave quite differently, as can be seen in figure 1.4. Attractors Even researchers and students new to the study o f chaos are familiar w i t h the notion o f order in chaos. The analyses o f nonlinear time series show that a deeper underlying order exists in these diverse types o f behavior. This order was discovered via graphical analysis o f chaotic time series. Yes, to under stand nonlinear systems, look at the pictures they generate. This deeper order is discovered by an investigation o f the attractors o f a nonlinear system. Attractors provide a qualitative assessment o f dynamic systems in motion (Mosekilde, A r a c i l , and A l l e n 1988, 21). Baumol and Benhabib (1989, 91) define an attractor as "a set o f points toward which complicated time paths starting in its neighborhood are attracted." Pool (1989, 1292) defines an attractor as "the set o f points in a phase space corresponding to all the different states o f the system." More simply, the term attractor is used because the system's temporal evolution appears to be consis-

60

Exploring Nonlinear Dynamics with a Spreadsheet

27

Fig. 1.4. S e n s i t i v i t y t o i n i t i a l c o n d i t i o n s

tently " p u l l e d " to identifiable mathematical points. The attractor functions as an abstract representation o f the flow, or m o t i o n , o f a system. I n short, the attractor stores information about a system's behavior over time. The attractor is used as a means for examining the structure o f the underlying order w i t h i n a nonlinear system. The examination o f an attractor is conducted by a mapping o f the data onto a phase space (Thompson and Stewart 1986). A phase space represents a graphic backdrop for presenting the motion o f time-based data. The examina tion o f an attractor is conducted i n a (/t + 1 phase space (Baumol and Benhabib 1989, 91). I n this case, t (time) represents the current value o f x, while t + 1 (time + 1) represents the next value o f x. The t is plotted on the horizontal axis and t + 1 is plotted on the vertical axis. This method o f plotting the data reveals the relationship between a previous period's mea sured result relative to the current time period's measured result. When plot ting a one equation system such as the logistic map we refer to the phase space as a phase plane. The attractors o f nonlinear systems thus represent a dynamic structure

28


that traces the longitudinal behavior o f a system. Studies o f these attractors, generated by various nonlinear equations, evidence an enormity o f shapes and patterns (Gleick 1987). These attractors thus represent the structures that describe and dominate a nonlinear system during its evolution. Each o f the three nonlinear regimes described above represent a unique attractor type. Steady state attractors converge to a point and remain there. Attractors from periodic regimes are called limit cycles and oscillate around a set o f denned mathematical points, creating a circular pattern. A chaotic or strange attractor takes on a multitude o f surprising shapes. Attractors can be generated using an electronic spreadsheet. This is accomplished by generating an XY graph as follows: 1. Create a chaotic time series for about sixty iterations in column A . This series should be the horizontal axis (0· 2. I n cell C I , place a zero. Then recreate the same chaotic series in column A , under the zero in column C. Make sure you put all values from column A , including the initial value, in column C. This series w i l l be the vertical axis (t + 1). 3. Delete the last cell at the end o f the column C (/ + 1) to ensure each column has the same number o f time periods. 4. Generate the graph to produce the attractor. When the logistic map is operating i n a chaotic regime it creates a h i l l shaped attractor, or parabola. I f the lines are deleted from the graph and only symbols are used, this underlying structure appears. This structure is the order in chaos. Given the nonlinearities that exist in social science data, social scientists may want to explore their data sets using phase space map pings. Researchers may find a wide range o f attractors that describe the order in the apparent chaos o f social science data. Conclusion The dynamism o f social systems suggests that each behavioral regime noted above can appear w i t h i n the long-term behavior o f a nonlinear system. Be cause dynamical systems are historical systems they can reveal many types o f behavior over time. Thus each behavioral type does not reflect permanent commitment only to that behavioral type, but rather reflects one possible type that may occur for a period during the life o f a system. Exploring the behavioral regimes o f nonlinear systems should provide social scientists w i t h a foundation for discovering such behavior in social phenomena. For example, notions o f periodicity have always been used in analyses o f society, as phenomena ranging from fads to vehicular traffic reveal

Exploring Nonlinear Dynamics with a Spreadsheet such oscillatory behavior. Analysts finding social systems that are truly to oscillate at varying frequencies. understanding the uncertain nature evolving systems.

29

may, however, have a more difficult time in a steady state. Social systems do seem Finally, chaos may provide a means for o f social systems as both historical and

CHAPTER 2

Probing t h e Underlying Structure in Dynamical Systems: A n Introduction t o Spectral Analysis Michael McBurnett

This chapter examines one method o f investigating the underlying dynamics of time-series data. It introduces the reader to spectral analysis, a tool for evaluating the frequency properties o f a time series. This is distinguished from the analysis o f the properties o f time series in the time domain, which is the subject o f most recent research in time-series analysis. Since this discus sion serves as an introduction to the topic, I have eschewed mathematical rigor. In this chapter, I focus on three basic types o f time series—periodic time series, chaotic time series, and a random time series—and explore their frequency properties with numerous examples. I also introduce noise into some o f the dynamic processes to illustrate how noise affects spectral anal ysis. In the final section o f the chapter, I introduce a time series whose properties are unknown and analyze its frequency properties. This time series is constructed from survey data collected by the Center for Political Studies. I also introduce some common problems that research has uncovered in the spectral analysis o f time series, particularly as these problems relate to the analysis o f chaotic time series. These problems involve the inability o f spec tral analysis to clearly discern cycles, even when they are known to be pre sent. Sometimes this is a data problem: the time series is too short or the signal-to-noise ratio is too high. Other times the signal resembles noise even though it is not. Unfortunately, the latter case is the more critical problem and the more likely problem when dealing with chaotic processes. The chapter is organized as follows. First, I provide the necessary mathe matical introduction to the spectral density and spectral distribution functions. I show how to interpret these measures and explain some o f the problems i n their use. Then I demonstrate by example how to use spectral analysis and introduce t w o specific cases: pure noise and periodic. Here, I show how the presence o f noise can interfere w i t h the resolving power o f the technique. T h i r d , I introduce several k n o w n chaotic time series and examine their respec31

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five spectra. This section shows that no two time series have identical spectra and that there is no single characteristic spectrum for chaotic dynamics. Fourth, I introduce an empirical time series constructed from surveys con ducted during the 1984 Democratic nomination race. These data are subjected to spectral analysis and compared to some o f the spectra from the prior section. I determine that spectral analysis is not conclusive in determining the dynamics o f this time series. I conclude w i t h a discussion o f the use o f spectral analysis and suggest the use o f some additional measures that can be used to quantify chaos i f it is present in a time series. Mathematical Background Inspection o f a time series may lead one to suppose that it contains a periodic oscillatory component with a k n o w n wavelength. This can be represented by Y, = R cos (cot + 9) + E,

(1)

where co is the frequency o f the oscillation, R is known as the amplitude o f the oscillation, 6 is the phase, and E, is a stationary random series. Sometimes the frequency,/, is expressed a s / = W/2TT, which is a measure o f the number o f cycles per unit time and is easier to interpret. I use this expression throughout in the interpretation o f data. The period o f a cycle, called the wavelength, is given by l / / o r ITTICD. Figure 2.1 shows a graph o f a time series w i t h / = 1/6 and wavelength 6. Equation 1 is extremely simple and, in practice, the variation in an observed time series may be caused by variation at several different frequen cies. For example, presidential popularity may show variation at yearly, quar terly, monthly, and even weekly frequencies. This means that the series shows variation at high (weekly), medium (monthly or quarterly), and low (yearly) frequencies. Equation 1 can be generalized to account for the combination o f variation in the observed series by

Y, = S t f ; COS (ü)jt + Oj) + E„

(2)

where Rj is the amplitude at frequency In the t w o equations shown thus far, it should be noted that neither is stationary i f R, 6, {Rj\, and {0,·} are constants since this condition implies that E{Y ) w i l l shift over time. It is customary to assume that {Rj} are uncorrected random variables uniformly distributed on (0, 2TT), which are fixed for a particular value o f the process (Chatfield 1992). This assumption allows the t

Probing the Underlying Structure in Dynamical Systems I

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treatment o f time series suspected to contain more than one oscillating compo nent as a stationary series. Using the trigonometric identity cos (tot + 6) = cos wt • cos 6 — sin cot • sin 6, equation 2 can be written as

y, = 2

(oj cos ci)jt + bj sin (o ) + E , jt

t

(3)

7= 1

where a- = Rj cos dj and bj = —Rj sin 0,. Clearly, from equations 2 and 3 we see that only a finite number o f frequencies are represented here (the index j counts only from 1 to k). W h y are there not more, indeed, an infinite number of frequencies? Wiener (1949) showed that when k—*«>, any discrete process measured at unit intervals can be represented as

Y. =

I cos (x)t du(a)) Jo

+

Jo

sin (at dv(a>)

(4)

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where u() are uncorrelated continuous processes defined for all a> in the range ( 0 , TT). This equation is called the spectral representation o f the process. Intuitively it helps to think o f Y, as a linear combination o f orthogo nal oscillating terms. One may also wonder w h y the upper l i m i t o f the integrals in equation 4 is •nrather than + ° ° . I f the process were continuous, the l i m i t w o u l d be + ° ° . We are concerned w i t h a discrete

process measured at unit intervals.

1

Hence,

there is no loss o f generality in restricting co to the range (0, TT), because

cos tat cos (TT — o))t

k, t integers where k is even, k, t integers where k is odd.

Variation at frequencies higher than TT cannot be distinguished from variation in a corresponding frequency in the interval (0, TT). The frequency