Chaos Theory in the Social Sciences: 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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Chaos Theory in the Social Sciences

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

3

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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Chaos Theory in the Social Sciences

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

5

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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Chaos Theory in the Social Sciences

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

7

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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Chaos Theory in the Social Sciences

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

9

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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Chaos Theory in the Social Sciences

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

11

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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Chaos Theory in the Social Sciences

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

Chaos Theory in the Social Sciences

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

Chaos Theory in the Social Sciences

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

Chaos Theory in the Social Sciences

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

Chaos Theory in the Social Sciences

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

Chaos Theory in the Social Sciences

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

Chaos Theory in the Social Sciences

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

32

Chaos Theory in the Social Sciences

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

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 1

2

3

A

5

G

7

8

9

10 11 12 13

14

33 L

r

15 16 17 18 19 20

Iterate Fig. 2 . 1 . S a m p l e s i n u s o i d a l f u n c t i o n w i t h w a v e l e n g t h 6 a n d f r e q u e n c y 1/6

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)

34

Chaos Theory in the Social Sciences

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
The Spectral Density and Spectral Distribution Functions The power spectral distribution function, denoted F{co), is the empirical mea­ sure that describes how the oscillations in a time series, i f they are present, are distributed w i t h respect to the frequency. This measure is the one most fre­ quently consulted when examining a time series for a periodic component. Suppose we have a stochastic stationary process. Then the spectral distri­ bution function is a continuous, monotone increasing, bounded function over the interval ( 0 , TT). The function may be differentiated w i t h respect to .

This is the (power)

spectral density function, which is simply referred to as the spectrum. When the derivative exists, and we have a stationary stochastic process w i t h an autocovariance function y(k),

we can express the relationship be­

tween the autocovariance function and the spectral distribution function as

(5)

Set k = 0 and we have

Probing the Underlying Structure in Dynamical Systems

"KO) = 0-2

35

(6)

The physical meaning o f the spectrum is that f(w) du> represents the contribu­ tion to the variance o f components with frequencies in the range (w,
36

Chaos Theory in the Social Sciences

n 0

1 .1

1 .1667

1 .25

1 .3

i

A

r .5

Cycles Fig. 2.2. C o m p a r i s o n o f spectral d i s t r i b u t i o n f u n c t i o n s f o r f i v e selected dynamic equations

The distinctions are quite dramatic. The spectral density for equation 1 is shown as a step function, which means that all the variance in the spectrum is accounted for by a single peak. This is also the case for the logistic equation with the driving parameter at 3.5, which produces a four-cycle. Its spectral distribution function has a single peak at 0.25 cycles per iteration and a corresponding step i n this figure at 0.25 cycles. The Lorenz equations produce a very different spectral density, which is mainly confined to the left region o f this figure. This figure shows that the Lorenz equations are dominated by a number o f short (high) frequencies spread over a relatively narrow range, which accounts for all the variance. The two remaining equations are the chaotic version o f the logistic equation and a random time series. These functions have spectral density functions that are indistinguishable from one another. Both show that all frequencies contribute about the same amount to the total variance in the time series. The chaotic logistic equation has frequen­ cies o f all cycles present in the series and these cycles are uniformly distrib­ uted across the entire spectrum. This makes it is easy to see why the random series and this equation look similar in this graph: each frequency contributes about the same amount to the total variance. This is a serious impediment to the sole use o f this technique for attempting to identify chaotic dynamics

Probing the Underlying Structure in Dynamical Systems

37

(Glass and Mackey 1988). It shows that deterministic chaos may not be distinguished from noise in some instances by spectral analysis. Despite this difficulty we can often learn about the dynamics o f a process from the fre­ quency properties o f a time series. Below I introduce examples o f the spectral distribution o f a variety o f time series. I n v e s t i g a t i o n of Selected T i m e Series The empirical investigation o f the spectrum o f a process involves the exam­ ination o f a graph o f the spectral density function over the range (0, TT). One first considers the theory underlying the process, and then examines the time series itself. I f one suspects that a periodic component is present, then the series is analyzed. Suppose we have a random process represented by a time series governed by a random process w i t h mean 0 confined to the range (—3.25, + 3 . 2 5 ) . Such a process can be generated w i t h any random number generator. For a purely stochastic process, we should suspect that no fre­ quency w i l l contribute more to the variance o f the spectral distribution func­ tion than any other. I n other words, the spectrum w i l l be essentially flat over its entire range, though some peaks may be present. Such a spectrum is shown in figure 2.3. It is common to rescale the vertical axis by taking the logarithm o f the spectral density estimates and to scale the x-axis by converting the frequency to cycles per unit time. The latter is accomplished by multiplying the fre­ quency by dlltr, where d is the number o f observations per unit time (d = 1 for discrete series). This follows from the fact that variation at frequencies higher than TT cannot be distinguished from a corresponding frequency in the interval (0, TT). Examination o f figure 2.3 shows that the random time series (N = 1,000) is featureless. Put another way, we could say that this spectrum has the same power at all cycles; it could be easily described by a line o f slope zero. Having the same power at all frequencies means that each component o f the spectrum contributes about the same amount to the total variance o f the function. This spectrum is the least interesting case but it provides a backdrop against which to check other, more interesting time series. Let us reconsider equation 1. The function Y = R cos (
r

38

Chaos Theory in the Social Sciences

-7 -8 -

Cycles per Unit Time Fig. 2.3. Featureless s p e c t r u m f o r a r a n d o m process. (/V = 1,000.)

Figure 2.4 shows the spectrum for equation 1, and the spike is present pre­ cisely where it is expected. This spectrum shows that the variance in equation 1 is accounted for by a single frequency that corresponds to the precise wavelength o f the time series. This example demonstrates convincingly the resolving power o f spectral analysis w i t h periodic data. 2

These two examples use a purely random process and a known periodic process to illustrate what to look for i n the spectrum o f a time series. In the next section, I introduce three specific time series and their respective spectra. First I show how the spectrum responds to the presence o f noise in the time series. Second, because our interest lies in the examination and characteriza­ tion o f (potentially) chaotic time series I demonstrate the analysis o f a selec­ tion o f time series w i t h k n o w n chaotic properties both w i t h and without noise present. T h i r d , I analyze the spectrum o f a time series derived from survey data that may have some chaotic properties. Spectra o f Selected T i m e Series In order to illustrate the spectra o f chaotic series, it is useful first to demon­ strate what happens when noise is present in a periodic time series. Social

Probing the Underlying Structure in Dynamical Systems

39

o
o

ÇL

o> o

~i .1

i r 25 .3 .16667.2 Cycles per Unit Time

Fig. 2.4. S p e c t r u m f o r a periodic f u n c t i o n

scientists are well aware that noise is present in all measurements o f social processes. The error in measurements can be due to random or systematic measurement error, the failure to account for a relevant variable, or unknown other means. I n statistics, measurement errors can be problematic (Achen 1986; Greene 1978; Kenny and McBurnett 1992). I n deterministic dynamics, the presence o f error can have a catastrophic effect (Peitgen et a l . , 1992). As a matter o f fact, all models o f dynamical systems w i l l contain some error. One source o f error is the computer. For many models, the only source o f error is that due to computer rounding error. I n physical experiments, error can creep into the measurements from imprecision in the instruments used to take mea­ surements. Social scientists are well aware that many measurements we make are imprecise. Inferences from surveys rely on sampling theory and the sam­ ple error increases as the sample size decreases. What we need to know is how stochastic error affects the resolving power o f the spectral distribution func­ tion. I first examine t w o variants o f a well-known equation to illustrate the resolving power o f the spectral density function in a periodic function, the periodic function w i t h an additive stochastic component, and a chaotic variant of the same equation. I then turn to an examination o f t w o additional w e l l k n o w n chaotic time series.

40

Chaos Theory in the Social Sciences

A Periodic Function The logistic equation Y, = 3.5 · y , _ , ( i -

y,_,)

(7)

with Y = 0.5 produces a time series that contains a four-cycle. This series contains 500 elements. The four-cycle is a stable attractor. The series con­ verges to the four-cycle by the third iteration. The points are .38281, .82694, .50088, and .87499. The series contains no noise except that due to computer rounding errors. Figure 2.5 shows the spectrum for the logistic equation above. A four­ cycle translates into .25 cycles per unit time, and the graph shows a very sharp peak at precisely that value. The remainder o f the graph is flat. What happens when error is present? The same time series produced by equation 7 is used again, but 20 percent error has been added to each observation. The error is normal, with variance equal to .07. This series also contains 500 data points. In this case,

.03 .025

.02 .015 .01 .005

.05

.1

.15

.2

.25

.3

.35

.45

Cycles per Unit Time Fig. 2.5. S p e c t r a l d e n s i t y of t h e l o g i s t i c e q u a t i o n w i t h p a r a m e t e r = 3.5. (The t i m e series c o n t a i n s a f o u r - c y c l e in t h e u n i t interval.)

Probing the Underlying Structure in Dynamical Systems

41

the spectrum is recovered precisely, with the sharp peak at .25 cycles per iteration, as before. The relatively large noise component manifests itself as background noise surrounding the sharp peak. Figure 2.6 shows that noise does not necessarily compromise the resolving power o f spectral analysis. What does the spectrum o f a series with multiple cycles look like? I f the driving parameter in equation 7 is changed to 3.6, the logistic equation produces a chaotic time signal. This means that the series contains 1, 2, 3, 4 , . . . , oo-cycles. The spectrum for this series should be quite different from the previous series, but the length o f this series is not sufficient to allow cycles whose period exceeds 500 to be determined from the data. Figure 2.7 shows the spectrum for this series and it differs dramatically from the previous two.

Here a sharp peak is seen again at .25 cycles, but this peak is straddled

by t w o additional peaks, which are straddled by t w o additional peaks, and these too are bounded by another t w o peaks, after which the resolution ceases to allow a clear identification o f the presence o f additional peaks. This chaotic signal contains multiple cycles as well as broad background noise and the analysis o f its frequency spectrum allows one to identify the presence o f some of them. Notice that the spectrum shows that the time series appears to be

.04 -

.02 .015 -

Cycles per Unit Time Fig. 2.6. Spectral d e n s i t y of t h e logistic e q u a t i o n w i t h parameter = 3.5. (The t i m e series c o n t a i n s a f o u r - c y c l e in t h e u n i t interval.)

42

Chaos Theory in the Social Sciences

.01 ® o

.00875 -

.9 .22 |

.0075 -

S

.00525 -005 -

>> c CO Q «

.00375 .0025 -

o CO


.00125 0 I

I

0

.05

""I

.1

I

I

I

i

.15

.2

.25

.3

1

.35

1

A

n

1

.45

.5

Cycles per Unit Time Fig. 2.7. S p e c t r a l d e n s i t y of t h e l o g i s t i c e q u a t i o n w i t h p a r a m e t e r = 3.6. (The t i m e series c o n t a i n s m u l t i p l e cycles.)

periodic, while it is k n o w n that this series is chaotic. We must recognize that spectral analysis does not allow the identification o f any cycles that take longer than the length o f the time series to complete. Indeed, one prefers a time series o f sufficient duration so that long cycles have ample time to complete several oscillations. W i t h social science data we have to recognize that this requirement is almost never met; hence, we run a constant risk o f failing to characterize correctly the periodicities present in the data. Note also that the use o f spectral analysis to develop a model specifica­ tion containing periodic components may also fail to characterize correctly the "true" model because the data fail to span the full range o f oscillations. I f one is prepared to assume that the length o f the available time series is sufficient to explore the range o f oscillation present in the true time series, then the examination o f the spectral distribution function is appropriate. The Lorenz and Rossler Equations and Their Spectra The most well-known system o f equations in chaos theory is the Lorenz equations. These equations comprise an early deterministic model o f the

Probing the Underlying Structure in Dynamical Systems

43

weather formulated by Edward Lorenz (1963). The system o f equations ( i n dot notation) is JC] = 10(— x , +

x x

= 2

3

28x|

X2

= —(8/3)x + 3

x) 2

-^1-^3

(^)

xxx

2

When integrated numerically they appear to possess extremely complicated dynamics (Hale and Kocak 1991). The spectrum for the Lorenz equations has been studied extensively. Here I reproduce the work o f Farmer et al. (1980). Figure 2.8 shows the spectrum for the Lorenz equations for the parameter values in equation 8 above. The equations were integrated numerically with the Runge-Kutta inte­ grator in Phaser for fifteen iterations with a step size o f 0 . 0 0 1 . The result is a discrete time series o f length 15,000. The variable J C , _ I (

t )

is used to investi­

gate the spectrum. This spectrum is essentially featureless, w i t h high power at low frequen-

44

Chaos Theory in the Social Sciences

cies, and it decays rapidly. The peaks that are present are not well defined and it appears that no periodicities are easily discernible. This result confirms results reported by Farmer et al. (1980) and is one characterization o f chaotic time series. The Rossler equations are another system o f dynamic equations k n o w n to display chaotic dynamics. The Rossler equations describe the spread o f disease and have been used effectively to model measles and whooping cough epidemics in children (Rossler 1976; Schaffer et a l . 1986; Schaffer 1987; Schaffer et al. 1990). The system o f equations is again written in dot notation. i , = ~(x x x

2

3

2

+ x) 3

= X] + -2x = .2 + x x t

3

(9)

2

— 5.7x3

The parameters given in this system o f equations describe what is called "the simple Rossler attractor" (Farmer et a l . 1980). The equations were integrated numerically with the Runge-Kutta integrator i n Phaser for 1,000 iterations with a step size o f 0 . 1 . The result is a discrete time series o f length 10,000. The spectrum for this system o f equations (using x ), as shown in figure 2.9, displays features dramatically different from the Lorenz spectrum. The Rossler equations contain many identifiable peaks corresponding to cycles o f particular length, w i t h broad background noise evident between the peaks. u

The logistic equation, the Lorenz equations, and the Rossler equations all display different spectra. Where periodicities are k n o w n to exist a priori, spectral analysis clearly identifies them as w i t h the logistic equations. The Lorenz and Rossler systems are k n o w n to contain aperiodic cycles. This is evident in the spectrum for the Lorenz system, which displays high power at low frequencies w i t h rapid decay in the spectrum. In contrast, the Rossler system has broad background noise and clearly identifiable peaks correspond­ ing to periodicity at particular frequencies. What we observe in these three examples, w i t h and without noise, is the full range o f possibilities in the spectral distribution functions o f chaotic data. The spectrum for the logistic equation exhibits period doubling, which is a well-known route to chaos. The Lorenz equations, in contrast, show no iden­ tifiable structure other than high power at l o w frequency with a decay in power as the frequency increases. This pattern resembles the spectrum for many economic time series. (See Granger and Newbold 1986, 66 for a similar spectrum derived from the Federal Reserve Board index o f industrial produc­ tion.) Finally, the spectrum for the Rossler equations shows some identifiable peaks at a variety o f frequencies that appear above broad background noise. W h y do we observe so much variability in these spectra, each o f which is

Probing the Underlying Structure in Dynamical Systems

45

Cycles per Unit Time Fig. 2.9. S p e c t r u m f o r t h e Rossler e q u a t i o n s . (This s p e c t r u m has d i s t i n c t features.)

k n o w n to be chaotic? The problem lies in the difficulty in distinguishing between signal and noise in chaotic series. As Glass and Mackey describe the problem: [b]road-band power spectra, perhaps with superimposed peaks, are often associated w i t h chaotic dynamics. Unfortunately "noise" is also associ­ ated w i t h broad-band spectra, and consequently the presence o f a broad­ band spectrum is not adequate to establish chaos as opposed to noise (1988, 48). The fact is that the spectrum provides us with only one piece o f evidence for chaotic dynamics and even that may be suspect. The examples used here were taken from time series whose dynamics are known to be chaotic. I f , for example, the driving parameter in the logistic equation were to be shifted downward by only 0.1 (to 3.5), the spectrum would show the single peak identified in figure 2.4, rather than multiple peaks surrounded by background noise.

46

Chaos Theory in the Social Sciences

Spectrum for the Public Opinion Series In 1984, the Center for Political Studies conducted a telephone poll o f the national electorate, using a probability sample over a very long period. This survey is k n o w n as the Rolling Cross-Section. Although it has some draw­ backs, such as a fairly large sampling error (Allsop and Weisberg 1988), it is the only national sample o f its type. This analysis uses that portion o f the survey containing those respondents interviewed after January 1 1 , 1984, and before June 12, 1984 (153 days). Those respondents were interviewed during the period in which all primary and caucus activity took place. The survey asked respondents to rate the chances o f the various Demo­ cratic candidates winning their party's nomination. As Bartels points out, most respondents failed to rate candidates' chances as i f they had probability theory i n m i n d (1988, app. A ) . Though Bartels used this survey for a quite different purpose, I make use o f his coding scheme, which measures each individual's perception o f each candidate's chance o f winning the Democratic nomination (1988, 3 2 2 - 2 3 ) . The individual responses were averaged according to the date o f the

Probing the Underlying Structure in Dynamical Systems

47

responses. This resulted in a measure o f the perceived mean daily probability o f each candidate's chance o f winning the Democratic nomination. I n order to see i f any interesting dynamical patterns were evident, I smoothed the data using a five-day moving average. Smoothing a time series that contains noise may allow a cyclical or otherwise interesting pattern to be identified visually, i f one is present. Smoothing reduces, but does not eliminate, the level o f noise in the time series. Figure 2.10 shows the phase portrait for Gary Hart's chances o f winning the 1984 Democratic nomination. For a review o f how phase portraits are generated, see chapters 1 and 8 in this volume. The dynamical time path indicates that the probability Hart w i l l w i n the nomination approaches the value .25. This matches Bartels's observation o f Hart's chances o f winning the nomination (1989, 251). Use o f a moving average does not alter the gross structure o f a time series, it merely serves to accentuate what is already present. The approach appears to be quite smooth, given the known level o f noise in the data. The time path progresses upward from the origin, then appears to approach a limiting value. People's perceptions o f Hart's chance o f

.04

.06

.08

.1

.12

.14

.16

.18

Lag10 Smoothed Probability of Jackson Nomination Fig. 2 . 1 1 . Phase p o r t r a i t o f p r o b a b i l i t y o f J a c k s o n n o m i n a t i o n series

time

48

Chaos Theory in the Social Sciences

winning the nomination appear to cycle about a particular value. Notice that Hart is always thought to have a chance o f winning the nomination. He is not eliminated from the competition; he is considered to be a contender through­ out the campaign. This pattern in public opinion matches the pattern in the electoral results closely. The expectation is that these data may contain a cycle w i t h long periodicity, as do the electoral results. Figure 2.11 shows the dynamics o f the probability o f Jesse Jackson w i n n i n g the nomination. The dynamics here are anything but smooth and orderly. The graph contains a shape reminiscent o f a figure eight. The dynami­ cal behavior cycles i n one direction and then reverses, and the two lobes revolve about separate and distinct values. Though the data are obviously noisy, the phase portrait shows that the dynamical pattern is bounded, that is, the dynamics remain near the attractor, and that this does not appear to be an ordinary cycle. Figure 2.12 shows the phase portrait o f the probability that Walter Mondale w i l l w i n the 1984 Democratic nomination. Though it is difficult to see in the figure, the dynamics indicate that the perception o f Mondale's chances o f winning increases rapidly at first, cycling about one value (near 0.50), and

c o '«

c

.7 -

-65 -

CD

S

.45 -

k_

0_ o

.3

.35

.4

.45

.5

.55

.6

.7

.65

Lag5 Smoothed Probability of Mondale Nomination Fig. 2.12. Phase p o r t r a i t o f p r o b a b i l i t y o f M o n d a l e n o m i n a t i o n series

time

Probing the Underlying Structure in Dynamical Systems

49

then drops to a lower value (near 0.35), about which the series continues to cycle. Toward the end o f the race the mean perception o f his chances rises beyond the first level (near 0.60) and continues to cycle about that value. This graph shows extremely complex behavior but it is hidden by the noise present in the data. Taken together, these figures reveal what may be a strange attractor resembling the Lorenz attractor from climate dynamics (Lorenz 1963; Schuster 1989). Strange attractors are associated with chaotic dynamical systems (Schuster 1989). The identification o f this type o f structure underlying the dynamics o f public opinion is important because o f the implications this has for forecasting the outcome o f nomination races as well as the study o f change in public opinion over time generally (Huckfeldt 1990). For example, i f the time series here can be shown to be chaotic, then we w i l l gain a deeper understanding o f why polls are so variable while election outcomes are rela­ tively easy to forecast (Gelman and K i n g 1993). The spectral distribution function for the Mondale time series has some features i n common w i t h the chaotic logistic spectrum. This spectrum is displayed in figure 2.13. It appears that all frequencies contribute about equally to the spectral density. The spectrum is dominated by broadband

i

Fig. 2.13. P o w e r s p e c t r u m f o r t h e M o n d a l e t i m e series ( s p e c t r u m is essentially featureless)

50

Chaos Theory in the Social Sciences

noise. As M o o n (1987) points out, spectral analysis may aid in identifying chaos i n an experimental time series, but the technique is not conclusive i f the system has "many hidden degrees o f freedom o f which the observer is un­ aware" ( M o o n 1987, 45). The Lorenz, Rossler, and chaotic logistic equations have decidedly dif­ ferent spectra. The analysis o f the Mondale series is based on the same principles, and the result reveals a spectrum that favors the chaotic logistic spectrum. Evidently there is no fundamental frequency for the Mondale se­ ries, which w o u l d appear as a sharp peak clearly extending above the back­ ground. Admittedly, the level o f noise in this time series is high and that is probably one source o f the problem in identification o f cycles, i f cycles are present. Some chaotic time series contain cycles, but in many cases (Lorenz equations, Rossler equations) these cycles are aperiodic and this fact mani­ fests itself as a broadening o f the spectrum. Spectral analysis o f the Mondale series does not eliminate a deterministic component in the dynamics. Conclusions Spectral analysis serves to illuminate hidden periodicities in data, when they are present. M a n y chaotic series such as the Lorenz system are aperiodic while some, such as the logistic map, contain many, indeed an infinite number of, cycles. The presence o f clear periodic behavior in a time series is easily discerned. This was demonstrated w i t h the Rossler equations. The Lorenz equations contain aperiodic cycles and the spectrum fails to sift these out o f the series, despite its great length. The logistic map, w i t h driving parameter at 3.6, contains infinitely many cycles, o f all lengths ( L i and Yorke 1975). Spectral analysis cannot discern any cycle whose length exceeds the length o f the data set (500 elements i n the logistic equation). This may be the problem for the time series constructed from the CPS survey data. I f the series has a cycle (or many) whose length exceeds the length o f the observed time series, we cannot discover its presence. I t seems intuitively reasonable to assume that the aggregated survey data contain no cycles longer than the entire nomination period. However, chaotic time series are k n o w n to have aperiodic cycles o f many lengths, so it seems reasonable to assume that i f they are present in the candidate time series they should have been observed, though the large stochastic component here is problematic. This analysis has highlighted one problem present in many frequency analyses o f chaotic time series: the results are ambiguous. The ambiguity arises on one hand from the difficulty in distinguishing between the stochastic component and the periodic signal when the signal-to-noise ratio is high, while on the other hand chaotic signals closely resemble pure noise. In some cases spectral analysis can discern between the two components and in others

Probing the Underlying Structure in Dynamical Systems

51

it cannot. W h i l e it seems that the Mondale (and the Jackson) series contains identifiable peaks corresponding to several frequencies, the series also con­ tains considerable noise. Noise introduces problems in the resolution o f peri­ odicity, especially when the signal-to-noise ratio is high. Glass and Mackey (1988) and M o o n (1989) clearly state that spectral analysis is only one tool to be used in the analysis o f time series that are thought to be chaotic. Here we have an example o f w h y no particular technique is to be relied upon exclu­ sively in the analysis o f suspected chaotic time series. A set o f tests need to be applied to the time series. Spectral analysis is a good place to start, but measures o f the dimension o f the series (the correlation integral) as well as o f the rate o f divergence o f nearby points on the attractor (the Lyapunov expo­ nent) are required for empirical identification o f chaotic motion.

N O T E S

I would like to thank Thad Brown and Jim Kuklinski for their cogent critiques of earlier versions of this paper, and Gottfried Mayer-Kress for suggestions regarding the spectra of a variety of time series. 1. All time series used here are discrete because they are produced by numerical integration or recursion. Numerical integrators rely on small step sizes, which are a discrete approximation to an integral solution. 2. A variety of smoothing (or weighting) techniques are commonly used with spectral analysis to accentuate periodicity (see Chatfield 1991; Wei 1990 for ex­ amples). None of these techniques are used here because smoothing a chaotic time series destroys the basic property of interest! None of the basic research in the natural sciences uses smoothing in spectral analysis because the interest lies not in modeling the process but rather in examining the extant time series for periodicity in its natural state. Hence smoothing is not used here, nor should it be used in general.

CHAPTER 3

Measuring Chaos Using t h e Lyapunov Exponent Thad A. Brown

A t some point in time, all scientific things must be measured and calibrated— even chaos. The Lyapunov characteristic exponent, A, is the clearest measure to prove the existence and to quantify chaos in a dynamical system or time series. Chaos exists when A is positive, and this indicates the system under investigation is sensitive to initial conditions. A dictionary would suggest chaos is a state o f complete confusion, or one lacking any organization. I n the field o f nonlinear dynamics, however, the term seems to imply specific prop­ erties o f turbulence i n a nonlinear system. But what is chaos? H o w can we test for it i n an empirical manner, and precisely, thereby separating it from other complicated yet fully predictable phenomena? As it turns out, the Lyapunov exponent, A, is a powerful measurement tool for separating chaotic behavior from the stable and predictable. Before turning to a discussion o f this mea­ sure, let us review some o f the main features o f chaos. Chaos Chaos exists when the long-term prediction o f a system is impossible. Chaos occurs when the uncertainty o f a system's initial state grows exponentially fast. I n time-series language, chaos fits the criterion that the autocorrelation function o f the time signal goes to zero in finite time. Because trajectories are unstable, errors o f estimation o f initial conditions or parameters, however small, can later accumulate into substantial errors. Thus, forecasts o f future behavior based on the past become problematic as current memory o f the past fails. This has both positive and negative effects on information about the underlying system. Nonlinearity is a necessary, but not sufficient, condition for chaotic mo­ tion. The observed chaotic behavior is due neither to external noise nor to an infinite number o f degrees o f freedom. The source o f irregularity is the non­ linear system's property o f separating initially close trajectories exponentially fast. Forecasting long-term behavior o f chaotic systems may be more akin to 53

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Chaos Theory in the Social Sciences

art than to statistics (Peitgen, J ü r g e n s , and Saupe 1992). Does this imply that it is impossible to predict the long-time behavior o f a political or economic system (or both)? Such systems are potentially chaotic because their initial conditions can be fixed only w i t h finite accuracy. Error may explode. Yet the long-time unpredictability exists only at the individual level (or, in the language o f dynamics, at the level o f individual trajectories). A t the level o f statistical properties o f the time evolution (averaged over different trajectories, say as they evolve from different nearby initial conditions), very definite predictions are possible. Said differently, when or where chaos exists can only be k n o w n by statistical prediction. Trajectories w i l l eventually move only on a small submanifold (chaotic or strange attractor to be described later) of the entire state space, w i t h predictable visitation frequencies o f the different parts o f the attractor. Is it possible, then, to estimate the time up to a shortterm prognosis o f an individual trajectory? Yes. It is the coexistence o f pre­ dictable and unpredictable that leads to the phrase order in chaos in the field of nonlinear dynamic systems. What are the attributes o f deterministic chaos? Systems o f nonlinear equations w i t h time dependencies are the easiest place to find chaos. They range from simple one-dimensional models such as the quadratic equation x —* ax( 1 — x) to more complex three- and four-dimensional equations (Hale and Kocak 1991). Chaotic systems exhibit perverse or complex motion i n that the time series never settles down to simple cycles or equilibrium. Orbits are inconsistent, wandering, and erratic, m i x i n g the properties o f the time evolu­ tion. The key to chaotic behavior is that the equation or the system o f equa­ tions o f motion exhibits a sensitive dependence on initial conditions. Often a series o f states can be seen when the value o f a parameter o f the model is changed and the system moves from one state to another. In time-series data where the exact equation o f motion can only be presumed, chaos is harder to find. Even more challenging is to infer the microscopic process that underlies a time series where chaos has been de­ tected. To know the equations o f motion in politics, economics, and social dynamics, we require just such discoveries o f microscopic laws. We w i l l turn to this discussion and point to some interesting approaches at the end o f this chapter. An Attractor The motion o f a dynamical system is represented by an orbit around an attractor. A n attractor is a closed invariant set that essentially captures all orbits starting in its domain o f attraction.'As a region o f phase space, an attractor exerts an appeal to a system, a point, or a subset o f points toward which any dynamical path w i l l converge. When the behavioral history o f a system is examined, the nature o f the attractor is the core o f inquiry. When a

Measuring Chaos Using the Lyapunov Exponent

55

rat, voter, or business firm is dropped into a completely different environment, a period o f erratic behavior is expected as adjustment takes place. This adjust­ ment is transient behavior that gives way to observed behavior in some equi­ librium state. A n equilibrium state is called an attractor. I f the attractor is a single point, asymptotically it represents stability because all trajectories that begin in some neighborhood o f phase space approach the attractor as time passes. O f course, there are other attractors besides equilibria. L i m i t cycles may exist. L i m i t cycles occur when the orbit is drawn toward a cyclic path, rather than toward a point fixed in space. Attractors can also be complex in that they may be quasi-periodic or strange. A n example o f a quasi-periodic attractor is an orbit on a torus, a doughnut generated by the cosines o f a pair o f incom­ mensurate frequencies. Chaos Characteristics Chaos is measured by the characteristics o f its dynamics: fractal structure o f the attractor; metric K entropy; and the Lyapunov characteristic exponent, A,. We w i l l briefly explore each. Chaotic A t t r a c t o r A chaotic attractor is characterized by exponential divergence away from any point w i t h i n the attractor (Ruelle and Takens 1971). The exponential diverg­ ing movement on a compact attractor means that a trajectory is constantly curling back on the attractor. The strange attractor causes the " f l o w " to con­ tract the volume in some directions and stretch it in other directions. Remain­ ing w i t h i n bounded dominion, the volume and backfolding mixing process produces a chaotic motion o f the trajectory at the strange attractor. I t is this folding motion which produces the fractal structure attributed to chaotic mo­ tion. The properties o f a strange attractor are: (1) the attractor set is such that it cannot be decomposed into smaller closed invariant sets (Ruelle 1980, 1989; Guckenheimer and Holmes 1983; StaniSic 1988). (2) There is a noticeable regularity i n the physical structure or dimension o f the attractor. A point, o f course, has no dimension; a line, one dimension; a surface, t w o dimensions; and so o n . B y contrast, a strange attractor has a noninteger dimension. A strange attractor usually violates Euclidean geometry. After a long series o f iterations, a fractal geometry stems from long-term stretching and folding o f trajectories on a manifold. Trajectories on the attractor end up like a Cantor set (the mathematical equivalent o f a croissant). mum

The fractal dimension has a fantastic interpretation. It delimits a m i n i ­ number o f degrees o f freedom necessary to describe chaotic behavior.

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Chaos Theory in the Social Sciences

The lower the dimensionality, the fewer independent variables are needed to describe the chaotic behavior. O f course, i n a chaotic system the dimension o f the attractor is less than the dimension o f the space w i t h i n which it is embed­ ded (see Feder 1988). A n d , finally, (3) the attractor is sensitive to initial conditions. Sensitivity refers to the amplification o f any arbitrarily small interval o f values by itera­ tion. Sensitivity to initial conditions is the nub o f chaos theory, and indeed may be the point where chaos theory touches science. When nonlinear deter­ ministic models—which are absolutely correct—are sensitive to initial condi­ tions, they cannot be used to predict very far into the future. Lorenz (1963), in his famous experiments on weather patterns, found that the higher iterations appeared statistically independent o f earlier values. The smallest error in the n decimal place becomes magnified and amplified under the numerical leftshift action o f the iterative function. , h

We assume that the volume o f an attractor is much smaller than the volume o f phase space. In a dissipative system the irregular stretching in some directions and contraction in other directions produces a final motion (once transients are over) that may be unstable within the attractor. Points that are arbitrarily near at the initial time step become exponentially separated at the attractor for sufficiently long times. This leads to positive Kolmogorov en­ tropy, the exponential separation that occurs in the direction o f stretching. A Fourier analysis o f motion on a strange attractor reveals a continuous power spectrum. Under conditions o f a linear model, a typical interpretation w o u l d be that there are an infinite number o f Fourier modes or oscillators. Under the conditions o f linear theory, such a reasoning has to take place i n an infinite-dimension phase space. A n interpretation can be either that the system is searching an infinite number o f dimensions in phase space or that the system is evolving in a nonlinear fashion on a finite dimensional attractor. Recognition o f nonlinearities has provoked a fundamental alteration from thinking about Fourier modes to thinking about the information dimension. The tool used to measure this information dimension is ergodic theory (Billingsley 1965). K Entropy A second measure o f chaos is K entropy. K entropy is also referred to as the Kolmogorov-Sinai entropy (Farmer 1982; Schuster 1988). Entropy measures the rate o f information production and the growth o f uncertainty. I n a system increasing in entropy, the number o f possible system states that evolve from some initial distribution over time also increases. In a chaotic system, infor­ mation about the system decreases over time when measured against the initial state. I f subsequent measurements are not made, indeed, the observer

Measuring Chaos Using the Lyapunov Exponent

57

knows less and less over time. W i t h a second set o f measures, however, we end up w i t h far more information about the system, even about the initial state, than w o u l d have been possible at the beginning (Shaw 1981). Consider that conditions where entropy increases are similar to a party system where voters were limited to two or three parties but suddenly are allowed to vote for twice as many. The disorder in the system increases because voters are no longer limited or confined to their prior political space. The increase i n disorder is coupled with an increase in our ignorance about the state o f the system. Before the sudden doubling o f the number o f parties we knew more about the position o f voters. Precisely then entropy, S, is proportional to —S, P log P where P is the set o f probabilities o f finding the system in state {/} and measures the informa­ tion needed to locate the system in a specific state {i}. Thus S captures our ignorance about the system. t

h

t

Lyapunov Characteristic Exponents The Lyapunov characteristic exponents, A,, o f dynamical systems measure the average rate by which the distance between close points becomes stretched or compressed after one iteration. Lyapunovs are generalized eigenvalues over an entire attractor i n that they give the average rate o f contraction or expansion o f trajectories on an attractor (Wiggins 1990). The A, allows further resolution by allowing us to see the (orthogonal) direction o f the growth or compression. For an attractor to be strange, it must have at least one positive Lyapunov exponent. The A, is closely related to the other measures o f chaos. Ruelle (1983, 1989) suggests that K entropy equals the sum o f the positive A,, implying that positive entropy exists in the presence o f chaos. The A, is linked to the information lost and gained during chaotic episodes (Ruelle 1980; Shaw 1981, respectively), and it is closely linked to the amount o f information available for prediction. The A, was "conjectured" to be related to the fractal character o f the attractor (Kaplan and Yorke 1979), and indeed this was proven true ( w i t h qualifications) for typical attractors (Russell, Hanson, and Ott 1980). The A, itself has a fractal, noncontinuous dimension, and that fractal quality is linked to the information available about the system (Kaplan and Yorke 1979; Young 1982; Ruelle 1989). In a one-dimensional system there are three alternative variations o f the Lyapunov. A, < 0 = ^ orbit is stable and periodic A, = 0 ^ orbit is marginally stable A | > 0 =^ orbit is chaotic

58

Chaos Theory in the Social Sciences

A Lyapunov-positive system is by definition chaotic. The positive value o f the Lyapunov exponents indicates orbital divergence and a time scale on which state prediction is impossible. Negative Lyapunov exponents set the time scale on which transients or perturbations o f the system's state w i l l decay to a periodic orbit. A zero Lyapunov indicates a marginally or neutrally stable orbit. This often occurs near a point o f bifurcation. In a more complex phase space, the Lyapunov spectrum is used to classify and describe the dynamics. The sign o f the Lyapunov is reported per k' dimension, again with the negative sign indicating convergence and a positive exponential sign indicating divergence. Hence, for a system denned in two dimensions, there are t w o Lyapunov exponents for the attractors o f the system, A, and A . The transformation from the Lyapunov exponent, A, to the Lyapunov number, e , can be used to give a more precise interpretation. The first Lyapunov number, e , indicates the error amplification; the second number, e , illustrates the degree o f contraction (—) or expansion ( + ) . The sum o f the first and second exponents, e , can be interpreted as a measure of how much the area is either reduced (—, —) or expanded ( + , + ) . For a system evolving on a fixed point where k = 3, the signs o f the Lyapunov are (—, —, — ) . I n such a system there is a continuous convergence on a fixed value. For a l i m i t cycle the Lyapunov characteristic exponents are (0, —, — ) , where convergence occurs on two coordinates, w i t h a marginal limit on the third. A two-torus attractor is indicated by (0, 0, —) and reflects instances where t w o dimensions are marginally stable and one converges. Strange attractors exist i n instances o f at least one positive Lyapunov exponent ( + , 0, —) and reflect instances o f simultaneous convergence and divergence. A purely chaotic state ( + , + , + ) represents utter divergence. Purely chaotic behavior is illustrated by a cantor-like set in a noninteger, fractal dimension. Intuitively, we can define a purely chaotic state where the three Lyapunov exponents are all positive. The sum o f the three Lyapunov numbers, Ai+A2+A3 approximates the maximal average by which the volume changes per time step. h

2

K

A1

n

A 1 + A 2

e

;

Since it is increasingly difficult to see the evolution o f a chaotic system as the divergence o f the trajectories on the attractor becomes more rapid, the Lyapunov provides the most useful indicator o f chaos. It should be mentioned that the usual methods o f statistical investigation w i l l not work when chaos exists. Spectral analysis and traditional time-series model fitting w i l l not work to reveal the underlying determinism. Running multiple experiments over a long time and comparing results at the smallest level o f observation run straight into the consequences o f the "butterfly effect." The slightest measure­ ment error w i l l be magnified, invalidating any comparison or goodness o f fit measures based on the expected versus observed observations.

Measuring Chaos Using the Lyapunov Exponent

59

Some Mathematical Underpinnings for A Schuster (1988) nicely summarized some o f the core mathematical properties o f the Lyapunov exponent. The Lyapunov exponent is a local quantity aver­ aged over an entire attractor. It measures exponential separation from some initial value, x , due to amplified error, e. The following illustration shows i n one dimension these properties: 2

0

N

e

xx+ 0

N

e

0

N

x

where ee * o>

x

ee ^ o>

N iterations N x

f (x )f ( o 0

N

= \f (x

N

+ e) —f (x )\.

0

+ £)'

The limits o f the previous expression

0

can be formally expressed as

N

+ e) N—t-x e - » 0

-f ix ) 0

"

which reduces to

. . .

lim 1

N

df (x ) dx 0

0

Schuster showed that is the average distance between closely adjacent points after one iteration. The Lyapunov also provides an unambiguous measure o f the presence o f chaos and gives us a measure o f the average information gain or loss per iteration. The Kaplan-Yorke conjecture proposed that it was possible to pre­ dict the dimension o f a strange attractor based upon a transformation o f the Lyapunov exponents o f the selfsame dynamical system (Kaplan and Yorke 1979; Russell, Hanson, and Ott 1980; Frederickson et a l . , 1983). In essence the units o f A, are bits o f information per iteration from the base o f the initial measurement. B y an application o f the chain rule applied to the following equation, we get

Jx

f

2

(

x

)

U =j f f [

x

{ x ) ]

L /'i/<w'c*o) =

= / ( * i ) / ' ( * o ) where x = x

Therefore, to evaluate A, we calculate

f(x ). 0

60

Chaos Theory in the Social Sciences

A. =

lim -

l i m — los

lim ^ A/-^»

log

n

/ ' U i )

log fix,)

E 1=0

B y Oseledec's multiplicative ergodic theorem (1968), this limit exists for almost all x . The average amount o f stretching depends upon the initial error as w e l l as on the initial value o f the system. Oseledec's theorem has been used in nearly all explications o f Lyapunovs to refine the definition o f their mathe­ matical underpinning (Eckmann and Ruelle 1985). 0

The implications are as follows: (1) the Lyapunovs are invariant o f the dynamics; (2) because they are independent o f the position o f the trajectory on the orbit, the Lyapunov can be used to classify the attractor; (3) they also define the limits o f predictability for any model. This implies a key interpreta­ tion o f the Lyapunov, since the largest Lyapunov governs the size o f any perturbation to the orbit around the attractor. When the system is Lyapunovpositive, the perturbation grows and predictability is lost. We can precisely k say now that 2 A, is the Kolmogorov-Sinai entropy, which dictates the rate i>0

o f loss o f predictability. Hence the computation o f Lyapunovs becomes cru­ cial since the system w i t h them defines its o w n predictability (Eckmann and Ruelle 1985; Abarbanel 1992). A map can illustrate how the change in the level o f information may occur after one iteration. D i v i d i n g the closed interval [ 0 , 1] on x into n equal intervals, x can occur w i t h a probability o f \ln. B y k n o w i n g w h i c h interval contains x , we gain the following information: 0

0

/

0

= - E ^ log ^ = log i=l 2

n.

2

The linear map, / ( x ) , changes the length o f an interval by a known factor, a | / ' ( 0 ) j . The loss o f precision leads to a loss o f information:

1

1

i

- log n n 2

-\og a 2

Measuring Chaos Using the Lyapunov Exponent

61

Hence,

A/=

- l i m i

2

log

2

\f'(x,)\.

The value o f | A / | is proportional to the Lyapunov, in that \(x ) 0

= log

2

· |A/|.

The larger the exponent, the greater the information loss. The gain or loss o f information reflects the amount o f uncertainty in the dynamical process. W i t h a loss o f information, the amount o f predictability decreases and the certainty in the future is diminished. Thus the Lyapunov exponent allows both a calcu­ lation o f a chaotic signal and an interpretation. Calculating A The calculation o f the Lyapunov exponents depends upon measuring the rate by which close or nearby trajectories split in phase space after small changes in initial conditions. When the equations o f motion are k n o w n , as i n a computer simulation, estimating the Lyapunov is not difficult. When data are used, difficulties arise. As a rule, chaotic systems contain at least one positive Lyapunov exponent. The trick is to calculate the Lyapunov w i t h i n the small­ est possible D-dimensional hypersphere that contains the attractor w i t h i n a specific number o f time steps. There are three control parameters: the embed­ ded dimension, the number o f time steps over which pairs o f points are evaluated, and the precision o f the estimate. A fc-dimensional volume segment grows by the average factor o f A, + A + . . . + \ at each time step. A g a i n , a A > 0 indicates that the attractor is chaotic and that a very small error or perturbation grows in the short run exponentially fast. 2

m

Two algorithms have been used to calculate Lyapunov characteristic exponents: Wolf's algorithm (1985, 1986) and the Eckmann-Ruelle algorithm (1985). The W o l f algorithm tracks a pair o f arbitrarily close points over a trajectory to estimate the accumulated error per time step. The points are separated i n time by at least one orbit on the attractor. The trajectory is defined by the fiducial and test trajectories. They are tracked for a fixed time period or until the distance between the two components o f the trajectory exceeds some specific value. In sequence, another test point near the fiducial trajectory is selected and estimation proceeds. The end product is that the stretching and squeezing are averaged. 3

Figure 3.1 shows

a representation

o f the Wolf computation o f a

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Chaos Theory in the Social Sciences

*o('o> y(t )

y(t )

0

2

Fig. 3 . 1 . S c h e m a t i c r e p r e s e n t a t i o n of t h e W o l f a l g o r i t h m t o c o m p u t e A,. ( F r o m V a s t a n o a n d K o s t e l i c h 1986. R e p r i n t e d w i t h p e r m i s s i o n . )

Lyapunov as presented by Vastano and Kostelich (1986, 102). The initial data point, y(t ), and its nearest neighbor, z ( / ) , are L units apart. Over A f , a series o f time steps from t to t , the two points y and z evolve until their distance, L ' , is greater than some arbitrarily small e. The y value at t be­ comes y ( f , ) and a new nearest neighbor, z , ( / , ) , is selected. This procedure continues until the fiducial trajectory y reaches the end o f the time series. The replacement o f the old point by its substitute point and the replacement o f the error direction by a new directional vector constitutes a renormalization o f errors along the trajectory. 0

0

0

0

0

k

0

k

4

The largest estimated Lyapunov exponent o f the attractor is then

Â, =

/VAf

where M is the number o f replacement steps (where some arbitrarily small e was exceeded) and N is the total number o f time steps that the fiducial trajectory y progressed. W o l f et al. (1985) provide the code for two pro­ grams: one for estimating the Lyapunov spectrum for systems o f differential equations and a second for estimating the Lyapunov from a time series. The code is relatively easy to implement in Fortran or to convert to Turbo C . The I B M PC computer packages developed by Schaffer (1988) and Sprott and Rowlands (1992) have implemented the Lyapunov exponent by means o f the W o l f algorithm. Figure 3.2 illustrates the Lyapunov exponent using the Wolf algorithm for an iterative logistic map. W i t h i n the areas o f the chaotic region, the Lyapunovs are positive. Note that the boundary o f the Lyapunov reflects the erratic nature o f chaos w i t h i n the logistic map. The geometric properties o f the Lyapunov exponent are known to be chaotic. + +

There are several folk wisdoms in estimating the Lyapunov exponent on data using the W o l f algorithm. Schaffer et al. (1988) suggest standardizing the

Measuring Chaos Using the Lyapunov Exponent

63

Fig. 3.2. I t e r a t i o n s of t h e l o g i s t i c m a p (a) a n d a c c o m p a n y i n g L y a p u n o v e x p o n e n t s (b). (From S c h u s t e r 1988. R e p r i n t e d w i t h p e r m i s s i o n o f t h e author.)

data into interval [ 0 , 1]. Such transformations allow for comparisons between different time series and permit a more ready interpretation o f the length scales. Second, Fraser and Swinney (1986) suggest verifying that the esti­ mated Lyapunov exponent is stable over a range o f parameter choices. I n ­ creasing the m a x i m u m admissible length scale between fiducial and test tra­ jectories eventually induces the value o f the Lyapunov exponent to drop. W o l f et a l . (1985) suggest that the m a x i m u m length scales should not exceed 10 percent o f the extent o f the attractor. The Eckmann-Ruelle algorithm offers some marginal gain over the W o l f approach and is considered by some the "current state o f the art" (Peitgen, J ü r g e n s , and Saupe 1992). I t uses a least-squares approximation o f the deriva­ tive matrices, an approach that was independently proposed by Sano and Sawada (1985). Figure 3.3 illustrates the Eckmann-Ruelle algorithm.

64

Chaos Theory in the Social Sciences

Fig. 3.3. S c h e m a t i c r e p r e s e n t a t i o n o f t h e Eckmann-Ruelle m e t h o d . ( F r o m V a s t a n o a n d K o s t e l i c h 1986. R e p r i n t e d w i t h permission.)

The Eckmann-Ruelle method calculates all positive Lyapunov exponents of the attractor. The fiducial trajectory, y , begins with y(t) and k nearest neighbors, z,(/) | / = 1 — k, all o f which are w i t h i n e o f y{t). Each point evolves through a specific period o f time, At, such that z,-(f) and z,.(j + At) are tracked along w i t h y(t) and y(t + At). A least-squares estimate is possible o f the spatial derivative Df (y(t)) o f the series using y(t) and the k points. A t each time step, all points w i t h i n e o f y(t) are used to calculate the best possible fit until all the data are used. 5

The Eckmann-Ruelle algorithm provides a useful way to calculate the entire range o f Lyapunov exponent values, while recovering more than one positive Lyapunov exponent w i t h the W o l f algorithm is substantially more complicated. Conclusion It might be that chaos at some time in the future w i l l be considered a sideshow of nonlinear dynamics. Issues that allow full investigation o f self-organization may simply assume that chaotic states and the nature o f prediction, and indeed theoretical understanding, must take chaos into consideration. The ideas behind chaos may also become a common heuristic device, a way o f understanding complexity, criticality, and nuance largely put aside by the earlier ideas underlying the behavioralism in the social sciences. There is an intuitive appeal o f nonlinear dynamical ideas in certain areas o f social and psychological sciences, where difficulty in measurement and systematic ob­ servations often yield qualitative judgment. As a colleague who studies East­ ern Europe suggested, we have to take the countries and people as they are, w i t h all their warts (complexity) and goofiness (unreliability). Chaos may, however, find a new home and more precise use and mea­ surement in more positive, prescriptive areas o f social inquiry. A t the end o f a

Measuring Chaos Using the Lyapunov Exponent

65

draft manuscript on infinitely repeated games o f incomplete information, McKelvey and Palfrey suggest: " . . . the resulting solutions have properties that are reminiscent o f chaotic dynamical systems, and one wonders whether such a solution describes in either a normative or positive fashion the behavior individuals w o u l d or should adapt in such games" (McKelvey and Palfrey 1992). Hubler and Pines (1993) investigate in game-theoretic logic how players attempt to model, control, and predict future states within chaotic environments. Indeed, they even suggest that competition between two players can lead to a chaotic state. In each instance, the ability o f a player to recognize the presence o f chaos presents a strategic advantage in adapting behavior to that chaos. In each instance, it w i l l be critical to identify the degree o f chaos, distinguishing between the "edge o f chaos" and full-blown chaos. Dynamical nonlinear systems are thought to predict different things than linear systems. Included are a set o f invariant measures such as the Lyapunov exponents or attractor dimensions. Determining their precise interpretation in the social sciences w i l l provide some interesting dissertations. As M a y (1992, 452) j o k e d : "chaos at last reconciled the Calvinist's God and his foreordained w o r l d w i t h the illusion o f free-will that we enjoy; sensitivity to initial condi­ tions makes our w o r l d and fate appear unpredictable, even though it is deter­ m i n e d . " I n low-dimensional political, social, and economic Lyapunovpositive environments, where does this leave choice theory, free w i l l , and individual responsibility?

NOTES Figures 3.1 and 3.3 are reprinted from "Comparison of Algorithms for Determining Lyapunov Exponents from Experimental Data" by J . A . Vastano and E . J. Kostelich, in

Dimensions and Entropies in Chaotic Systems, ed. G . Mayer-Kress (Berlin: Springer¬ Verlag, 1986), by permission of the authors, Elsevier Science, and Springer-Verlag. 1. Formally, a closed invariant set A C R" is an attracting set if there is some neighborhood U(A) such that (x,t) e (7 for; > Oand <S>(x,t)-^>A a s f —» °° for every x G U -U t

0


2. Some research work concerns local Lyapunov exponents that specifically aim at short-term forecasts of orbital instabilities (Abarbanel 1992; MaCaffrey et al. 1992; Nychka et al. 1992; Wolff 1992). These approaches attempt to describe orbital instabilities for a fixed number of time steps ahead rather than over an infinite number of time steps ahead such as with global Lyapunov exponents. These approaches are especially interesting in instances where estimating the Lyapunov exponent is from experimental data. The difficulty with data is that we may not know the exact equation of motion nor the correct embedding dimension for the phase space. Hence we will not know the Jacobian matrices necessary to calculate the global Lyapunov exponents.

66

Chaos Theory in the Social Sciences

Local Lyapunov exponents may hence provide a useful approach to detecting chaos in some cases. 3. B y definition a fiducial distribution makes probability statements about unknown parameter values. 4. It should be noted that every replacement data point should be in the direction of the previous point on the test trajectory. The replacement point, z,(f,), should be located as near as possible to the value y ( / , ) in the direction z (f,). The search for such 0

a value is restricted to a cone of width © and height rj about the fiducial trajectory y. The angular-width value of 0 is adjusted as needed. Wolf (1985) has suggested that the resulting errors in the estimation of the Lyapunov are not sensitive to the choice of 0 or to approximate values of e. Usually, 0 = .349, the value of TT/9. See Vastano and Kostelich 1986 for an excellent discussion of the implementation of both the Wolf and Eckmann-Ruelle methods. 5. Using a least-squares estimate of the Jacobian at y(i), given a flow x = f(x),

the

variation equations u = Df(x)u are integrated using Benettin's algorithm to compute the Lyapunov.

CHAPTER 4

The Prediction Test for Nonlinear D e t e r m i n i s m Ted Jaditz

One o f the more interesting problems o f modern empirical economics is what one might call the forecasting paradox: Standard linear statistical models o f economic phenomena invariably fit very well in sample. However, results for out-of-sample prediction are typically much worse than one might expect given the in-sample fit. This problem has been known in the profession since the late 1940s; see, for example, the discussion in Malinvaud (1970, chap. 4) of some early efforts toward predicting postwar consumption patterns. Given the difficulties that economists have i n forecasting, many in the profession are quite receptive to the possibility o f nonlinear determinism in economic data. Some claim to have found evidence o f chaos. (The most notable examples are Barnett and Chen [1988] and DeCoster and Mitchell [1991].) We call this the chaos hypothesis. I f true, the hypothesis could help to explain the forecasting paradox. I f the data-generating mechanisms are nonlinear, then i t is not surprising that linear models don't forecast very w e l l . Second, i f the data generator is chaotic, then long-run point prediction is hopeless anyway, for reasons that we discuss below. Direct tests o f the chaos hypothesis are problematic. Certain tests, such as the calculation o f dimension and entropy numbers, are quite useful at identifying underlying nonlinear determinism in the experimental sciences. However, these tests are difficult to apply to economic data, and the results to date are highly controversial. Economic data sets tend to exhibit certain features, such as regime shifting and volatility clustering, that make it difficult to sort out whether the underlying data are nonlinear deterministic or non­ linear stochastic. See Jaditz and Sayers (1993a) for a brief overview. A great deal o f literature is emerging that suggests that an approach based on out-of-sample prediction may be able to tell us whether it is likely that the underlying data-generating mechanism is nonlinear. Even i f the underlying process is chaotic, certain special nonlinear forecasting techniques ought to do significantly better at short-term forecasting than the linear tech­ niques typically used by economists. We refer to this hunt for forecastable 67

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Chaos Theory in the Social Sciences

nonlinear structure as the prediction test. This paper w i l l discuss the basis for the prediction test and evaluate the prospects for its application to economic data. We illustrate by applying the test to monthly inflation data. We ask whether nonlinear methods offer statistically significant improvements i n forecast performance over the standard linear models typically employed by economists. To summarize the discussion, the prediction test for nonlinearity is ex­ tremely promising. There are certain difficulties in applying the test and interpreting the results: It is not at all obvious what measure o f forecast accuracy should be used, and certain care must be taken in interpreting the statistical tests o f forecast improvement. Despite these difficulties, out-ofsample forecast improvement is perhaps the most persuasive evidence that one can offer in favor o f the hypothesis o f nonlinearity. Having said all that, the evidence for deterministic nonlinearity in eco­ nomic data is weak. The methods that work very well at predicting even noisy chaos do not offer substantially improved forecasts o f economic time series. This suggests that it is rather unlikely that low-dimensional chaos underlies economic time series. Preliminaries: W h a t Is t h e Chaos Hypothesis? The prediction test works because nonparametric forecasting methods can exploit the subtle dependence that characterizes certain nonlinear determinis­ tic systems. To explain w h y the prediction test works, we must digress some­ what to discuss more precisely the characteristics o f nonlinear systems, and to point out the features o f these systems that allow us to form accurate short-run forecasts. Following is a very brief discussion o f what we mean by a chaotic, deterministic system. We refer the reader to Brock (1986) for a more technical introduction from an economic perspective, and Eckmann and Ruelle (1985) for an introduction from the perspective o f physical systems. D E F I N I T I O N 1. (Brock 1986) The time series {a,}f has a smoothly deterministic explanation if a system exists (h, F, X j such that h: 3t" —* fft andF: 3?" —» 3t" are "smooth" (i.e., twice differentiable almost everywhere) and =l

0

X, =

(1)

F(X,_,)

a, = h(X,)

(2)

To interpret, the relationship X, = F(X,_ ) is an unknown law o f motion involving the unobserved n vector o f state variables. Given an initial starting t

The Prediction Test for Nonlinear Determinism

69

value X i n period 0, the state evolves to F(X ) = X , in period 1, then F ( X , ) = X in period 2, and so on. The sequence o f values o f the state variable {X , X , , X , • · • } defines an orbit o f the system. We introduce some additional nota­ tion: let F'{x) be denned as the f iterate o f F. For example, F (x) = F(F(x)), F (x) = F(F(F(x))), and so on. Thus, X, - F'(X ). I f we knew the precise initial state o f the process and the transition function, F, then in principle we could predict the subsequent evolution o f the process perfectly accurately. Note that the function h can be interpreted as a "measuring device," which provides information about the current values o f the state variables. I n this scheme, we need not observe the state variables directly. O f particular interest is the case where F() is bounded on a closed subspace o f and admits an attractor: Heuristically, the attractor is a parsimonious description o f the long-run behavior o f the system. The long-run behavior o f the system can be envisaged as orbits moving along the attractor. 0

0

2

0

2

th

2

3

0

D E F I N I T I O N 2. A set A is an attractor for the system (h, F, X ) relative to 0

a set U if the following

hold.

(a)

For all x E. A, F(x) G A . This means that once an orbit enters the attractor, all subsequent motion of the system is on the attractor. (Formally, the set A is invariant with respect to the law of motion F.)

(b)

The attractor isn't a union of disjoint sets. (The set A is

(c)

(d)

indecompos-

able.) The orbits generated by F are dense in A. This means that the trajectory visits all neighborhoods of the attractor, so that a single realization of a time series will fully describe the system if that realization has enough observations. (The set A is topologically transitive.) Finally, for the law of motion F, A = n j l F'(U). All orbits that start anywhere in U eventually converge to the attractor. 7

To illustrate these definitions, consider the tent map, whose properties were first described by Saki and Tokumaru (1980). The tent map is a simple nonlinear iterative map from [ 0 , 1] to [ 0 , 1]. The transition function F for the tent map is given by the equations

2(1 - X , )

i f X , < 0.5 i f X, > 0.5

(3)

The phase curve and a sample time path are given in figure 4 . 1 . The attractor for the tent map is just the phase curve, the kinked line segment in (X,, X ,

+ 1

)

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Chaos Theory in the Social Sciences

space from (0, 0) to (0.5, 1), to ( 1 , 0). Starting from any point on the attractor X„ the subsequent value X — F(X,) w i l l also lie on the attractor. Thus, even though a typical orbit o f the tent map appears haphazard to the eye (and to many statistical tests!), it is in fact generated by a very simple rule. I f one knew the rule, one could predict subsequent observations to a high degree o f accuracy. t+l

The Prediction Test for Nonlinear Determinism

71

Note that attractors need not be as simple as this example. One way to describe an attractor is to give a measure o f its complexity. One such measure is the dimension. For the tent map, the attractor is a one-dimensional curve in a two-dimensional space. Hence we say that the attractor has dimension one. Attractors can be highly irregular; many interesting systems exhibit attractors o f fractional dimension. The dimension is generally less than or equal to the m i n i m u m number o f coordinates necessary to represent the state space o f the process. See Eckmann and Ruelle 1985 for further discussion o f this impor­ tant concept. Let us add a trivial complication. Suppose that our observer function is a

t

= h(X ) = Xf

(4)

t

In other words, we do not observe the state o f the system, but we observe some function o f the state o f the system. O f course, i f we knew h(-), we could invert it and thus observe the state o f the system directly. However, we typically do not know enough about h(-) to be able to invert i t . Can we infer anything about the true state o f the system from the observed time series? The answer to this question is given by the Takens Embedding Theorem. Takens's (1985) theorem states that we can learn about the underlying state variables, X„ from "embeddings" o f the observed a,'s. Given a sequence o f observations, {a,}, define the embedding or m-history as: a? = (a„ a, , +l

. . . , a

l + m

^)

(5)

Each m-history consists o f m consecutive observations, so that the first three points in a sequence o f two-histories are ( a , , a ) , (a , a ), and ( a , a ) . There w i l l be a total o f T - m + 1 o f these, or T - 1 histories in the two-history case. Takens (1985) shows that the mapping from the m-histories o f a se­ quence a, to the state vectors X, given by 2

op = {h(X ), t

h(F(X,)),

h(F*(X ), t

. . . , h(F>

2

3

3

4

(6)

is a diffeomorphism ( i . e . , a differentiable map w i t h a differentiable inverse), provided that the embedding dimension, m, is sufficiently greater than dimen­ sion o f the process, F. ( I f the process is n-dimensional, Takens shows that m £t 2n + 1 w i l l certainly w o r k . ) This means that knowing the m-histories, af, is essentially equivalent to knowing the current state o f the system, X,. To illustrate, suppose we observe 250 iterates o f the tent map filtered through the observer function h(X,) = Xf. Plot the corresponding t w o histories, and you get a picture that looks like figure 4.2: the two histories map out a one-dimensional curve in the plane. The plot o f the two-histories o f

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Chaos Theory in the Social Sciences

0.8·-

h(X(M))

0-F 0

1 0.2

1 0.4

| 0.6

-.. -

• _ 0.8

! 1

h(X(t)) Fig. 4.2. E m b e d d i n g o f t h e o b s e r v e d values a ,

the observed values, a,, is in many ways similar to the phase diagram for the unobserved variables, X It is stretched somewhat due to the effects o f the observer function, h, and perhaps it would be stretched or folded differently for a different observer function. However, Takens theorem states that many o f the fundamental topological properties o f the attractor i n the unobserved state space o f the process are preserved in and can be inferred from embeddings o f the observed data. Takens theorem is the most important result in nonlinear science. r

One more feature o f the tent map deserves comment. The tent map is also chaotic. We w i l l say that the deterministic system (h, F, X ) is chaotic i f it exhibits sensitive dependence on initial conditions. 0

DEFINITION initial conditions a point y on the || F'(x) - F'(y) ||

3. A dynamical system exhibits sensitive dependence on if for all x on the attractor, and for all e, there exists a 8 and attractor such that for some t, we have || x — y |j < e and > 8.

In plainer words, the time paths o f the process started from t w o nearby states w i l l diverge eventually. Figure 4.3 illustrates two trajectories o f the tent map, using starting values that differ by 1 0 ~ . Even though the t w o starting values are quite close, by the eleventh iteration, the paths have diverged enough to lose all subsequent resemblance. 4

Sensitive dependence to initial conditions makes long-term prediction o f their future states virtually impossible. Suppose we only know the current

The Prediction Test for Nonlinear Determinism

73

state o f a chaotic process to some fixed degree o f accuracy, due to the impreci­ sion o f our measuring device. Suppose we observe a noisy chaos. D E F I N I T I O N 4. A noisy chaos is a dynamical system (h*, F, X ) where F and X follow definition 1 above, with the observer function "polluted" by the addition of observer noise: 0

0

a, = h*(X ) t

= h(X ) + e,, t

E , a white noise random variable.

(2')

Thus, even i f we knew the transition function, F, forecasting would be com­ plicated by the imprecision w i t h which we observe the current state o f the system. Suppose that we knew that the current state, X„ was in an interval [a, b]. Then we would predict that X , would lie in the interval F([a, b]). I f the process is chaotic, uncertainty about the current state is rapidly magnified, and thus our long-term forecasts o f a noisy chaotic process can do no better than to predict that the process w i l l be somewhere on the attractor. On the other hand, i f the interval [a, b] is small e n o u g h — i f we know the current state to a fair degree o f precision—then our prediction o f the near future states may have a reasonable degree o f accuracy. +1

The chaos hypothesis is just the suggestion that economic data are gener­ ated by a noisy chaos. I f this is true, then there is a nonlinear function, F, that is the data generator underlying economic time series. To implement the prediction test, we use embeddings o f the observed data to try and identify features o f the attractor for this function, F. I f the attractor is simple enough,

74

Chaos Theory in the Social Sciences

i f the observer function does not obscure the underlying structure, and i f the amount o f observer noise is small, we may be able to learn enough about the attractor to aid in short-term forecasts. A n I n t r o d u c t i o n t o t h e Prediction Test The underlying nonlinear structure o f a deterministic system suggests that certain specialized nonlinear forecasting techniques may be able to out­ perform linear techniques at short horizons. Casdagli (1992) describes how near neighbor algorithms can be used to forecast nonlinear processes. Given the current m-history, a™, we may be able to predict the next observation, a

t + l

, by looking at past m-histories, af, s < t, that are sufficiently close to ap.

I f there is a deterministic function underlying the data, and i f the noise-tosignal ratio is small, then i f af is sufficiently close to af, close to a

s + ]

a

t + l

ought to be

. Casdagli shows that i f we have a simple nonlinear process on a

bounded attractor, and i f we have enough observations to get a complete picture o f the attractor, this method can generate highly accurate forecasts o f nonlinear systems. We start w i t h observations on a time series {x^

which we partition

t=lT

into a fitting set 3* and a prediction set 9 \ such as 9= = {x, : m + 1 < t < N }

(7)

f

3> = {x, : N < t < 7"} f

for some N < T. The aim is to use the information in 5F to predict observa­ f

tions i n 9*. For a given lag length, m , construct for each observation {x},= \j,

an

m+

m-history, xf_

x

x

T-\

x

~

( l-\>

x

x

l-2>

· · · > t-m)-

(8)

Thus we have a set o f ordered pairs, { ( * , , xT-\)} = +i,TP° prediction set, we calculate the distance between x™_ and Casdagli suggests using the norm to calculate distances, t

m

l

|| * || = max,-1

|.

r e a c r i

x

n

tr

i < »e V J£

(9)

We then select the k nearest pairs, (x , JCJL, ) , to estimate the parameters i n the local regression, s

x

s

= a

k

+ x?_ ß x

k

+

e. s

(10)

The Prediction Test for Nonlinear Determinism

75

The estimated parameters &Q and d are used to calculate the prediction, k

k

x = a + xT-ikt

(11)

k

The prediction is then used to calculate the prediction error, x, — x = e . Citing the Takens theorem, Casdagli suggests setting the lag length, m, equal to In + I , where n is the dimension o f the process. When the dimension of the process is not k n o w n , we can estimate i t , using (to pick one possibility out o f several) the procedure sketched in Brock and Baek 1991. We should point out how this procedure compares to the global linear predictors typically used by economists. For the global linear predictor, we use every observation in the fitting set to estimate the parameters o f the regression. I f the underlying process is sufficiently nonlinear, the global linear predictor w i l l do a poor j o b o f approximating the relationship between the histories and the futures, and the resulting forecasts w i l l be inaccurate. The essence o f the prediction test is to compare the forecasts generated by the linear methodologies preferred by economists to the forecasts gener­ ated by the nonlinear, nearest neighbor algorithms preferred by physicists. I f the near neighbor algorithm gives more accurate forecasts, then we have evidence that the underlying process is nonlinear. To evaluate forecasting performance, we use the Root Mean Square Error ( R M S E ) o f the forecast. Given a time series, { y y , • • • , y } , and a set o f forecasts for that series, { y , , y , • • • , y }, we define the R M S E as t

1 (

2

2

tk

r

T

0.5

(12)

RMSE =

Many authors further normalize the R M S E by dividing by the standard devia­ tion o f the data that we are trying to predict: Define the normalized R M S E as NRMSE = RMSE/cr

r

This normalized R M S E attains a value o f 1 i f the method o f prediction is no more accurate than forecasting the unconditional mean o f the prediction set. This normalization is quite useful as an aid to the interpretation o f the results. We w i l l follow custom and report only the normalized root mean square error. Another example is in order. To evaluate this procedure on a short data set, we generated 600 observations from the tent map, using a starting value of 0.1234567891. For our example, we set the fitting set to be the first 500 observations, and the prediction set to be the last 100 observations o f the

76

Chaos Theory in the Social Sciences

process. We first attempted to forecast the process w i t h the global linear predictor. We estimate an A R ( 3 ) model, 3

X,

=

(13)

a

We use the estimated parameters from the fitting set to generate predictions for the data in 3 . The results for the linear model are summarized in table 4.1 and are quite poor. First, the linear model does not fit the data very w e l l . The regression R is only about 0 . 0 1 ; the regression explains only about one-half of 1 percent o f the total variation o f the process. The only significant coeffi­ cient is the regression intercept. Out-of-sample forecast performance is dis­ mal. The normalized R M S E for the A R ( 3 ) model is 1.024, which is worse than the result obtained i f we just use the unconditional mean o f the fitting set as our forecast. <

>

2

I f we apply the near neighbor methodology, we obtain far superior re­ sults. In figure 4 . 4 , we plot the normalized R M S E as a function o f the number of near neighbors used in the regression. For a fitting set o f 500 observations, we can set the number o f near neighbors used in the regression from one to 500. The m i n i m u m o f the N R M S E plot is the best near neighbor predictor. The plot is minimized at k = 5: our best prediction o f an element in SP is obtained when we comb through the fitting set and calculate our prediction based on the five observations in the fitting set that are most similar to the history o f the observation that we are trying to predict. A regression calcu­ lated using only those five observations w i l l , on average, explain nearly 98

TABLE 4 . 1 . S u m m a r y o f Regression Results f o r t h e T e n t M a p Coefficient Constant

Value

Test Statistic

0.560 -0.059 -0.019 -0.029

12.99 -1.31 -0.42 -0.65

2

Regression R Fitting Set Mean Forecast NRMSE Regression Forecast RMSE Number of Observations

0.012 1.008 1.024 500

Note: The test statistic is the standard OLS test statistic for the hypothesis that the corresponding coefficient is significantly different from zero.

The Prediction Test for Nonlinear Determinism

11

1.2 t

0

100

200

300

400

500

Number of Near Neighbors Fig. 4.4. N R M S E p l o t f o r t h e near n e i g h b o r f o r e c a s t s o f t h e t e n t m a p

percent o f the variation in the process. The forecasts generated by the near neighbor methodology are stunningly more accurate than the forecasts gener­ ated by the global linear predictor, and this provides very convincing evidence of the nonlinearity present in the tent map. It bears mentioning that the N R M S E plot for the tent map is quite typical of the shape o f the plot when significant nonlinearity is present. Here, we observe that the plot is minimized for a relatively small number o f near neighbors, and the plot is smoothly upward sloping from the m i n i m u m out to the m a x i m u m number o f near neighbors, where we are essentially replicating the global linear predictor. For real economic data, the difference in forecast performance between the linear and nonlinear predictors is seldom so clear cut. The normalized R M S E for the nonlinear predictor is typically much closer to the R M S E o f the linear predictor. We are then left to ask whether the forecast improvement generated by the nonlinear models is sufficient to justify the inference that the data generator is nonlinear. It turns out that answering this question is sur­ prisingly difficult. The Problem o f Forecast Evaluation There are a number o f conceptual problems that must be addressed before one can sensibly compare and evaluate forecasts generated by different methods. The first problem is: how should we measure forecast accuracy? Almost all authors focus on a mechanical measure o f forecast accuracy, such as Root

78

Chaos Theory in the Social Sciences

Mean Square Error ( R M S E ) or Mean Absolute Deviation ( M A D ) . Each method o f measuring forecast accuracy has its aficionados, and one can make a purely technical statistical argument for either. However, in economic con­ texts, it is very often useful to measure forecast accuracy in terms o f an underlying objective function. For example, i f one is forecasting the move­ ments o f stock prices, the forecast R M S E or M A D are poor yardsticks o f forecast accuracy. There are two reasons for this. The first reason is that one isn't interested in stock prices per se, but rather in the profits that one could make by f o l l o w i n g an optimal trading rule based on the forecasts. Measured in these terms, an accurate forecast is one that allows the trader to make large (risk-adjusted) rates o f return. Indeed, the empirical finance literature is full o f anomalies that are i n some sense "statistically significant," but economically negligible. It is w e l l k n o w n , for example, that there is an element o f mean reversion in stock prices: a stock that experiences an especially large loss in one week can be expected to rebound somewhat the next. What is debatable is whether the effect is large enough to allow one to make profits in excess o f transactions costs by trading on this information (see, for example, Conrad et al. 1990). I f one cannot design a trading rule that uses this information to generate profits, then the anomaly is not economically significant, no matter how statistically significant it appears to be. The second reason is that there are types o f forecast improvements that these measures are poorly suited to identify. R M S E and M A D are global measures o f forecast accuracy. It is possible to have a method that forecasts poorly most o f the time, but very well some o f the time. The method may still be useful, provided that one can tell when it is going to work and when it w i l l not. Global measures o f forecast accuracy may not be sensitive enough to identify these occasional successes. Having said all o f that, we w i l l continue to focus on the R M S E measure of forecast performance. We w i l l follow the usual practice o f measuring forecast performance by normalized R M S E . The second and much more difficult problem in forecast evaluation is: how much better does the forecast have to be before the improvement is "significant"? The term significant improvement is thrown about quite fre­ quently i n papers on forecasting, and almost all o f the time the use o f the term is not justified. One step toward turning a heuristic procedure into a statistical test is to utilize some recently developed tools for comparing the accuracy o f competing forecast methods. A recent paper by Mizrach (1992) has derived a statistic to test the hypothesis that one forecast has a smaller mean square error than another. The test is a refinement o f an approach due to Granger and Newbold (1986) and Meese and Rogoff (1988). To review the test, let { e , , } be the forecast residuals from method one, and let {e } be the forecast residuals from method t w o . 2t

The Prediction Test for Nonlinear Determinism

79

Granger and Newbold test whether var(e,,) = a] is significantly less than var(e ,) = a\ by looking at the orthogonalized residuals u, = e — e „ v, = i,t 2,t- The correlation between u, and v, is just o-\ — < J \ . Thus, i f the correlation between u, and v, is significantly greater (less) than zero, then o~ is significantly greater (less) than cr . 2

e

+

l t

2

E

2

2

Mizrach's refinement is to generalize the test to biased, heteroscedastic forecast residuals, using Newey-West (1987) hardware. Mizrach shows that given that the t w o sequences are a m i x i n g , the statistic is distributed standard normal asymptotically

R

=

\ 2 «,v,

(

(14)

Vf

V

X w(j') i=k

S (i) uvuv

where

w(0 = 1 -

k + 1

T - i

2

T + i

k = k(T), w i t h l i m _ ^ , r

u ,v u,v,, i+

k(T) fl/2

i < 0

i+t

0,

3

= i n t ( J " ) + 1,

for example. The a - m i x i n g assumption is worth explaining. A sequence o f random variables is independently distributed i f no single observation or group o f observations contains any information that can be used to forecast any aspect o f any other observation in the sequence. A sequence o f random variables is a m i x i n g i f (loosely speaking) the degree o f dependence between observations declines d o w n the sequence. That is, while information in observation x, (say) can be useful in predicting some aspect o f x (say), observation x, contains very little information about x, for sufficiently large s. This is essentially an assumption that anomalies or quirks in the data have only a transitory impact. t+x

+s

80

Chaos Theory in the Social Sciences

To use an economic example, it is essentially the assumption that anything that happened to inflation in (say) 1950 has very little effect on the inflation rate o f (say) 1990. It is essentially the assumption that there is little long-run persistence i n our data. We now have a procedure to test the hypothesis that method one ( R M S E ) generates more accurate forecasts on a given data set than method two ( M A D ) . The test involves applying the Mizrach statistic to test the null hy­ pothesis that the M S E for forecast method number one is equal to the M S E for forecast method number t w o . I f the test statistic is sufficiently large in abso­ lute value, we reject the hypothesis that the two forecasts are o f equal accu­ racy, and thus we have a winner i n the forecasting competition. However, there is a conceptual difficulty w i t h this approach that we should not pass over without comment. The difficulty has to do with certain inescapable problems w i t h economic data. To explain: suppose that we have two competing methods o f forecasting, method one and method t w o . H o w should one decide whether forecast method one is better than forecast method two? To correctly answer this question, one would like to have access to repeated sampling from the data generator, and then look at the distribution o f one's preferred measure o f forecast performance for each method. The answer would depend on the expected values o f the measures o f forecast performance for a typical time series from the data generator. In principle, one could do studies o f the statistical size o f the test under the null hypothesis o f no nonlinearity, and the statistical power o f the test at detecting various types o f nonlinearities. This has not yet been done, and I would venture to guess that one reason for the lack o f results is that the near neighbor forecasting methodology is very computationally intensive. Calcu­ lating the near neighbor forecast for a prediction set o f 500 observations and a fitting set o f 1,500 observations at an embedding dimension o f twenty could easily take a day on a personal computer based on the Intel 80486 chip running at 33 M H z . A convincing Monte Carlo study could require 1,000 iterations for several different data generators. However, given enough com­ puting power, we could conduct such a study o f the reliability o f the method on simulated data sets. On the other hand, the major implication o f the forecasting paradox is that we do not yet have reliable models o f economic phenomena. Economic and financial data exhibit exotic dependencies that are often very difficult to model. There are, for example, controversies over the appropriate model for volatility clustering in economic data. A number o f approaches have been tried, including Auto-Regressive Conditional Heteroscedasticity ( A R C H ) models, unconditional mixture models, and regime-shifting models. N o single model seems to dominate the others for all applications. Worse, even i f

The Prediction Test for Nonlinear Determinism

81

we confine our attention to a single class o f models, one must still demon­ strate that the parameterization chosen for the simulation is sensible. These difficulties all occur because, unfortunately, economics is not (for the most part) an experimental science. Economists (particularly macroeconomists) typically observe only a single, short realization o f their datagenerating process. A time series collected monthly since World War I I (like the Consumer Price Index) has fewer than 600 observations. Data sets col­ lected quarterly (like G N P figures) have less than 200 observations. We cannot do controlled experiments to assess how the economy would respond to controlled shocks. This means that there is, in general, no way to determine whether the economic data we observe are typical or representative o f the underlying process. I n terms o f the prediction test, this means that the best we can expect is for the test to tell us whether method one outforecast method t w o on the given sample. We have no way to determine how the methods w o u l d compare on alternative samples from the underlying data generator. There­ fore, assessing the reliability o f our measure o f forecast performance on real data is a deep problem. These are important caveats that must be kept i n mind when assessing our test results. 1

A n Illustrative Example: The Inflation Rate To illustrate the use o f these techniques, we apply them to a data set generated from monthly observations on the Consumer Price Index for the period Janu­ ary 1947 to February 1993. I take the first differences o f the logs o f the C P I , which results i n the monthly inflation rate. Setting aside the last 20 percent o f the data, we estimate an A R ( 1 2 ) model on the first 442 observations. For the observations * in our filling set, we obtain estimates o f the parameters in the regression 12

y,

= « + 2 Vs-i

ft-i

+ £,·

(15)

The lag length, while plausible, is selected for convenience. I n sample fit o f the model is quite good. The R for the regression is 0.56, indicating that the regression explains a sizable fraction o f the total variation o f the process. This R would correspond to an in-sample R M S E o f 0.44. In the regression, the standard measures o f parameter variability indicate that many o f the coeffi­ cients are significantly different from zero at the 95 percent confidence level. Summary statistics for the regression are given in table 4.2. 2

2

M o v i n g to out-of-sample forecasting, we use the estimated coefficients from the A R ( 1 2 ) to forecast the rest o f the series. For observations o f y, in our prediction set, and their corresponding histories, we calculate forecasts using

82

Chaos Theory in the Social Sciences TABLE 4.2. S u m m a r y o f Regression Results f o r t h e I n f l a t i o n Rate Regression Coefficient Constant X.-2

*,-3 X,_4 *.-3 X.-6 X.-1

Xt-S %l-9 X-l-10

Xt-l 1 X*-I2

Value

Test Statistic

0.00046 0.36696 0.13934 0.13432 0.04699 -0.05750 -0.05846 0.05797 0.09294 0.12793 0.10052 -0.07539 -0.02392

2.13 7.50 2.69 2.59 0.93 -1.19 -1.21 1.20 1.91 2.66 2.19 -1.65 -0.54

2

Regression R Fitting Set Mean Forecast NRMSE Regression Forecast RMSE Number of Observations

0.558 1.011 1.005 442

Note: The test statistic is the standard OLS test statistic for the hypothesis that the corresponding coefficient is significantly different from zero.

the O L S estimates (a, b) o f ( a , /3) from equation 15 above to form the forecasts 12 £

S, = a +

+ r

(16)

i= l

These results are an illustration o f the forecasting paradox. Even though the model fits reasonably w e l l in-sample, the out-of-sample forecasts are spec­ tacularly inaccurate.The normalized R M S E for the fitting set is 1.005. To compare, i f we just use the unconditional mean o f the fitting set as our predictor, our normalized R M S E is 1.011. The regression estimate offers a reduction i n R M S E o f less than one-half o f 1 percent out-of-sample, over the unconditional mean predictor. The conclusion is that the global linear A R ( 1 2 ) model is a very poor representation o f the underlying data-generating process. One then might cast about for a reason w h y performance is so poor. A useful start w o u l d be to estimate the dimension o f the inflation data, to look

The Prediction Test for Nonlinear Determinism

83

for evidence that there may be a low-dimensional attractor underlying the data. Table 4.3 gives the estimated dimension o f the process, calculated at varying lag lengths, following the procedure given in Brock and Baek 1991. To remove spurious linear dependence in the data, we calculate the estimates of dimension from the residuals o f the A R model. Also in table 4 . 3 , we have two other common measures o f nonlinear structure, the BDS statistic (Brock, Dechert, and Scheinkman 1986) and the Kolmogorov Entropy (calculated following the method o f Brock and Baek 1991). A l l three measures o f depen­ dence indicate significant evidence o f nonlinear structure in the residuals o f the linear model. To serve as a comparison, we repeat the procedure on simulated data from noisy chaos. We generated 600 observations from the Henon process (500 from the fitting set and 100 from the prediction set), which is a simple, low-dimensional chaos (Henon 1976). We "polluted" the Henon data w i t h normally distributed pseudorandom numbers with a standard deviation o f 50 percent o f the standard deviation o f the Henon data. This corresponds to a noise-to-signal ratio o f 50 percent, which is fairly high by the standards o f physical systems. When we estimate a linear model to the Henon data, we observe results that are somewhat similar to the results we obtained for the CPI data. In-sample fit is acceptable, w i t h a regression R o f about 0.202. (This corresponds to an in-sample R M S E o f 0.798.) Out-of-sample forecast performance is worse than the in-sample fit, with an out-of-sample nor2

TABLE 4.3.

Nonlinearity Diagnostics f o r t h e Regression Residuals f r o m Inflation Rate Data E

Embedding Dimension 2 3 4 5 6 7 8

=

0.81

E

Estimated Dimension

Test Statistic

BDS

Entropy

Test Statistic

1.569 2.312 3.049 3.794 4.553 5.284 5.945

-2.47 -2.51 -2.47 -2.39 -2.27 -2.19 -2.17

5.71 6.66 6.75 6.88 7.25 7.47 7.72

0.584 0.580 0.569 0.547 0.553 0.550

-2.13 -2.11 -2.19 -2.39 -2.19 -2.12

—

-

=

0.9

BDS

Entropy

Test Statistic

6.05 7.01 7.08 7.22 7.65 7.85 8.00

0.506 0.502 0.490 0.467 0.476 0.480

-2.15 -2.13 -2.24 -2.50 -2.26 -2.09

—

—

Note: The dimension of the process is estimated as the elasticity of the correlation integral with respect to the link scale e. Under the null of independent and identically distributed data ( I I D ) , the expected value of this statistic is equal to the embedding dimension. The test statistic should be distributed as a standard normal random variable under the null of I I D . The entropy statistic is a measure of sensitive dependence to initial conditions. Under the null of I I D , the expected value of the entropy is minus the log of the correlation integral at embedding dimension 1. The test statistic should be distributed as a standard normal under the null. The BDS test is a portmanteau test of I I D . Again, the test statistic should be distributed as a standard normal under the null of I I D .

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Chaos Theory in the Social Sciences

malized R M S E o f 0.913. This is still a bit better than the R M S E o f the unconditional mean forecast, which is 0.996. When we estimate the B D S , dimension, and entropy numbers for the residuals o f the linear model applied to the Henon data, we observe results that indicate that, i f anything, the Henon residuals exhibit less nonlinear dependence than the inflation data. (These results are summarized in tables 4.4 and 4 . 5 . ) When we forecast the Henon data using the near neighbor method, we obtain substantial forecast improvements. The R M S E plot for the polluted Henon data is given in figure 4.5. The R M S E plot is minimized at k = 5 1 . Regressions based on 51 near neighbors yield normalized R M S E o f 0.78. Note that this R M S E is about 15 percent lower than the R M S E for forecasts based on the linear model. When we apply Mizrach's test o f forecast perfor­ mance we observe a test statistic o f 2.13, which is significant at the 95 percent level o f confidence. I n this case, the nonlinear forecast algorithm generates predictions that are significantly more accurate than the forecasts generated by the linear model. This is evidence that there is significant nonlinear determin­ ism underlying these data.

TABLE 4.4. S u m m a r y o f Regression Results f o r t h e N o i s y Henon Data Regression Coefficient Constant

X.-2 X.-3 X.-A

x,-s X.-6 X.-1

x,-, X.-9

Xi -10 Xl-I

1

X/ -12

Value

Test Statistic

0.399 -0.120 0.094 -0.287 -0.059 -0.107 0.001 -0.006 -0.084 0.095 -0.012 0.042 0.038

5.63 -2.45 1.92 -5.84 -1.16 -2.12 0.01 -0.12 -1.66 1.87 -0.25 0.85 0.78

2

Regression R Fitting Set Mean Forecast NRMSE Regression Forecast RMSE Number of Observations

0.798 0.996 0.913 500

Note: The test statistic is the standard OLS test statistic for the hypothesis that the corresponding coefficient is significantly different from zero.

The Prediction Test for Nonlinear Determinism

85

TABLE 4.5. N o n l i n e a r i t y D i a g n o s t i c s f o r t h e Regression Residuals f r o m N o i s y H e n o n M a p Data

e = 0.81 Embedding Dimension

£ = 0.9

Estimated Dimension

Test Statistic

BDS

Entropy

Test Statistic

1.728 2.627 3.499 4.269 4.984 5.687 6.204

-1.39 -1.22 -1.19 -1.33 -1.48 -1.57 -1.80

4.52 3.17 2.90 2.57 2.54 2.57 3.07

0.796 0.782 0.787 0.771 0.762 0.705

-0.60 -0.84 -0.70 -0.96 -1.06 -1.90

—

—

2 3 4 5 6 7 8

BDS

Entropy

Test Statistic

4.69 3.45 3.14 2.60 2.40 2.32 2.57

0.701 0.690 0.706 0.696 0.688 0.650

-0.68 -0.87 -0.53 -0.71 -0.81 -1.40

—

—

Note: The dimension of the process is estimated as the elasticity of the correlation integral with respect to the link scale e. Under the null of independent and identically distributed data (IID). the expected value of this statistic is equal to the embedding dimension. The test statistic should be distributed as a standard normal random variable under the null of I I D . The entropy statistic is a measure of sensitive dependence to initial conditions. Under the null of I I D , the expected value of the entropy is minus the log of the correlation integral at embedding dimension 1. The test statistic should be distributed as a standard normal under the null. The BDS test is a portmanteau test of I I D . Again, the test statistic should be distributed as a standard normal under the null of I I D .

Compared to these results, the results for the inflation data are rather disappointing. The R M S E plot for the inflation data is given in figure 4.6. The N R M S E is minimized at k = 85, where N R M S E = 0.975. This is approximately a 2.5 percent improvement over the N R M S E for the AR(12) model. We check whether this improvement i n forecast performance is signifi-

1.2 j 1.1 •• 1 •• NRMSE 0.9¬ 0.8 • 0.7 •• 0.6 -I 0

1 100

1 1 200 300 Number of Near Neighbors

1 400

Fig. 4.5. N R M S E p l o t f o r t h e near n e i g h b o r f o r e c a s t s of t h e noisy Henon data

1 500

86

Chaos Theory in the Social Sciences

1.2-*

0.8·· 0.7 •• 0.6 -J 0

1

1

1

1

1

100

200

300

400

500

Number of Near Neighbors Fig. 4.6. N R M S E p l o t f o r t h e near n e i g h b o r f o r e c a s t s o f t h e i n f l a t i o n data

cant, using the Mizrach test. The test statistic for this comparison is 1.77. The improvement in forecast performance is not statistically significant by the usual criteria. The lack o f forecast improvement is especially troubling when we con­ sider the computational burden o f the near neighbor approach. The AR(12) model fits the data w i t h 13 parameters, counting the regression intercept. For a fitting set o f 500 and a prediction set o f 100 observations, the near neighbor methodology requires on the order o f 500· 100 = 50,000 regressions, and the calculation o f 750,000 parameters, all to yield an insignificant forecast i m ­ provement. Discussion

The prediction test is a very enticing approach to testing for nonlinear deter­ minism i n a data set. For data from systems known to be chaotic, near neighbor techniques offer spectacular improvements in forecast performance, relative to the performance o f the global linear predictors favored by econo­ mists. The near neighbor methodology is also useful for predicting noisy chaos. These developments are well documented in Casdagli (1992) and references therein. Numerous authors have applied near neighbor algorithms to economic data sets, w i t h no significant forecasting success in evidence in the literature.

The Prediction Test for Nonlinear Determinism

87

Recent efforts include Diebold and Nason 1990, LeBaron 1992, Meese and Rose 1990, Mizrach 1992b, and Jaditz and Sayers 1993b. When the method is applied to economic data, the forecast improvements are generally not statis­ tically significant. Even when there appears to be ample evidence o f nonlinear dependence i n a data set, nonlinear methods are unable to exploit the depen­ dence to offer improved forecast performance. The question is, why does this occur? In our previous discussions, we have touched upon some possible rea­ sons w h y we may be failing to identify nonlinearities. One could question our measure o f forecast performance or our method for testing whether our forecast improvement is significant. Other explanations are also possible. For example, the data set available may be too short to allow us to conclusively identify whether nonlinear determinism is present. I t would be very difficult to detect the presence o f a moderately high-dimensional process in samples as short as the one at hand. High-dimensional chaos may be observationally equivalent to random behavior i n time series this short. However, one could just as easily reject the hypothesis that economic data generators are chaotic, i n favor o f stochastic alternatives. I n their study o f near neighbor forecasting o f current exchange rates, Diebold and Nason make two points on the apparent paradox o f dependence and lack o f forecastability. The first point is that in economic time series, the conditional variance tends to change over time. The standard tests for nonlinear determinism are known to be sensitive enough to pick up this dependence. A discussion o f this point is carried out in Brock, Hsieh, and LeBaron 1991 in the context o f the BDS statistic. W h i l e volatility clustering is easily detected, dependence in variance is not useful for the prediction o f the mean. Thus there may be substantial dependence in the time series that cannot be exploited to improve level predic­ tion. A second possible explanation for the poor forecast performance is that tests for nonlinear dependence may be picking up regime shifts, statistical outliers, or other underlying structural instability. M a n y economic time series are subject to regime shifts. For example, in our inflation data there is gener­ ally conceded to be a regime shift occurring about 1980, when the Federal Reserve Board switched from a policy o f pegging nominal interest rates to a policy o f pegging money supply growth. Occasional regime shifts or lowprobability outliers may be o f limited utility for short-horizon forecasting by any pure time-series method. Hence we reach a paradox. Conventional in-sample measures o f non­ linear dependence suggest that economic data exhibit substantial nonlinear structure. Forecasting algorithms seem to be unable to utilize any o f that structure to improve forecasting. Hence, claims o f nonlinear determinism in economic data are still controversial.

88

Chaos Theory in the Social Sciences

The prediction test does, however, offer hope for a possible resolution o f the dilemma. I f nonlinear methods generate forecast improvements in eco­ nomic time series on the order o f the improvements observed in the forecasts o f physical systems, the case for nonlinear determinism w i l l be irrefutable. Such improvement has not yet been demonstrated. I t is certainly worthwhile to continue the search.

NOTES

1. It is possible to argue that the problem is less severe in financial economics. Tick-by-tick price and volume of sales data are available for thousands of stocks, bonds, and related financial instruments. On the other hand, these many time series are not independent observations from the same data-generating process, nor are they independent in cross section. Thus, many of the same problems remain.

CHAPTER 5

From Individuals t o Groups: The Aggregation of V o t e s and Chaotic Dynamics Diana Richards One

takes the sum of particular wills . . . [and] . . . take[s] away from these

same wills the pluses and minuses that cancel one another . . . [and] . . . the general will remains as the sum of the differences. —Jean Jacques Rousseau

The aggregation o f individual preferences into a group choice is one o f the most significant questions in political science. H o w citizens combine and weigh their interests and desires toward a societal agreement is the foundation of democratic theory. However, the individual-group connection is not as straightforward as early democratic theorists assumed. The process o f reach­ ing a societal agreement between a group o f individuals, each with their o w n preferences and abilities to act strategically, is no longer subsumed under a simple additive relation where the group interest is merely the sum o f individ­ ual interests. It is now well accepted that the individual-to-group connection is capable of serious pathologies, regardless o f the voting scheme used (e.g., A r r o w 1963). But establishing the possibility o f pathologies, such as identifying examples o f voting paradoxes, does not imply that the connection between individual and group is fully understood. We know some conditions under which democratic social choice falls short o f our ideals, but we do not know why it fails or what set o f tools is appropriate to address the "black box" o f the aggregation from individual to groups. This chapter demonstrates that the process o f aggregation from individ­ ual to group is in the realm o f chaotic dynamics. In particular, the nonequilibrium cases o f social choice, where group dynamics fail to reach a stable agreement point, exhibit chaos. This observation accounts for the wide diversity o f potential group outcomes and the nonadditive aggregation func­ tion. These findings also suggest that the tools o f chaotic dynamics may shed light on w h y the instability occurs, where social choice w i l l be most unstable, and how the instability can be reduced. These are future topics. The first task is to establish that the micro-macro connection manifested in social choices exhibits chaos in its nonequilibrium cases. The chapter is organized as f o l 89

90

Chaos Theory in the Social Sciences

lows. The first section examines the social choice debate and its importance for democratic theory. The next section introduces chaotic dynamics using the example o f the logistic function to illustrate the intuition behind examining a chaotic process using symbolic dynamics, in particular the iterated inverse image approach. The third section reviews the multidimensional spatial voting model and the fourth applies the tools illustrated i n the second section to the case o f multidimensional spatial voting. To suggest the robustness o f this connection, the fifth section briefly outlines existing results on addi­ tional cases o f individual-group aggregations, including simple voting and equilibrium-adjusting market mechanisms. A l l these related aggregation schemes also have connections to chaotic dynamics. The sixth section explores an implication o f chaotic dynamics in social aggregation functions: the presence o f an underlying structure created by the interaction o f individuals. Chaotic processes are unusual in that although they exhibit complex, seemingly random possibilities, they are in fact highly con­ strained. The final section concludes by returning to the issues raised by democratic theory and to the implications o f the individual-group connection being chaotic. F r o m Individuals t o G r o u p s Democratic theory, particularly in its populist form, depends on a fundamen­ tal assumption that one can move coherently from individual preferences to a societal choice (Dahl 1956; Riker 1982a). I f societal choices have little rela­ t i o n , no relation, or an inverse relation to the desires o f its citizens, then the meaningfulness o f democracy is called into question. Society becomes an entity that is independent o f the individuals that compose i t . As Riker (1982a) points out, democracy depends on more than the ends it achieves; it is based on the validity o f the means by which it achieves its ends. Voting, as the mechanism by which individual preferences are aggregated into a democratic choice, is at the center o f the legitimacy o f democratic means. Early democratic theorists assumed that aggregating individuals' votes was a coherent process. In some cases, the "general interest" was assumed to be equivalent to the citizens' interests. Rousseau assumed that the social contract created a "moral and collective body" that contained the w i l l o f the people in what he referred to as the "general w i l l . " Similarly, Hegel spoke o f the "national spirit" as a coherent single entity. Where addressed specifically, the process o f moving from individual to group was viewed as an additive relation: an aggregation or sum. Rousseau outlined a scheme where "pluses and minuses cancel one another." Bentham viewed the connection as a sum o f the utilities o f individuals. Even as recently as 1958, Truman ( 1 9 7 1 , 260) spoke o f public opinion as "an aggregate o f the more or less rational opinions held by the individuals who . . . make up the ' p u b l i c . ' " 1

From Individuals to Groups

91

Even these early theorists had already conceptualized the individualgroup connection in terms o f a mathematical function: a mapping rule from a set o f individual preferences (or utilities, in the case o f Bentham) into a societal choice or a societal preference ordering. Rousseau, Bentham, Truman, and countless others interpreted this "function" in additive terms— the simplest possible mapping. Yet this function need not be additive. As A r r o w (1963, 4) points out, there are an infinite number o f possible mappings from individual utilities to a social utility function: the sum o f individual utilities, or their product, or the product o f their logarithms, or the sum o f their products taken two at a time, and so on. A n d even i f one assumes that interpersonal comparison o f utilities is nonproblematic, there is an implicit value judgment in choosing the aggregation rule. The assumption that the connection from individual to group is a straightforward additive relation became weakened by the discovery o f pre­ sumably isolated voting paradoxes. As democracies spread and interest in voting was becoming more common, examples were discovered, such as those by Condorcet, Borda, Dodgson, and Nanson (see Black 1958), where the aggregation o f votes led to "peculiar" social outcomes. These outcomes are peculiar in the sense that a different outcome results i f the voting order is changed, or the addition or withdrawal o f what should be superfluous options causes unexpected and radical changes in the outcomes, such as reversing the social ordering by making the winner the new loser (see, e.g., Black 1958; Farquharson 1969; Fishburn 1973; Ordeshook 1989). The discovery o f these examples was not consistent w i t h what was assumed to be an additive rela­ t i o n , for additive relations do not show such variability and sensitivity. How­ ever, the implications for democratic theory remained small as long as the examples were viewed as interesting, but isolated, curiosities. The discovery o f cases where the aggregation o f individual preferences was problematic led to debates over which voting rule was the "best" in the sense that it w o u l d not be subject to a breakdown in the coherence o f the mapping from individuals to group. As early as the eighteenth century, Borda and LaPlace were debating and advocating certain voting rules as better in terms o f avoiding paradoxes (Saari 1985a). However, A r r o w ' s (1963) General Possibility Theory demonstrated that no procedure exists for passing from a set o f individual preferences to a pattern o f social outcomes that is consistent w i t h a set o f minimally desirable conditions o f any democratic process. A l ­ though problems w i t h the individual-group connection did not occur for every set o f individual preferences (as Black's [1958] single-peakedness condition demonstrates), the possibility o f unstable group outcomes did occur w i t h more than a negligible probability (e.g., N i e m i and Weisberg 1968; Riker and Ordeshook 1973; Ordeshook 1989). A further blow to the assumption o f a nonproblematic additive relation between individuals and groups came when voting involved two or more

92

Chaos Theory in the Social Sciences

issues (Davis and Hinich 1966). When the topic to be voted on contains— either explicitly or i m p l i c i t l y — t w o or more components, then stability in a group outcome is nearly impossible (Cohen 1979; Cohen and Matthews 1980; McKelvey 1976, 1979; Schofield 1979, 1980; Plott 1967). I n this case, any social outcome can arise from a democratic process, since every outcome can be defeated by some coalition o f voters. I n addition, voters have a disturbing potential for manipulating the agenda in order to influence societal outcomes (Gibbard 1973; McKelvey 1976). We now know that the connection between individuals' preferences and group outcomes is not as simple as originally thought. What was once as­ sumed to be a trivial exercise in aggregating or summing individual prefer­ ences in an additive way has become a "black box" that has the potential to transform straightforward individual preferences into outcomes that can be very complex and sensitive to small changes. The function that maps individ­ ual preferences to group choices is not linear and additive; in fact, as w i l l be seen, the mapping is nonlinear. A nonlinear relationship implies that an i n ­ crease in one variable does not cause a uniform increase in the other variable: uniform changes in individual preference orderings do not imply uniform changes i n the social orderings, as is evident in the winner-turn-loser paradox and the inverted-order paradox. There is a domain for functions o f this type: chaotic dynamics. Since the social choice function is nonlinear, it is natural to expect chaos theory—the theory o f nonequilibrium, nonlinear dynamics—to apply to the function that aggregates individual preferences into a social choice (see also Schofield 1993). Chaotic D y n a m i c s Before examining the dynamics o f the mapping from individual preferences to social outcomes using chaos theory, it is worthwhile illustrating the charac­ teristics, methods, and formal definition o f chaos with a simple example from outside o f choice theory. This section introduces the concepts o f chaos theory by exploring the w e l l - k n o w n logistic function. The example is important because it illustrates the definitions and techniques used later in the chapter. The approach uses symbolic dynamics (Devaney 1989) and, in particular, the iterated inverse images (Saari 1991). Using this approach makes demonstrat­ ing Devaney's (1989) three requirements o f chaos, sensitive dependence on initial conditions, topological transitivity, and dense periodic points, rela­ tively straightforward. I n a subsequent section, these techniques w i l l be ap­ plied to the social choice function o f multidimensional spatial voting. One o f the simplest functions that exhibits chaos is the logistic equation, discussed in chapter 2. The logistic function illustrates the three important characteristics o f a function exhibiting chaos: (1) sensitive dependence on initial conditions, (2) topological transitivity, and (3) dense periodic points.

From Individuals to Groups

93

Devaney (1989) defines the presence o f chaos in a function by the satisfaction of these three conditions: D E F I N I T I O N 1. A function exhibits sensitive dependence on initial conditions if two points that are arbitrarily close separate by a distance 8 by the iteration of that function (Devaney 1989, 49). D E F I N I T I O N 2. A function exhibits topological transitivity if the mapping has points that move from one arbitrarily small neighborhood to another arbitrarily small neighborhood (Devaney 1989, 49). D E F I N I T I O N 3. A function has dense periodic points if, given a periodic point, there is another periodic point arbitrarily close by (Devaney 1989, 15, 42). These conditions combine for the formal definition o f chaos (Devaney 1989, 50): D E F I N I T I O N 4. A function initial conditions, topological

is chaotic if it satisfies sensitive dependence transitivity, and dense periodic points.

on

These three conditions have all been informally illustrated w i t h the example o f the logistic function i n chapter 2. Sensitive dependence on initial conditions is evident i n the effect o f a new initial condition. Topological transitivity implies that one can move from one region of x to another region—i.e., that outcomes do not merely repeat the same few values but have a diversity o f potential outcomes. Dense periodic points are evident in the bifurcation diagram. Peri­ odic points are nearby other periodic points; there is a consistency in nearby trajectories. Although all three conditions are intuitively evident in the ex­ ample o f the logistic function, the proof that these conditions are in fact satisfied requires more. Establishing these three conditions formally becomes straightforward when the tools o f symbolic dynamics are used. Symbolic Dynamics The intuition behind the presence o f cycles and chaos in a function becomes clearer w i t h the tools o f symbolic dynamics (see, e.g., Devaney 1989; Saari 1991). This technique permits a cataloging o f the permissible sequences o f the function. It accomplishes this by abandoning point precision, as the previous discussion emphasized, and examining instead how regions map into regions. In particular, this approach reverses the focus to look at how sets o f points map to other sets, rather than at where a single point maps. To illustrate this approach, divide x, = [ 0 , 1] into two regions, labeled /?,

94

Chaos Theory in the Social Sciences

and R , as shown in figure 5 . 1 . The image o f these regions is shown on the x axis by reflecting across the x, = x line. One is interested i n which regions map to which regions via the function. In this case, the function is the one-to-one mapping o f the logistic function. Note that from figure 5.1 it is apparent that region 1 maps to region 2 because for some values o f x, G R , the function results i n some values o f x, G R . Similarly, some values o f x, G R map to region 2. B y a similar logic we can describe the rules o f mapping under the logistic function for all the regions, summarized as 2

t+)

l + i

x

+i

2

2

i) i f x, G R ii) i f x, G R

then x G R; then J C , , G R or R .

x

t + l

2

2

+

x

2

These "rules" on the sequences o f mappings outline some o f the permis­ sible sequences. Each o f these sequences is called a w o r d , and the set o f all permissible words under a particular function is called the dictionary. Each sequence describes the iteration o f a single starting point via that function. Se­ quences that are identical for the first n entries and that differ after the n + 1 entry designate initial points that are nearby. I n other words, the distance between t w o points is measured by the extent to which their sequences match (see, e.g., Saari 1991). The possibility o f cyclic sequences arises because o f the feedback region o f R . B y iterating w i t h i n R cycles o f all lengths are possible; the mapping back to /?, allows for the completion o f the cycle. The rule on the mapping is simply that R must be followed by R \ R can be followed by either /?, or R . t h

2

2

x

2

2

2

For example, a four-cycle sequence is possible: (/?,, R , R , R , R, R , . . . ) . Finding this four-cycle sequence is not obvious w i t h the forward approach used in choice theory. The iterated inverse image approach (Saari 1991) identifies where the cycle must appear. First, find all points in R that map to the first target region o f 7", = R , this is given b y / (T ) = f~ *(R ) H R . However, the goal is not to map to any point in R , but to the subset o f points in R where the next iterate stays in R . This means we need to refine the starting region o f /?, to the refined target o f T = f~ (R ) D R C 7",. These p o i n t s / ( T ) C f~ (T ) are all the points that start in R , go t o R , and on the second iterate remain in R . 2

2

2

t

2

x

_ l

2

t

{

2

2

2

2

l

2

- X

2

2

l

2

{

x

2

2

The same argument ensures that the third iterate is in R . Here the refined target region is T = / - ' ( / - ' ( / ? ) <~l R ) C T , a n d / - ' ( T ) <= f~\T ) C f~ (T ) C R,. A t each stage, the set o f initial points is refined to include only those points that hit the refined target—this is the region that ensures that more o f the steps o f the proposed cycle occur. The final stage, T = / ~ ' ( . . . ( / (/?,)) C 7" , defines the set o f inverse iterated points f~ (T ) C R . This is the set o f all points in R where the image o f the fourth iterate is i n 2

3

2

2

2

3

2

l

}

4

_ 1

l

3

x

4

t

From Individuals to Groups X 1 =4gx (1-X ) 1 ~y t+

1.0

: :

/

/

/

/

/

JC. . . . . . .

/ /

/

/ 1/

/

/

/

/

/

/

/

......

/

/

/

'

\ /

\

V.

t+1

t

t

\: \ : :' x

95

\.

/

.

•

0.0

R

F?i x

2

10

t

Fig. 5 . 1 . M a p of t h e l o g i s t i c f u n c t i o n

4

/?,. Trivially, there is at least one point x G R so that f (x) period-four orbit. x

= x. This is the

The possibility o f cyclic points and o f a set o f nonperiodic or chaotic points arises because o f the presence o f a feedback in the logistic function. It is the fact that region one maps to region t w o , which maps back to region one, that allows for the abundance o f permutations o f sequences. A similar feed­ back process occurs in nonequilibrium social choice, evident in the presence of an intransitivity among three alternatives. Using symbolic dynamics, Devaney's three conditions for chaos can easily be shown to be satisfied, verifying that the logistic function is indeed chaotic. To establish sensitive dependence on initial conditions, one must demonstrate that two points that are arbitrarily close w i l l separate by a dis­ tance 8 by the iteration o f the function. This requires that one can construct two sequences o f symbols 1, 2 that are identical for n entries and that eventu­ ally differ i n their n + 1 entry. Let the distance between regions 1 and 2 be 8. Assume that the « entry o f these two sequences is 1. Then the n + l entry can be 1 for the first sequence and 2 for the other, separating the points by 8 and demonstrating sensitive dependence on initial conditions. t h

t h

l h

v

Topological transitivity requires that the mapping have points that move from one arbitrarily small neighborhood to another arbitrarily small neighbor-

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hood. Then any point that begins in one and ends in two satisfies this condi­ tion. It is straightforward to come up w i t h a permissible sequence based on the mapping rules o f the logistic such that the two sequences end in different symbols. The condition o f dense periodic points requires that, given a periodic point, there is another periodic point arbitrarily close by. Recall that distance between the iteration o f points is evident by the extent to which two sequences match. Given a sequence o f arbitrary length, it is possible to find another sequence that matches the beginning n entries o f the first sequence, demon­ strating the existence o f two periodic points arbitrarily nearby. This is obvious in terms o f the permissible sequences o f 1, 2. For example, the initial condi­ tions leading to (1222222) and (1222221) would be nearby points. The satisfaction o f these three conditions verifies that the logistic func­ tion is chaotic. Devaney (1989, 50) summarizes these conditions in terms o f three characteristics that a chaotic function possesses: unpredictability, indecomposability, and an element o f regularity. The presence o f sensitive depen­ dence on initial conditions makes long-term prediction o f a chaotic process impossible. Chaotic systems are indecomposable because they cannot be bro­ ken d o w n into two subsystems that do not interact; this arises because o f topological transitivity. T h i r d , although a chaotic process exhibits complex, apparently random, behavior, there is an element o f regularity, in that periodic points are dense and nearby points behave similarly. M u l t i d i m e n s i o n a l Spatial V o t i n g These same techniques can be applied to the case o f voting by considering the social choice function as a mapping rule. This section summarizes some o f the notation and main results o f multidimensional spatial voting (see Kxehbiel 1988 or Strom 1990 for a complete survey), and can be skipped by those familiar w i t h the standard notation and approach. The choice among alternatives is often modeled spatially (Davis and Hinich 1966), where the alternatives to be voted on are arranged on a contin­ uum, such as the liberal-conservative spectrum or the amount o f a budget allocation. The one-dimensional spatial approach has led to several interesting implications, notably Black's (1958) median voter theorem, stating that i f all voters' preferences are single-peaked then a stable outcome emerges at the median voter, and Downs's (1957) results on party competition and the ten­ dency for parties to move toward the center o f the distribution o f voters. The single-dimension continuum can be extended to include voting over several policy dimensions, such as when an issue implicitly contains two or more components or when issues are linked and must be decided as a package. Assume an alternative space o f issues with m issue dimensions. Assume

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a set N = { 1 , 2, . . . n) o f voters where each voter has preferences over the issues in the policy space. Each voter's most-preferred outcome is referred to as his or her ideal point. Each voter also has preferences over the entire issue space that are typically assumed to be a monotone decreasing function o f the distance between a proposed point and the voter's ideal point. Then each voter's indifference curves are spheres around his or her ideal point. These utility functions are referred to as Type I utility functions and can be summa­ rized as *>,yo||*-x,||<||y-*,.||,

(1)

or, alternative x is preferred to (or indifferent to) alternative y by voter i f and only i f the distance between x and /'s ideal point x is less than (or equal to) the distance between proposal y and i's ideal point. Simply put, the rule on preference among alternatives compared to a voter's ideal point is simply "the closer the better." For simplicity, I restrict attention to simple majority rule. A set o f voters, M, is a majority-rule winning coalition i f t

\M\ >

n

^ *

i f n is odd and \M\ > " ^ ^

i f n is even.

L e t / d e n o t e the social choice function for simple majority rule. Then f(x) = {y I y *} is o f points in the policy space that are majority-preferred to x. In other words, f(x) is the set o f points that beats x by some majority coalition. Iff(x) = 0 , then there are no proposals that beat alternative x and x is the winner. A n y x £ X for which f(x) = 0 is called a core point. Note that i f a core point exists then the iteration o f the social choice function w i l l reach an equilibrium at outcome x. >

t

n

e

s e t

However, the existence o f an equilibrium under the social choice func­ tion o f majority rule in a multidimensional policy space is extremely rare. In order to achieve a majority rule equilibrium outcome, severe restrictions on the voters' ideal points are required (Black 1958; Plott 1967). For nearly all voter preferences, majority rule does not lead to any stable winning social outcome, but continues to wander over the policy space. Every issue can be beaten by some other proposal that is also preferred by a majority o f the voters. I n addition, the movement o f winning proposals can exhibit intransitivity or cycles, where, by a sequence o f majority votes, the social outcomes w i l l be alternative b over a, c over b, and then a over c! Moreover, any point in the policy space can be reached by some sequence o f majority rule deci­ sions. When an equilibrium breaks d o w n , it breaks down completely and includes the entire policy space (Cohen 1979; Cohen and Matthews 1980; McKelvey 1976, 1979; Schofield 1979, 1983). These results are typically

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referred to as the chaos theorems, because o f the negative interpretation that all order and stability are lost: agendas can be constructed between any two points, sequences o f votes do not reach a stable equilibrium consensus, and any final group choice may have little relationship to the voters' preferences. However, the use o f the term chaos was semantic. The following section demonstrates that the connection between multidimensional spatial voting and chaos is not merely semantic, but theoretic. Chaotic D y n a m i c s in M u l t i d i m e n s i o n a l Spatial V o t i n g The previous section briefly summarized some o f the troubling aspects o f multidimensional voting and pointed out the inherent and pervasive instability in voting. B y viewing the social choice process as a nonlinear mapping, symbolic dynamics can be used to establish that the sequences generated by a group o f voters in a multidimensional policy space do in fact exhibit chaotic dynamics (Richards 1992, 1994). The chaos theorems o f social choice can be understood as chaotic nonlinear dynamics; the connection is not merely se­ mantic, but is theoretic. This implies that outcomes and agenda paths are not completely random and all order is not lost. This section demonstrates the existence o f chaotic dynamics in m u l t i ­ dimensional spatial voting using an approach analogous to the example o f the logistic function shown in the second section. However, there is an important technical distinction. In the case o f the logistic function, the mapping was one-to-one and continuous. I n the case o f multidimensional social choice, the "function" is instead a correspondence: each policy point maps to a set o f points that beat it by majority rule. However, the general logic remains the same: divide the space into regions and examine how regions map into other regions. The rules o f the mappings catalog the feasible dynamics and allow for the establishment o f Devaney's three conditions o f chaos. The Importance of Cycles 2

Before examining the inverse image o f the social choice function, two pre­ liminary facts must be established. First, every multidimensional spatial vot­ ing context without a core has at least one three-cycle among alternatives. Second, i f a cycle o f any length greater than three exists, then a three-cycle also exists. These results are similar to previous findings (e.g., Cohen 1979; Schofield 1983), but are presented using concepts needed for the subsequent results. First, the notion o f a cycle must be clarified. A n alternative y G X can be reached from a point x G X i f it is possible to move from x to y by iterations of / . I f an x G X can be reached from itself by k iterations o f / a n d x ¥= p(x) for 1 ^ j < k, then a £-cycle exists.

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(1) For t w o or more issue dimensions, three or more voters, type I utility functions, and majority rule, then i f the core is empty, there exists a sequence o f alternatives (a, b, c) such that b £ f(a), c £ f(b), and a £ f(c). To see that this is true, suppose that a three-cycle exists among three alternatives, a, b, and c. Then a, b, and c form a nondegenerate triangle i n R . Label the three perpendicular bisectors o f ab, be, and ca as H , H , and H, which must intersect in a space o f m — 2. Denote H% as the open halfspace that includes b and that by construction contains a majority o f voter ideal points. Denote / / £ . and H+ similarly, such that each halfspace contains c and a, respectively, and contains a majority o f voters. A cycle exists among three points a, b, and c i f such a partitioning is possible (fig. 5.2). This partitioning creates the "pinwheel" o f social choice preferences such that b £ f(a), c £ f(b), and a £ / ( c ) . Therefore, the question o f the existence o f a three-cycle for a given configuration o f voters is the question o f whether voters' ideal points can be partitioned into the appropriate pinwheel o f hyperplanes. m

ab

ca

bc

b

a

Assume voters' ideal points are located i n the policy space such that a core exists. Denote the core point as v*. Then v* £ f(x) for all x v* and / ( v * ) = 0 . The agreement set o f any winning coalition o f voters must contain v* or there w o u l d be a winning coalition that could beat v*, violating the construction o f v* as a core point. Therefore, in partitioning the voters into the sets H+ , H£ , and H+ , v* must be an element o f H+ , and o f H£ , and o f H+ , b

c

a

b

c

a

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Chaos Theory in the Social Sciences

i m p l y i n g that H+ f l D H+ ¥= 0 . However, for a three-cycle pinwheel partition to exist, the hyperplanes must be such that H£ D H£ ^ 0 and H+ D f l H+ = 0 (see fig. 5.2). Therefore, i f a core exists, a three-cycle partition is not possible. b

a

b

c

h

a

Conversely, assume voters' ideal points are located in the policy space such that a core does not exist. I f the core is empty, then the halfplanes H+ , Hg , and H+ must be able to be placed such that a majority o f voters are contained in each H and such that H£ f l / / £ . D H+ = 0 . I f this were not the case, then there w o u l d be a point that must be included in every winning coalition's set and this point w o u l d be a core. Given that there is some H such that H+ D H£ Pi / / + = 0 , then there must be an H such that the three lines intersect in a single point. Either the three lines already intersect i n a single point or H~ includes all o f the cone H^ D H . I n the second case, H has the flexibility to move toward the vertex o f the cone (toward H 's "—" region), since this can only result i n more, rather than fewer, voter ideal points. Therefore, H can be moved until it intersects H and H in a single point at the base o f the H+ f l Hfc. cone. Therefore, i f there is no core then a three-cycle partition is possible. b

c

a

+

b

a

ca

b

c

ca

a

b

bc

ca

ca

ca

ab

bc

b

(2) I f there is a &-cycle for a k s 3, then there is some a', V, and c' such that V = f(a'), c' —f{b'), a n d a ' = f(c'), i . e . , there is a three-cycle. To see that this is true, note that the existence o f a &-cycle implies that the core is empty (Cohen 1979; Schofield 1983). B y (1), i f the core is empty then a three-cycle exists. The Iterated Inverse Image of the Social Choice Function The previous discussion established that when voters' ideal points are such that no equilibrium exists, then the majority-rule social choice function cre­ ates a mapping from a to b, from b to c, and from c to a. As in the example o f the logistic function, the existence o f regions that feed back into previous regions allows for the existence o f cycles o f all lengths and for the demonstra­ tion o f chaos. Therefore, one can characterize the permissible sequences o f the social choice function in terms o f mappings o f regions. This is accom­ plished by examining the iterated inverse image o f the social choice function. The policy space is partitioned into regions, and by examining the nested sequences in terms o f target regions, one can outline the permissible se­ quences under the iteration o f the social choice mapping. This approach differs from the typical approach to spatial voting in two

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ways. First, rather than examining the mapping o f single points—as in the " w i n set" formed by the indifference curves through a single point—this approach examines the mapping o f regions into regions. B y abandoning the point precision o f previous approaches, one can characterize the general dy­ namics o f the social choice function (Saari 1991). Second, the iterated inverse approach differs from previous analyses o f spatial voting by focusing not on the social choice function, but on its inverse, / " ' ( * ) · Previous approaches examined the set o f points that beat an existing status quo point, namely, the set/(jc). The iterated inverse image approach reverses this logic and examines the target region o f x—namely, the set o f points that map to x via the social choice rule. I f y is the target region o f x, then x can follow y in a social choice sequence. The iterated inverse image approach outlines which sequences o f outcomes are possible under / b y examining the iterations o f target regions. This allows for a general cataloging o f everything that can happen under the social choice rule in terms o f permissible sequences o f target regions. P R O P O S I T I O N 1. For two or more issue dimensions, three or more voters, type I utility functions, and majority rule, if a three-cycle among alternatives exists, then there are cycles among alternatives of all periods greater than or equal to three. P R O O F . Let a, b, and c denote three alternatives in a three-cycle, w i t h H , H , and H the respective perpendicular bisectors (as i n fig. 5.2). Label a closed region B bounded by a hyperplane parallel to H and such that b G B. Define a set R(B) as the open set o f all points defined by reflecting across H (see fig. 5.3). By construction a G R(B), since H is the perpendicular bisector and b G B. Regions C, R(C), A, and R(A) are defined similarly, such that cEC,bE R(C), a £ A, and c G R(A) (see fig. 5.3). ab

bc

ca

ab

ab

ab

Assume a sequence {A, B, B, . . . , A, . . . } . First we need to find those points i n A that map to T, = B in the first iterate. This is given by the cone f~ (T ) = R(B) f l A. N o w we need to refine/~ (T ) to only those points that remain in B i n the second iterate. That is, we want to find the points in A that map to the refined target region o f T = f~ (B) f l B CT . The open cone B n H G f~ (B), since all points i n this cone map to B by approaching the H hyperplane, so T = B D H . The same argument ensures that the third iterate is i n f l . Here the refined target region is T — f~ (f~ (B) D B) C T . A t each iteration, the set o f initial points is refined to include only those points that remain i n B. Then = f~ . . . (f~ (B) f l B); one can stay in B as many times as necessary since the boundary H is open. The final stage defines the set o f inverse iterated p o i n t s / ~ '(7*,) C A—i.e., the set o f points i n x

l

x

x

l

2

X

l

bc

bc

2

bc

l

3

l

l

2

l

bc

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Chaos Theory in the Social Sciences

R(B)

\

A

ROM

Fig. 5.3. Three-cycles w i t h i n t h r e e a l t e r n a t i v e choices

t h

A where the image o f the i iterate returns to A. This means the final target region must be refined to T = H f l f~ (A). This is an open cone and is nonempty. Therefore, one can construct a sequence that begins in A, iterates w i t h i n B for any number o f iterations, and then returns to A. x

t

bc

Since sequences o f the type {A, B, B, . . . , A, . . . } are permissible under this social choice rule, it is straightforward to see that cycles o f all length greater than three can be constructed: vary the number o f iterates w i t h B from t w o (a three-cycle) to fl = j (an i + 1-cycle). Note, however, that unlike the logistic function, the presence o f a three-cycle feedback does not imply cycles o f all lengths in the social choice function. "One-cycles" ( i . e . , equilibrium points) are not implied. Although region fl can follow region fl, in fact, subsequent points in fl cannot return to previous points. As seen in the above discussion, mappings w i t h i n fl must approach the open boundary o f H. Therefore, there is no periodic point o f one. 3

bc

Proposition 1 makes clear that the mapping rules for the social choice function allow region B to follow A, C to follow fl, and A to follow C. In addition, note that each region can follow itself, merely by moving closer to its perpendicular hyperplane. Therefore, the mapping rules on the majorityrule social choice function can be summarized for x, as the status quo alterna­ tive i) i f x, G A then x ii) i f x, G fl then x, iii) i f x G C then x

t + l

+x

t

t+l

G A or fl or C; G A or fl or C; G A or fl or C.

(2)

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103

Chaotic Dynamics Recall from the example o f the logistic function that the presence o f chaotic dynamics is established by demonstrating satisfaction o f three conditions: sensitive dependence on initial conditions, topological transitivity, and dense periodic points. A s in the case o f the logistic function, demonstrating that the social choice function is chaotic becomes a straightforward exercise when the permissible sequences in terms o f symbolic dynamics are used. P R O P O S I T I O N 2. For two or more issue dimensions, three or more voters, type I utility functions, and majority rule, if an intransitive social choice cycle exists among any alternatives, then the social choice function exhibits chaotic dynamics. Recall that to demonstrate sensitive dependence on initial conditions, one must demonstrate that two points that are arbitrarily close w i l l separate by a distance 8 by the iteration o f the function. Let the distance between regions A and B be 8. Then sensitive dependence on initial conditions implies that one can construct t w o symbolic sequences that are identical for n entries and that differ on the n + 1 entry. Assume that the n entry o f these two sequences is A, then the n + 1 entry can be A for one sequence and B for the other, separating the points by 8 and demonstrating sensitive dependence on initial conditions. t h

l h

t h

Recall that topological transitivity requires that the mapping have points that move from one arbitrarily small neighborhood to another arbitrarily small neighborhood. Then any point that begins in A and ends in B satisfies this condition. Clearly such a point does exist, by the permissible sequences outlined in (2). Recall that the condition o f dense periodic points requires that, given a periodic point, there is another periodic point arbitrarily close by. Distance between the iteration o f points is evident by the extent to which two sequences match. Given a sequence o f arbitrary length, it is possible to find another sequence that matches the beginning n entries o f the first sequence, demon­ strating the existence o f two periodic points arbitrarily nearby. By examining the social choice correspondence o f multidimensional spa­ tial voting i n terms o f the iterated inverse images, the demonstration o f the existence o f chaotic dynamics becomes straightforward. The existence o f a three-cycle intransitivity arising from a nonequilibrium preference configura­ tion creates a feedback process among alternatives. It is this nonlinear feed­ back that, as in the case o f the logistic function, creates complex behavior in the mappings, allowing for countless cyclic outcomes and nonperiodic, cha­ otic sequences.

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Chaotic Dynamics i n Other I n d i v i d u a l Group Aggregations Multidimensional spatial voting is one subset o f the larger class o f aggregation schemes from individuals to groups. The multidimensional case is particularly interesting however, since virtually all voter preferences lead to chaotic dy­ namics. This section addresses the question o f the general robustness o f chaotic dynamics i n the aggregation from individual tastes to social outcomes. To do this, I briefly review two other aggregation schemes where the existence of chaotic dynamics has been established. These include Saari's (1984, 1985a, 1987, 1989) work on simple voting and the findings o f chaotic dy­ namics in market mechanisms. S i m p l e Voting Unlike multidimensional spatial voting, in simple voting individuals rank a finite set o f discrete one-dimensional alternatives. Each voter has a preference ordering over the alternatives and the social choice is determined using a given voting rule, such as pairwise majority votes, two-thirds rule, or the Borda count. As in multidimensional voting, one would like the function mapping individual preferences to a group choice to be coherent in some sense. Arrow (1963) formalized the notion o f coherence in terms o f five conditions: (1) every individual preference ordering that is complete and transitive is admissible, (2) i f all voters prefer an outcome x over an outcome y then the social ordering must be x over y, (3) i f x and y are the only two outcomes that the group can consider, then the social preference between x and y depends only on individual preferences over {x, y } , and not on individual preferences over a larger set, (4) no single voter is decisive for every pair o f outcomes, and (5) the additional requirement that the social choice ordering must also be complete and transitive. A r r o w showed that no voting rule exists that satisfies these conditions. N o matter what voting rule is used, the mapping from individual to group w i l l not be trouble-free. Paradoxes in individual-to-group mapping w i l l occur for every voting rule. 4

The mapping from individual to group in the case o f simple voting also exhibits chaos, as demonstrated by Saari (1984, 1985a, 1987, 1989, 1990, 1992). Using symbolic dynamics, and for any number o f alternatives and any positional voting procedure, Saari catalogs all possible social choice rankings over all possible subsets o f candidates. He shows that any social choice ordering—no matter how paradoxical (such as the social ordering a > b, c > a, a > b = c)—can be achieved by some profile o f voters. A n y outcome is

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possible, paradoxes are abundant, and it is impossible to construct a voting rule that avoids these paradoxes. This occurs because all sequences are possible under the individual-group mapping rules. The permissible mappings in any positional voting procedure create the existence o f chaotic dynamics in simple voting contexts. 5

Market Mechanisms In addition to voting schemes, individual preferences can also be aggregated using market mechanisms. As A r r o w ( 1 9 6 3 , 1 - 2 ) and Riker and Ordeshook (1973) point out, voting and the market are special cases o f the general category o f collective social choice. Both market mechanisms and voting schemes take the preferences and desires o f many individuals and "amalgam­ ate" them into a single social choice. I n the case o f voting, the choice is a selection o f a candidate or policy. In the case o f the market, individual preferences over quantity and prices o f goods translate into a single market price. I n this way, the market incorporates individual preferences and results in a social choice in terms o f the allocation o f goods among the individuals (Riker and Ordeshook 1973). However, i n their acts o f voting, individuals are consciously choosing and anticipating a social outcome. I n the act o f a market mechanism, the act is less consciously oriented toward the ultimate social outcome (Riker and Ordeshook 1973, 82). Nonetheless, the collective out­ come o f a market mechanism, such as the price o f a commodity, is still achieved by an aggregation rule that takes individual preferences over the commodities and leads to a social outcome by the combination o f individual actions. As in voting, there are various rules by which market mechanisms can respond to individual preferences. Here I focus only on the most general form. The basic idea is that one has an exchange economy with a fixed number o f commodities, a fixed amount o f goods, and a set o f agents w i t h given initial endowments. Each individual has preferences over each good, where an individual's demand is given by his or her utility function. The combined demand o f all the individuals is the excess demand function. As­ sume that total buy is not equal to total sell. The market mechanism is a force that w i l l move prices to an equilibrium point where total buy equals total sell and the market clears. Hypothetically, this is modeled as being adjusted by a Walrasian auctioneer, whose mission is to search for the market clearing price (see, e.g., Varian 1992). Prices move iteratively in the direction o f excess demand: i f the i commodity is i n demand, then its price increases until demand equals supply and it clears. I n general, these models are described as tâtonnement models or price adjustment mechanisms. Depending on the de,

h

106

Chaos Theory in the Social Sciences

mand functions, the individuals' preferences w i l l combine through the excess demand function and the market rule to yield an equilibrium price and thereby allocate goods among individuals. However, it is w e l l k n o w n that such market mechanisms need not con­ verge to a stable social outcome. For example, economists have long known that price adjustment mechanisms are not stable for two or more commodities. The presence o f chaotic dynamics in iterative market adjustments has also now been established (see, e.g., Day and Pianigiani 1991). Saari and Simon (1978) and Saari (1985b, 1992) show that for most prices there need not be any relationship among the demand functions. The outcome o f price models can lead to many outcomes in terms o f price and allocation, and the social outcome is very sensitive to changes in the set o f agents, the set o f commodi­ ties, or the set o f preferences. In addition, no market procedure satisfies three minimal conditions in moving from aggregate excess demand to a market price (Saari 1991). Majumdar (1992) and Bala and Majumdar (1994) show that w i t h t w o commodities, two agents, and the price o f one good fixed, chaotic dynamics w i l l result, as it w i l l in a competitive economy with two commodities. C o n s t r a i n t s o n Instability: The Order in Chaotic D y n a m i c s

Interpretations o f a chaotic process in terms o f the colloquial use o f the word chaos—as disorderly, random, or conflictual—ignore the unique aspects o f chaotic dynamics. A chaotic process is disorderly: it results in an infinite array o f seemingly random possibilities; it is sensitive to very small shocks; it never reaches a stable equilibrium; and it is unpredictable in the long term, no matter how long its behavior is empirically observed. However, the other side of chaos is the presence o f an underlying order in the dynamics. Although the outcomes o f a chaotic process are so complex they appear random, in fact, chaotic processes incorporate specific constraints on these outcomes. It is this mix o f complexity and order that makes chaos theory an unusual theoretical framework. In this section we explore some preliminary evidence o f this structure in the case o f multidimensional spatial voting and discuss its i m p l i ­ cations for understanding the individual-group aggregation process.

The Constraints o n O u t c o m e s and Paths It is well known that in a multidimensional setting, any outcome in the policy space—even points that are very far away from the voters' ideal points—can be reached by a sequence o f majority-rule votes. However, this result is often

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interpreted to imply more than it technically does. While it is true that any point can be reached from any other point, this does not imply that any point can be reached from any other point by any path. In effect, for any two origin and destination points, " y o u can get there from here," but the routes from here to there may be quite constrained. The symbolic dynamics approach emphasizes the constraints on feasible sequences. The mapping rules outline what is permissible and what isn't permissible i n the social choice dynamic. I f majority rule was in fact total disorder, then every region would map to every other region even as regions were refined to smaller and smaller sets. Although the construction o f the fourth section d i d have the property o f the dictionary being the universal set, it also contained only three partitions. It is obvious that as the regions are refined, rules on the mapping w i l l emerge and the dictionary w i l l no longer be the universal set. These constraints were already evident in terms of mappings within a region. Although, for example, B maps to B, in fact, points must approach the hyperplane with each iteration: not all points in B map to all other points in B in one iteration. A refinement o f the region B would high­ light these mapping rules. Another example o f the order in a chaotic process is evident in the condition o f dense periodic points. Recall that this condition states that for any periodic point there is another periodic point arbitrarily close by. I n the case o f the logistic function, the order in terms o f nearby dynamical behavior was evident in the bifurcation diagram, where two-cycles transformed to fourcycles, then a chaotic region, and so on. Bifurcation diagrams can also be constructed o f the multidimensional spatial voting social choice dynamics. Like the bifurcation diagram o f the logistic function, the bifurcation diagrams of the social choice function show complex patterns (fig. 5.4). The bifurcation diagrams o f figure 5.4 show the outcomes in one o f the issue dimensions (y on the vertical axis) as a function o f the winning coalition choosing either an extreme point (near the tip o f the coalition agreement set) or an incremental point (near the status quo p o i n t — i . e . , the base o f the coalition agreement set). For many steps over this parameter continuum, a sequence o f eighty points was generated and the last fifty points were plotted in a horizontal line. For example, part b o f figure 5.4 shows a three-cycle for some o f its region. As seen by comparing parts a and b o f figure 5.4, a different fine structure emerges as a result o f the voting rule and the individual preferences. Each set o f voters produces a unique bifurcation diagram, con­ stant to transformations that preserve the configuration o f the voters' ideal points (such as rotating or a linear transformation). Although these diagrams are o f limited use (because o f the nature o f the parameter choice to make the function one-to-one), they do illustrate the fine structure to the dynamics o f the social choice function.

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150-r—

0.0

1 200-

0.2

0.4

0.6

0.8

1.0

-200

0.0

0.2

0.4

0.6

0.8

1.0

maximum feasible step < — > minimum incremental step

maximum feasible step <-—> minimum incremental step

Fig. 5.4. T w o s a m p l e b i f u r c a t i o n d i a g r a m s . D i s t r i b u t i o n o f v o t e r ideal p o i n t s is: (10. 10) (20, 10) (35, 20) (20, 80) (70, 30) a n d (60, 30) (60, 70) (70, 50) (90, 7 0 1 ( 1 2 0 , 1 1 0 ) .

The Structure of Outcomes Given that the social choice function for multidimensional spatial voting is a chaotic process, we know from the definitions o f chaos that there must be a strict structure to the feasible outcomes. There are suggestions o f this struc­ ture in a simulation o f iterations o f majority-rule decisions. Figure 5.5 shows the distribution o f outcomes w i t h five voters deciding by majority rule over a two-dimensional issue space. Since majority rule is not one-to-one, for the simulation a single point had to be chosen as the winner. I t was arbitrarily set as the midpoint o f the largest winning coalition's agreement set. Note that the choice o f this point is arbitrary—it merely serves as an anchor on the movement o f winning coalitions. The simulation was begun w i t h an arbitrary initial point, a . a is beaten by a,, which is the alternative that is preferred to a by the largest winning coalition, where a, is the midpoint o f that coalition's agreement set. N o w a, is also not stable and can be beaten by a new set o f points by a reformation o f coalitions. a is the point that beats a, and is the midpoint o f the largest coalition that can outvote a,. Continuing this iterative process provides a sample o f the distribution o f alternatives under the majority-rule social choice. 0

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the logistic function, there is an order to the dynamic o f the majority-rule social choice function. This order has been anticipated by previous studies. For example, there has been a proliferation o f new solution concepts that seek to capture some o f the order to voting outcomes. As a sampling, some o f the best k n o w n are the generalized median set, the minmax set (Kramer 1977), and the uncovered set (McKelvey 1986). It is not necessary to review or define any o f these con­ cepts, but they do have some relevant common properties. First, these sets are typically constructed as descriptions o f likely outcomes. For example, Kramer (1977) notes that over time, sequences roughly converge onto the minmax set. Ferejohn et a l . (1980, 1984) suggest a probabilistic convergence o f outcomes toward the generalized median set. Second, these sets are all "centrally located"—i.e., they tend to be centered around a region in the geometric center o f the voters' ideal points. For example, Feld et al. (1985) demonstrate that agendas that move toward the generalized median set are easier to con­ struct than agendas moving away from the center o f the voters. T h i r d , they are all compact solution sets. For example, the minmax set is a polygon bounded by voter median lines. The generalized median set and the uncovered set are circles o f a particular radius. The order suggested by the theory o f chaotic dynamics has also been noticed by experimentalists (e.g., Fiorina and Plott 1978; W i l s o n 1986). They find that outcomes are not only not scattered over the policy space, but tend to be clustered i n patterns. For example, W i l s o n (1986, 404) concludes that "one point is clear; the outcomes selected here are not scattered throughout the alternative space." Rather than covering the entire policy space, most experi­ mental studies find that outcomes cluster w i t h some deviation near the center of the voters' ideal points. I suspect that the experimentalists are capturing portions o f the structure o f the chaotic dynamics in their results. What is happening in the dynamics o f social choice is that the individuallevel preferences i n a context o f interaction ( i . e . , multidimensional majority rule) are creating their o w n structure, which in turn constrains feasible out­ comes. Although there are no external constraints on individuals' votes (in terms o f preference orderings, etc.), the voters' actions combine to create a system where each voter, in effect, constrains the other. Game theorists are familiar with this intuition in the important role that the rules o f the game have on structuring feasible outcomes (see, e.g., Schelling 1978). Even i f individ­ uals have complete autonomy over their actions, the context o f their inter­ action creates structural limits on what individuals can do and therefore on the group outcomes. Note the parallels between the order implicit in the chaotic dynamics o f the individual-group connection and the concept o f structure in social science. In the social sciences, structure is something generated by the interaction o f

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unit-level behavior, yet conditions and constrains system-level options. How­ ever, structure in social science is not an external, concrete entity (as an institution might be). "Structure is not something we see. . . . [SJince struc­ ture is an abstraction, it cannot be defined by enumerating material charac­ teristics o f the system. It must instead be defined by the arrangement o f the system's parts and by the principle o f that arrangement" (Waltz 1979, 80). "Structure is a system-wide component that makes it possible to think o f the system as a w h o l e " (Waltz 1979, 79). It is this structure that makes a chaotic system indecomposable. The whole is complex (disorder), but it is united by a common endogenous structure (order), whereby the system cannot be broken d o w n into additive subsystems. Chaotic systems are disorderly. They are characterized by an acute sensi­ tive dependence on initial conditions, where slight changes in individual-level input lead to radically different social outcomes. This characteristic in turn leads to an impossibility o f long-term prediction based on empirical observa­ tion. The dynamics appear random and, no matter how long the dynamic is observed, no repetitious, deterministic patterns w i l l be evident. Prediction must shift from point predictions to boundary predictions, from long-run to short-run. Chaotic dynamics are also disorderly because o f the vast diversity o f achievable outcomes. Outcomes are not limited to a few, often-reached, repeated outcomes. A n y outcome can be reached, from somewhere, via some route, by some length o f time. As others have already established, group outcomes are possible anywhere in the policy space. But chaotic systems are also orderly. They incorporate rigid and u l t i ­ mately describable constraints on mappings and on routes from group choice to group choice. Previously made choices do constrain future options. Paths are closed, but paths are also continually opened. Social choices are not completely unrelated to individual preferences; there are constraints, but these constraints are "strange." Chaotic systems are also orderly in that they display a fine structure in the distribution o f points and the paths. I have alluded to this complex structure in the patterns o f the bifurcation diagrams and o f the distribution o f outcomes from the dynamic. But the structure is multifaceted. It is not merely a circle or a polygon. It is a unique endogenous structure created by the aggregation o f individual-level behavior in a given context. Chaotic systems are also orderly in that there is a "consistency o f weirdness": stable regions adjoin stable regions and chaotic ones are near chaotic ones. There is a smoothness in the dynamical profile o f the social choice function, as made clear by the condition o f dense periodic points. We have long known that the aggregation o f individual choices is prob­ lematic. I f we can understand this mapping as a chaotic process then we have some concepts and tools available that allow for the incorporation o f a m i x o f order and disorder. The process from individual to group is not a regular,

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predictable, determinate process. Yet neither is it an incoherent, completely random process. I t is this blend that is so unique—both to chaotic dynamics as a general framework and to the aggregation o f individual preferences in particular. The individual-group connection has order and constraints that it creates itself as a result o f the dynamics o f the social choice function. How­ ever, the individual-group connection is also capable o f infinite variety and a perpetual evolutionary dynamic, where—even assuming deterministic individual behavior rules—group outcomes cannot be predicted in the long run. Chaos, Social Choice, a n d D e m o c r a c y The presence o f instability in social choice has already been discussed in terms o f the implications for democracy. These implications fall into two categories: (1) the view that the instability makes certain forms o f democracy, such as populism, infeasible (Riker 1982), and (2) the view that the instability may help the political stability o f pluralist democracy ( M i l l e r 1983). I w i l l briefly outline some o f the implications for each o f these views in light o f the individual-group connection being a chaotic process, although I w i l l not at­ tempt to resolve the implicit normative debate. View 1 : Incoherent Social Choices Given a social choice function that results i n outcomes that can be anywhere and that are not consistent w i t h individual preferences, Riker (1982) makes the case that liberal democracy, rather than democracy in its populist form, is more effective. Liberal democracy does not attempt to represent its citizens' preferences, but is democratic in that citizens have the opportunity to remove their representatives. Populism, on the other hand, seeks to aggregate citi­ zens' preferences on specific issues, achieving a social choice by combining and weighing citizen's views. However, based on the pervasive presence o f instability in the social choice function, Riker asserts that liberalism is the only democracy that works, since it bypasses the instability problems o f the social choice function. Populist democracy, being dependent on an imperfect and troublesome aggregation scheme, w i l l only result in arbitrary decisions and continually changing social outcomes. What i f these aggregation schemes, including the individual preferencegroup choice connection, are chaotic? The instability was previously known. The new information is that the aggregation function that populism relies on is a chaotic process. Returning to the technical characteristics leads to several implications. First, group outcomes w i l l appear arbitrary. Studying the relationship

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between outcomes over time without knowing the mapping rule, the choice w i l l appear random. N o amount o f empirical observation w i l l allow for pre­ diction. However, as seen in both examples, these outcomes are far from random. Outcomes are strictly related to individual preferences. They are not arbitrary, in the sense that the same set o f individual preferences results in different outcomes at different times. But outcomes are very sensitive to slight changes in these preferences and the connection between individual and group is complex. Complex does not mean random or arbitrary, however. Outcomes in the logistic function were complex, but they were quite "regular" in terms of resulting from a strict deterministic mapping. Although the mapping is certainly not random or arbitrary, this may not be enough for populism to work. Perhaps the very presence o f any instability must be removed. I f we understand that the aggregation function is chaotic, what does this tell us about removing the instability? First, it says one can't solve the problem by breaking it down into smaller problems. I f there is instability in the whole group, trying to break it down into stable subsystems w o n ' t w o r k . This tends to imply that i f population w o n ' t work in its purest form then neither w i l l a type o f "hierarchical populism." Chaotic systems are indecomposable. Second, it tells us w h y the instability is occurring. I t is not merely unstable; i t is unstable because it is in the chaotic regime, which in turn implies that it is nonlinear and there is some "functional folding" or feedback occurring. As seen in both the logistic and spatial voting, it is this feedback process that creates the instability. In theory, i f one could "unfold" the social choice function then the instability would disappear! B y understanding the aggregation process as chaotic, one can examine why it is occurring. I f one knows w h y the mapping is chaotic, then one can systematically, rather than haphazardly, outline ways to reduce the instability, rather than identifying isolated cases where instability does and does not occur. 8

View 2 : Instability Fosters Political Stability The case for the instability o f social choice allowing for a context that sup­ ports the stability o f pluralism is made by M i l l e r (1983). He reasons that the instability o f the social choice process in fact helps promote pluralism. Plural­ ism views preferences as largely determined by the issue group to which an individual belongs, w i t h society divided along one or more lines that partition the individuals into sets. Where the arrangement is pluralistic, these cleavages are in a crosscutting rather than a reinforcing pattern: individuals that disagree on one issue are likely to agree on other issues. In addition, intensity over issues is also dispersed. The argument o f pluralism is that pluralistic prefer-

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ences contribute to the stability o f the political system. M i l l e r makes the argument that the instability o f the social choice function contributes to the stability o f pluralism. A s M i l l e r (1983, 744) states: "not only does each competitor ' w i n some and lose some,' but most wins and losses are them­ selves reversible. Thus the competitors can never be confident o f their victo­ ries, nor need they resign themselves to their defeats." It is this indeterminacy and instability o f winners created by the aggregation process that induces losers "to continue to play the political game, to continue to w o r k w i t h i n the system rather than try to overthrow i t " (1983, 744). Pluralism only guarantees that different people w i l l w i n or lose on different issues. The instability o f the social choice function guarantees that losers on a given issue can hold out in hope o f being winners on the same issue. The presence o f an instability in the social choice mapping may promote the stability o f a democratic system, but it does not imply any sort o f effi­ ciency. As Rae (1982) points out in his metaphor o f disequilibrium, demo­ cratic decision making becomes an "Escher staircase": leading always up but only coming back to its o w n foundation. But this instability may be too costly to justify any advantages in promoting political stability i f the outcomes are completely arbitrary and unrelated to individual preferences ( M i l l e r 1983, 744). M i l l e r d i d not resolve this issue, but suggested (1983, 745) that "most o f us w o u l d view political outcomes in the real w o r l d o f pluralism as consider­ ably unpredictable but clearly confined w i t h i n certain bounds o f 'political feasibility.'" A chaotic aggregation process w o u l d certainly support M i l l e r ' s conjecture. A social choice mapping in the chaotic domain w o u l d be empiri­ cally unpredictable. This is one o f the characteristics o f chaos and it arises from the sensitive dependence on initial conditions. Second, the chaotic func­ tion also does contain very definite bounds and restrictions—this is the order in the complexity o f outcomes. Outcome and paths are highly constrained, although the variety o f outcomes is endless, as evident in both the logistic functions and the spatial voting example. The topological transitivity o f chaos gives a variety o f outcomes that one w o u l d n ' t otherwise get without changing preferences. Finally, outcomes are not arbitrary or completely unrelated to public opinion. Just like the logistic function, preferences map to outcomes in a strict deterministic fashion. However, in a chaotic process, one gets an indeterminacy from a determinate mapping (Huckfeldt 1990). A deterministic mapping (majority rule) from individual to group leads to indeterminate, complex, evolutionary outcomes. This is quite unusual. Even i f one assumes deterministic choice and behavior rules on the part o f individuals, one gets an indeterminacy o f social outcomes! B y my interpretation o f M i l l e r , this is the best combination for democratic political stability: constraints, bounds, and a nonarbitrariness in the individual-group mapping, combined w i t h an un­ predictability and evolutionary variety o f outcomes over time.

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Conclusion The purpose o f this chapter was to reraise the question o f the "black b o x " o f the individual-group connection. We know that this connection is not the simple additive relation that early theorists once assumed. We know it is "strange," as evidenced by complex outcomes, instability, and sensitivity. I n this chapter I sought to demonstrate that this black box o f aggregation from individual to group is i n the realm of chaos theory. The function that maps preferences to outcomes and that maps outcomes to outcomes through demo­ cratic voting is chaotic—i.e., a unique type o f mapping w i t h a blend o f disorder—in terms o f sensitivity, unpredictability, and evolutionary instability— and o f o r d e r — i n terms o f strict mapping rules, regularity in regions o f insta­ bility, and bounds or constraints on outcomes and paths. I showed this w i t h the example o f multidimensional spatial voting: the voting context w i t h the most pervasive presence o f instability. However, I suspect this applies to most aggregations, as I suggested by pointing to similar findings by Saari on voting and by economists in terms o f aggregation through price mechanisms. The normative question o f whether this is good or bad for liberalism, populism, pluralism, or democracy i n general remains open. But it suggests several future tasks that need to be completed. I have shown that multidimen­ sional spatial voting is a chaotic process. This only sets up the problem o f understanding what is causing the chaos. M u c h work needs to be done on w h y the social choice function is chaotic. A l s o , much work remains on specifying the general conditions o f instability and ways to reduce the instability. But i f we know what type o f problem we are dealing w i t h , then we can begin to systematically understand the origins o f the instability. Only then can we begin to open the black box o f the aggregation functions and to resolve the normative debates i n democratic theory.

NOTES Epigraph from The Social Contract and Discourse, Book 2, chapter 3. Quoted in Frölich and Oppenheimer 1978,15. 1. Obviously Bentham's scheme raises problems of interpersonal comparison of utility. This is a related issue of such magnitude that it will not be discussed here.

2. Occasionally I will use the term social choice function rather than social choice correspondence nontechnically, as it is used throughout the voting literature, to designate a mapping from individual preferences to a social choice. 3. Sarkovskii's theorem (that a three-cycle implies the existence of cycles of all lengths) applies to the logistic function since it is one-to-one and continuous. These conditions do not hold for the social choice correspondence. 4. Completeness requires that all alternatives can be ranked.

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5. However, Saari (1990) demonstrates that the Borda count reduces the number of paradoxes. 6. O f course, this distribution is only one of many potential examples, but it illustrates the presence of order. 7. The connection with fractals is suggested by the complex patterns in the distribution of outcomes. 8. This is the subject of a future research project.

Part 2 Chaos Theory and Political Science

CHAPTER 6

Nonlinear Politics Thad A. Brown

That chaos may be a part o f elementary politics is evidenced by highly exploratory work i n the fields o f electoral behavior, game theory, axiomatic choice theory, and conflict analysis. The social dynamical processes that may induce chaos, methods o f investigating large-scale collective behavior, and some implications for political science research are outlined in this essay. Deterministic Chaos At the turn o f the century, Poincare, and later Birkoff, noted that fully deter­ ministic dynamics do not necessarily provide explicit predictions on the evo­ lution o f a dynamical system (Poincare 1892; Birkhoff 1927). The subtlety o f their observations was not fully appreciated until much later (Cartwright and Littlewood 1945; Levinson 1949; Smale 1967; Ruelle and Takens 1971). Today, chaos is a component in understanding dynamic macrobehavior in many sciences. Chaos exists when the long-term prediction o f a system is impossible because uncertainty in a system's initial state grows exponentially fast over time. Chaos fits the criterion that the autocorrelation function o f the time signal goes to zero in finite time. Because trajectories are unstable, errors o f estimation o f initial conditions or parameters, however small, can later accu­ mulate into substantial errors. This implies that forecasts o f future behavior based on the past become problematic as current memory o f the past fails. Nonlinearity is a necessary but not sufficient condition for the generation o f chaotic m o t i o n . O f course this implies some feedback mechanism exists. The observed chaotic behavior is due neither to external noise nor to an infinite number o f degrees o f freedom. The source o f irregularity is the nonlinear system's property o f separating initially close trajectories exponentially fast. Because the time evolution is self-independent from its own past history, predicting the long-term behavior o f chaotic systems is an interesting exer­ cise. The process does not come to rest i n a stable equilibrium but instead 119

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comes to occupy large patches o f state space. Analysis is difficult because stable and unstable states are strewn together in extremely complicated ways. The critical feedback process central to the dynamics under investigation therefore can be docile or unruly, depending solely on the tuning parameter values o f the equation(s) o f motion. Indeed, for all practical purposes the distinction between determinism and nondeterminism disappears. That is, even equations that are solely deterministic become nondeterministic in their long-term realizations. One may assume it impossible to predict the long-term behavior o f political systems that are potentially chaotic because their initial conditions can be fixed only w i t h finite accuracy and subsequent error increases exponen­ tially fast. For politics, this means political actors w i t h i n a chaotic environ­ ment who are governed by the same laws and are identical in every measur­ able way may evolve differently over a long time. A g a i n , this would occur despite identical initial configurations. But this unpredictability o f the long-time behavior exists only at the level of individual trajectories. A t the level o f statistical properties o f the time evolution (averaged over different trajectories, say as they evolve from differ­ ent nearby initial conditions), very definite predictions are possible. Trajecto­ ries w i l l eventually move only on a small submanifold (chaotic or strange attractor) o f the entire state space, w i t h predictable visitation frequencies o f the different parts o f the attractor. It may be possible to estimate the time up to a chaotic event, offering a short-term prognosis o f a political actor's trajec­ tory. So far the theory o f chaotic dynamics has uncovered three major routes for the onset o f chaos as the external system parameters are changed: the Ruelle-Takens scenario via quasiperiodicity, Feigenbaum's period doubling, and Pomeau-Manneville's intermittency route. Each has its o w n characteristic signature (e.g., in the autocorrelation function). Such signatures may provide important indicators for incipient changes. Other indicators may come from the fact that, generically, a system develops characteristic large-amplitude fluctuations as it approaches a phase transition. W i t h i n measured turbulent flows (mostly found in the physical and life sciences), chaos is k n o w n to exist. Thus, for any theory o f dynamical politics to be viable, it must come to grips with political turbulence and hence incorporate chaos. Gaps in our understanding o f political dynamics have become partic­ ularly obvious and acute during recent political events in Europe, Russia, Japan, and the M i d d l e East. These events and rapid macropolitical changes came as unexpectedly to any political scientist as the 1989 San Francisco earthquake came unpredicted by any geologist, or the 1993 rains in the M i d ­ west by any meteorologist. It is not that developments such as those in Russia or Eastern Europe

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could have been predicted i n detail i f better models had been available; the point is that despite the unpredictability o f such developments, there may be definite patterns i n their occurrence. Furthermore, there may be well-defined relations between the time scales on which short-term prediction is possible and the characteristics o f chaotic evolution at long times; there may be indica­ tors o f incipient macropolitical changes; and there may be a wealth o f macro­ scopic variables that are predictable at long times even though the system state is not. In the field o f nonlinear dynamic systems, such coexistence o f predict­ able and unpredictable properties has led to phrases like deterministic chaos (Schuster 1988) and order in chaos (Prigogine and Stengers 1984). A common example is weather forecasting. Here the solutions o f the relevant (Lorenz) equations o f motion are so sensitive to small changes in initial conditions that predictions beyond t w o or three days are intrinsically impossible. In some places, like Los Angeles, the weather is pretty much the same from day to day. On those days, the utterance o f any local weather person w i l l be as correct in predicting the weather as forecasts o f the entire U.S. National Weather Bureau. A n inversion layer produces smog; the end­ less summer continues. Then one day the predictions break d o w n . The mas­ sive computers around the world miss the formation o f a Pacific storm that flattens the coast o f Baja, California. A t the same time, it is possible to predict average temperature, amount o f rainfall, and so o n , for the month o f Novem­ ber in the year 2022 at any desired location for any given set o f external parameters. The equations that govern weather can simultaneously behave both ways; order shifts to disorder and back through the mathematical pas­ sages o f chaos. Nonlinear Political Interactions Politics, whether at the communal, state, national, or international level, results from the interactions o f individuals. These individuals are members o f respective political collectives; while some may be equipped with more power and attain an elite status, even the elites are interdependent. Political leaders base their decisions on interactions with their advisors; they form impressions o f their advisors, who form impressions o f them. Relationships change as these impressions and experiences inform subsequent interactions. Legislators chat in the halls before key votes and discuss issues with staff, advisors, and adversaries. A t times they also check with constituents back home. What legislators say and do influences their standing in a legislative assembly and in their districts. Their standing in each one influences the other. The process is reflected down to the level o f the typical voter who decides, say, to support a Republican or a Democratic candidate in an election, based on interactions w i t h friends, family, or acquaintances in her local area. Even the individual

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voter's thoughts and actions may be the product o f subtle self-interactions, as in Elster's multiple self. A useful event that illustrates the impact o f social dynamics might be Black Monday: October 19, 1987. Might is not a trivial adverb: the most widely accepted cause o f the market's collapse, programmed trading, has been shown to have only a weak effect on market volatility (Grossman 1987), and as a consequence we really don't know what happened. Brock (1991) interprets the market collapse as being the product o f a herd behavior induced by the attenuation o f normal information channels available to traders. His parable goes as follows: In a market functioning normally, traders regularly upgrade their information by calling on industry and security analysts. Communication between the traders and industry an­ alysts is open. Prices are a function o f both past price performance and volume, as well as expectations for future earnings and dividends. When a negative shock hits the market, prices drop. Traders use the behavior o f other traders, price, and information from analysts to decide their next move. I f the negative shock is large enough, the information channels between traders and analysts j a m — t o o many phone calls. A trader is left w i t h simply following the herd; the only existing information is the movement o f prices and the observed behavior o f other traders. As Brock suggests, a sell-off gets started when a positive feedback loop emerges in an otherwise insipid market. The individual traders' reward structures are likely to magnify this process into a serious collapse (LeBaron 1989). Banerjee (1992) further argues that herd behavior can occur where a decision maker pays more heed to information others are perceived to have. The reason is simple: the visible actions o f others may be based on informa­ tion that the decision maker lacks. Private information is overridden by others' signals and the herd takes off. The event o f an economic landslide suggests even catastrophic change, let alone small shifts in preference, can result from group interactive effects. Unfortunately, attempts to analyze complex political behavior often tend to attribute observable cooperation or conflict to the individual. Behavior in politically intricate situations is attributed to higher levels o f political informa­ t i o n , a deeper understanding o f the game's rules, or superior strategies, and o f course lots o f information (or at least enough information so as not to be duped). Nearly all published theory on the iterative prisoner's dilemma fits into this category. What must be remembered and used to calculate strategies includes: discount values, prior encounters, likely future encounters and asso­ ciated probabilities, w h o might use what piece o f information to figure out what to do, the cost o f keeping information, and so on. O f course, nobody thinks this way. A n d even i f anyone d i d , there is no guarantee that a success-

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ful strategy at one time w i l l not be utterly unsuccessful at another. I n real political interactions are we left to laws o f chance? For decades we have seen glimpses o f the complex individual-group nexus that defines politics. For instance, we know political beliefs are often based on socially cued ideologies by proxy. That is, individuals may faithfully reflect and follow over time the beliefs o f others who possess developed abstract formulations o f politics (Campbell et a l . , 1960; Converse 1964). That most voters lack much useful political information or fail to behave rationally is rarely debated anymore (Ferejohn and K u k l i n s k i 1990). But the end o f such debates doesn't mean we really understand the mechanism that induces even the most basic forms o f political behavior. Turnout is one example. Rational choice theorists still cannot understand why anyone would vote, given the costs relative to proportional influence. Empirical electoral analysts cannot figure out precisely w h y aggregate voting rates appear to be going d o w n — o r are they going up this year? As a good friend in industry says, there is something wrong w i t h this picture. Social dynamics result from sets o f local interactions between group members and their interactions with the environment. Logically, such signal­ ing fits w i t h i n a variety o f classical social psychological perspectives that suggest individuals adapt to social environment (Asch 1951; Festinger 1957) or form impressions about others based on behavioral experiences (Heider 1958). I n politics we know that decision makers modify their o w n behavior and the influence o f social environments through self-selection (Schelling 1978; Finifter 1981), avoidance (MacKuen 1990), migration ( B r o w n 1988), or a generalized contagion process (Huckfeldt and Sprague 1987, 1988, 1993). Formally treating interactive political behavior w i t h i n massively diverse collectives is tricky. Interactive behavior is peculiar in that it can neither be predicted nor analyzed by observing sets o f individuals cross-sectionally, or even the time series from a given individual or group. Social dynamics and the concomitant social behavior cannot be reduced to individual behavior i n the sense that isolated individuals cannot induce the variety and richness o f global collective behavior prevalent in any political system. Social and political behavior is by definition holistic and synergetic (Haken 1978, 1983) and must be the product o f interacting individuals who can communicate and modify their behavior as a consequence o f their interactions. A n y time series is a rough statistical characterization o f a collective process. A power spectral analysis alone, for instance, cannot decide the dynamical rules that model the time series. Both a deterministic logistic map w i t h m a x i m u m parameter values (Mayer-Kress and Haken 1981) and a stochastic system can generate a white noise signal such that a flat spectrum cannot tell w h i c h model is correct. Figuring out what drives the show for

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interactive social behavior poses some interesting problems. I n political dy­ namics there are likely to be spatial and temporal phase transitions. For instance, we know that transition from a pattern selection phase to fully developed turbulence occurs via the intermittency (Kaneko 1989). Nonlinear interactions w o u l d almost demand that abrupt and widespread events occur unexpectedly. H o w to proceed is the question. A m o n g the possible ways to investigate dynamical politics are simula­ tions o f intriguing spatial arrays with cellular automata. Cellular automata are formal dynamical systems w i t h many discrete degrees o f freedom. The beauty o f automata is that while the rules o f interaction are surprisingly simple, complex nonlinearity can be induced by the iterative nature o f interaction. As nonlinear systems, cellular automata can display the full array o f dynamics o f any real, l i v i n g system, from fixed points to cyclical behavior to chaos. Can political life be the product o f simple, primitive models? Cell-space models have been used i n physics (Herrmann 1992), chemistry, and biology (Gutowitz 1991; Forrest 1991), and to a more limited degree in the economic and social sciences to investigate the ecological structure o f behavior (von Hayek 1937; Schelling 1971, 1978; Cowen and M i l l e r 1990). In politics, they may represent the potential to "program" the information available and used by entities (voters, groups, elites, nations, or whatever), and hence what we can discern from cellular automata may give us further insight into the informa­ tion dynamics in real politics. Rules can reflect essential existential states o f being: love and hate. Love attracts and induces assimilation; hate repels and induces egestion. Rules can be deterministic or probabilistic. In both cases, cellular automata are formed on a lattice. Cellular automata can clearly produce patterns in space and time that can be m o v i n g fractal clusters i f the underlying dynamics are chaotic. We can determine i f the underlying dynamics are chaotic by watching the spread o f small changes in the configuration o f the system on a lattice. A lattice is usually a (/-dimensional array o f discrete cells whose behavior is governed by local, uniform, and time-independent rules. Boundary condi­ tions may be periodic. The rules determine the behavior o f individual cells as they interact w i t h others at the next time step. Future states o f a cell are normally a function o f the values o f neighboring cells and the cell itself. The radius o f a cellular automata is the number o f neighbors whose state at time t affects the state o f any cell at t + 1. The state o f the cell is a variable, s „ (where / = 1 , 2 , . . . , N), which can be any measurable political trait. I f s, = 0 we can say the voter is a Democrat; i f s = 1 the voter is a Republican. The state o f the entire system, its configuration, is defined by the collection o f values («!, s , • • • , % ) . From these values, the average fraction o f partisans at any time step, t,f(t) = ((\/N) X, .$,-), its variance, and so on, can be easily calcu­ lated, as well as the asymptotic fraction o f partisans based on a long period. t

2

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125

Although individuals remain in fixed physical locations, their partisan­ ship may change as a result o f interactions with other voters on the lattice. This is a primordial political w o r l d , but well reflects many o f the central ideas in contextual models o f politics. B y using political rules on a lattice, enough interactions between decision makers can replicate real political situations. Finding useful rules that map different forms o f political behavior is the key to making automata theory viable for the social sciences. A simple two-dimensional interaction law in which each individual inter­ acts in the same way with its neighbors is the deterministic majority-rule law. This law makes each site adopt the value prevailing among the cell and its four nearest neighbors on the lattice. The interaction rule for such electoral behavior is given by

Gfait

+ 1), · - · , s (t N

+ 1» = l

l

^

X

Sj

(t)

> 3,

where G, are given functions o f time t and the current values s (t),. . . ,s (t); \i ~ j \ = V(*,- — xj) + (y, yj) is the Euclidean distance between cells i and j , w i t h location (x y,) and (x yj), respectively. The summation is over all cells j whose distance from i is less than or equal to a, including s, and its four nearest neighbors (fig. 6.1). Thus, the voter at cell / w i l l be Republican at time t + 1 i f , among i and its four nearest neighbors, at least three are Republicans at time /; otherwise, it w i l l be Democratic. Since the summation condition x

2

t<

_

2

jt

Fig. 6 . 1 . A r e p r e s e n t a t i v e lattice site. /', a n d i t s n e i g h b o r s o n a square lattice w i t h D = 2 a n d lattice c o n s t a n t a. (The c o o r d i n a t e s x , and / , d e s i g n a t e t h e physical l o c a t i o n o f site /.)

N

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Chaos Theory in the Social Sciences

always examines five cells, no ties can occur and the interaction law is symmetrical w i t h respect to partisan influence. To see how the rule works consider i, starting from some given initial state, 0, its four nearest neighbors, and their respective nearest neighbors, over t w o time periods, as shown in figure 6.2. In the first time step, the interaction o f the four neighbors o f i w i t h the peripheral cells leads to a conversion o f the four neighbors into Republicans, leaving i Democratic. I n the second time step, the central cell i also becomes Republican. As this simple illustration shows, over time a voter is affected by voters outside the range o f the defined radius o f the interaction law. Had we chosen the voter to the Northwest o f / to be Democratic instead o f Republican at t = 0, then i w o u l d have remained Democratic at t = 2. A more general statement would be that the state s,(f) w i l l be influenced by the initial states s-(0) o f all cells j for which the inequalities

bj - yt + ( J - *,·)! x

bt - (*,· - •*,·)!

T A

'-- ta

hold (r = 0, 1, 2, . . . ). Despite the simplicity o f this political interaction rule and its explicitly local nature, it shows that the long-time behavior o f the voter depends on the initial configuration o f the whole system. A more com­ plicated rule, or one w i t h global characteristics, would be at least as sensitive. B r o w n and McBurnett (1992) have used a stochastic version o f the

f=0

f=1

f= 2

Fig. 6.2. First t w o s t e p s of t h e t i m e e v o l u t i o n of a r e s t r i c t e d s e g m e n t of t h e l a t t i c e , c e n t e r e d a t s o m e s i t e /', under t h e d e t e r m i n i s t i c m a j o r i t y l a w . T h e values in t h e circles d i s p l a y t h e p a r t i s a n s h i p o f t h e d i f f e r e n t s i t e s ; the symbol * means that the state cannot be updated w i t h o u t addit i o n a l k n o w l e d g e o f t h e i n i t i a l c o n f i g u r a t i o n b e y o n d w h a t is s h o w n f o r t = 0.

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above model to simulate multiple political realities. I n the model t w o oppos­ ing groups are arrayed on a 128 x 128 periodic two-dimensional lattice o f initially randomly distributed sites. From such stochastic rules it is k n o w n that highly ordered systems evolve from what are essentially random configura­ tions (Clifford and Sudbury 1973), though the specific evolved configurations are themselves apparently random and are highly sensitive to initial conditions (Durrett 1988). When a stochastic voting rule is applied to a random initial configuration, the system behaves in a fashion that is completely dependent on the initial concentration o f Democrats or Republicans. Small chance events early in the history o f a political system (or a nested subsystem) may funda­ mentally alter the numerical power balance between or among political actors. W h i l e the asymptotic state o f nearly all stochastic models such as that described above is unanimity, the finite time path provides insights into inter­ esting political dynamics (Brown and McBurnett 1992; McBurnett and B r o w n 1992). For instance, a natural clustering o f like voters emerges as an outgrowth o f the interaction rule. Over time these clusters become a randomly built political landscape. ( A time series describing this process is given in fig. 6.3.)

15,000

J3 Q. 0 1

10,000

5,000 5,000

10,000

15,000

20,000

Time Path Fig. 6.3. T i m e series of t h e n u m b e r o f Republican v o t e r s g o v e r n e d b y a s t o c h a s t i c v o t e r r u l e as s i m u l a t e d o n a 124 x 124 square lattice w i t h a p e r i o d i c b o u n d a r y . Initial c o n d i t i o n s are r a n d o m . T i m e s t e p s 1 t o 20,000 are s h o w n ( B r o w n a n d M c B u r n e t t 1992).

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The steady gain o f one group comes at the expense o f the second. The outcome o f which group w i l l dominate, however, is by no means predictable. A t any point the fortunes o f a leading group can be reversed. A n equilibrium is not predictable i n advance, in that the trajectory o f the individual voter exists w i t h i n a highly irregular political environment. The time series from these and other experimental data appears to have a fractal, noninteger dimen­ sion, though more experiments are needed to determine i f this is true ( B r o w n , Pfeifer, and McBurnett 1992; McBurnett and Krassa 1991). Wolfram (1986) has offered the most complete classification o f cellular automata. Cellular automata appear to be classifiable into four general types of behavior: Class I : cellular automata evolve into a fixed, homogeneous state and these structures evolve to fixed l i m i t points; Class I I : evolution to a simple periodic structure represented by l i m i t cycles; Class I I I : chaotic, aperiodic patterns as found when strange attractors are present; Class I V : complex localized organizations as represented by systems that exhibit very long tran­ sients. Distinguishable classes o f cellular automata related to politics can only begin to tell us what the subtle, underlying structure might be. For instance, electoral systems may represent an example o f Class I V behavior by exhibit­ ing very long transients, a great deal o f local order, and a global order that may emerge. Bureaucracies, on the other hand, may be reflected by Class I dynamics where both local and global scales o f order exist. Yet the statistical behavior o f such systems would still be unknown. Simple classification o f dynamics is not sufficient to give us greater insight into political and social dynamics. While lists o f political systems (and problems) classified by dy­ namical behavior would be an improvement over current methods, these classifications could not tell us precisely how the reality, process, and order­ ing o f numerous forms o f qualitatively differing systems came about or, more importantly, how they w i l l evolve. I f political dynamics are likely to be complex or chaotic, how would we know? One set o f measures w o u l d be based on the properties o f the aggregate time series o f the number o f partisans, as well as on their asymptotic fraction. The correlation integral, the Lyapunov exponents, and spectral analysis are now commonly used to evaluate the presence o f chaos (Peitgen, J ü r g e n s , and Saupe 1992). O n a spatial lattice, for the trajectories to be chaotic, slight perturbations in the initial conditions w i l l have to induce a changed configura­ tion in the values o f cells. H o w two trajectories quickly become different is not the issue because the exact rule o f interaction is k n o w n . We must instead estimate how rapidly and how far the trajectories w i l l diverge. The notion o f chaos in politics may indeed include the speed w i t h which trajectories sepa­ rate. In order for the concept o f closeness o f trajectories to have meaning, a measure o f just what fraction o f the voters are different for the initial and

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perturbed configurations is needed (Herrmann 1992). The Hamming distance measure provides a useful definition o f distance in phase space

A(0 =

S

|<x,.(0 - p,.(f)|,

where a (t) and p,(f) are two time dependent configurations in phase space and N, as mentioned above, is the number o f cells indexed by <'. The dynamical behavior w o u l d be chaotic when in the thermodynamic l i m i t A(t) gets big for large times i f A(0) —» 0. t

W h i l e the w o r k o f B r o w n and McBurnett suggests that even simple political interaction models show a sensitivity to initial conditions, much more analysis is needed. To determine an exact dynamical behavior requires using in a simulation the same sequence o f random numbers applied to t w o configurations o f a lattice. Meaningful outcomes can only be determined after averaging over many initial configurations o f equal distance and across differ­ ent sequences o f random numbers. O f course, i n deterministic systems, once a set o f configurations is identical they w i l l stay identical. Once knowledge o f the behavior o f cellular automata models is gained, analysts w i l l be i n position to calibrate lattice rules to match both time signatures and spatial movements that match some categories o f interesting political mass behavior. Strategic Interaction Models o f social interaction investigated by means o f cellular automata may open up for political science issues as yet unapproachable. We know that w i t h i n strategies for solving common collective choice problems, cooperation (or conflict) may appear to be stable or even frozen i n place while it can also instantly disappear with the slightest nudge or noise. Cellular automata can model conditions where a large number o f actors make many decisions, as in any complex social community where individuals interact with imperfect information. However, knowing the rules that are important to investigate requires an adept touch at this point, because w i t h i n the structure o f automata theory, the number o f &-state, r-radius rules is rather large (k *). I t is probably best to look to familiar ground. k2r+

Determining how levels o f cooperation or conflict are reached and spatially diffuse is a core element in iterative prisoner's dilemma research (Rapaport and Chamma 1965; Axelrod and Hamilton 1981; Axelrod 1984). As is well k n o w n , the prisoner's dilemma is a two-person, zero-sum game that can model the political evolution o f either cooperation or conflict. Each

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player on every round can choose between cooperation (C) or defection ( D ) , and the outcome is determined by a standard payoff matrix, V:

C D

C R,R T,S

D S,T P,P

where T > R > P > S and 2R > T + S. In a one-shot play o f the game, each player's best strategy is to defect, since D gets the largest payoff, T, while C w o u l d get 5. The paradox is obvious. I n finitely many repeated plays o f the game, all Cournot-Nash (noncooperative) equilibria have the property that no matter how long the sequence, a noncooperative outcome occurs in each period. I n games o f an indeterminate duration, however, the player's utility is calculated on the expected long-run payoff o f repeated one-shot plays. In essence, it is best to figure out how to cooperate, since i f both players use D both end up w i t h a lower payoff since R > P. Axelrod explored some spatial properties o f the game w i t h an eye on the territorial stability o f tit-for-tat (Axelrod 1984). When the iterative prisoner's dilemma was examined w i t h i n difficult environments, Richards (1990) showed the clear potential for chaotic behavior. In exploring evolutionary games played w i t h neighbors on a spatial lattice, the most common approach is to set each lattice cell to a single strategy. The strategy may be either fixed (Axelrod 1984) or allowed to evolve based on a genetic algorithm or other optimization h i l l - c l i m b i n g routine (Axelrod 1987; M i l l e r 1989). As before, the lattice mechanism is quite simple. A l l cells are updated simultaneously; the score o f a cell is determined to be the average it obtains playing the neighbors defined by the given radius; the score o f the cell is then compared to the scores o f the neighborhood and the cell itself; the cell adopts the strategy o f the highest score w i t h i n the sites defined by the radius. Ties, when they occur, are usually broken by adding a touch o f noise to the scores. As was observed above, the scores o f the defined neighbors are dependent on the strategies o f their defined neighbors, and so on. Thus, the contextual influence on any cell extends beyond the defined radius. When the model is restricted to set strategies, it clearly qualifies as a cellular automata. To illustrate, let's examine one set o f recent findings on the iterative prisoner's dilemma. I n a paper, N o w a k and M a y (1992) use fixed strategies o f cooperation (C) and defection ( D ) to study how cooperative behavior evolves and is maintained. A s mentioned above, nearly all prior work demanded that individuals or groups remember past encounters, with some discounting o f retrospective evaluation for prospective payoffs. Such recollections are neces­ sary because o f the highly complicated strategies employed. Using fixed strategies means that memory is not a requirement for behavior.

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131

O n a two-dimensional lattice o f varying sizes, interacting players face the following payoffs: R = 1, T = b (b > 1), S = P = 0. The parameter b identifies the relative advantage o f defection versus cooperation. The entire lattice is occupied w i t h C's or D's and again the scoring is the sum o f encounters w i t h neighbors. A t the start o f a next generation (a time-step) each cell is occupied by the player with the greatest score among the previous owner and the defined neighbors. The radius and the boundary conditions are varied. The rules among the n players on the n x n lattice are deterministic. Updating o f sites occurs i n discrete time and in a synchronous fashion. 1

2

As reported by Nowak and M a y (1992), extremely complex patterns are achieved even when strategies are fixed that in essence involve no strategies at all. What they show is that ecological structure o f cooperation stems from the magnitude o f the advantage given to defecting, the value o f b. For instance, the initial configuration starts with 10 percent defectors when b is between 1.75 and 1.8, indicating the prevalence o f cooperation. When the b parameter is moved to values between 1.8 and 2, the lattice evolves. Nowak and M a y (1992, 827) interpret their results as indicating that C and D "coexist indefi­ nitely in a chaotically shifting balance, with the frequency o f C being (almost) completely independent o f the initial conditions." This means that the asymp­ totic C fraction, f , roughly .318, remains constant for nearly all conditions tested. The inference is that o f nonconvergence on defection commonly re­ ported in iterative prisoner's dilemma games. c

Despite the assertion o f spatial chaos, the appearance o f complex spatial patterns does not imply a Class I I I cellular automata. A g a i n , Class I I I implies that disorder exists on both a local and global scale. In a game-theoretic situation, as devised by Nowak and May (1992), strategies are fixed. Because strategies are not available to each player (cell), the rigidity o f the game may preclude deterministic chaos. Yet the difficulty o f characterizing the patterns that emerge may well imply that the complexity found reflects the interesting Class I V behavior. In a recent paper, Huberman and Glance (1993) examined N o w a k and May's findings relaxing the assumptions o f synchronous updates o f sites. They report that the spatial complexity and richness o f evolved actors disap­ pear and the population moves quickly to converge on defection. Bersini and Detours (1993) also report that asynchrony induces significant stability and structure in cellular automata based models. Clearly, more analysis is needed. A x i o m a t i c Theory Since the seminal work o f A r r o w (1963), the indeterminate characteristic o f politics in majority-rule settings has caused a search for political equilibrium. Indeed, democratic theory once could only be understood with the proviso that neither global nor local equilibrium could exist within even the simplest

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multidimensional voting system. A n y outcome o f any majority-rule system could cycle among alternatives at any time (McKelvey 1976, 1979; Schofield 1978; Cohen 1979). A n y status quo could be overthrown by some other alternative drawn randomly from the entire issue space (Riker 1982). Even in political settings packed w i t h elites, such as legislatures, decision making w o u l d be in constant disarray; policy decisions w o u l d be utterly unpredict­ able. Governments based on populist democratic voting theory should be without order or structure. Human subject experimental work on the spatial properties o f voting systems found outcomes clustered around a central tendency (Fiorina and Plott 1978). I n experimental voting games where any outcome was feasible, different groups converged on similar collective choices rather quickly. So while the presence o f political disequilibrium was well k n o w n , the micro­ scopic causes and dynamics that might induce political order within multi­ dimensional voting environments or disorder were not. Serious progress has been made. Using simulation techniques, Browne, James, and M i l l e r ( 1 9 9 1 , 1992) have proven the link between cyclical behav­ ior under conditions o f majority rule. In related w o r k , Richards ( 1 9 9 1 , 1992) has formally linked the L i and Yorke (1975) period-3 theorem to disequilib­ r i u m in multidimensional choice. Both efforts focus on the dynamical nature o f decision makers and environments and the interaction that follows pro­ posals to alter the status quo. In their initial w o r k , Browne, James, and M i l l e r (1991) confirmed that the McKelvey-Schofield-Cohen models exhibited deterministic chaos. I t was by no means clear that the "chaos" confirmed by axiomatic theories o f choice was comparable to dynamical chaos. Using three-dimensional, noncooperative fixed ideal points in smooth and continuous policy space, they proved that the position o f the status quo is iterated by decision makers attempting to maximize expected utility based on their o w n preferred positions. Their re­ sulting analysis shows a Lyapunov-positive dynamical system (see Browne, James, and M i l l e r 1991). Richards (1991) employs the Li-Yorke period-3 cycle theorem to prove the link between chaos and conditions in multidimensional choice under conditions o f majority rule. The Li-Yorke theorem is an extension o f the Sarkovskii theorem (1964), which states: I f f(x)

has a point x that leads to a

cycle o f period TT, then it also has a point x' that leads to a T-cycle for every T <— TT. Here, T and TT are elements o f the following sequence: 1<—2<—4<—8<— . . . 2

m

. . . 22

m

• 9 *- 2 · 9 «- 2

2

1 6 . . . 2 m

• 7 <— 2 · 7 <- 2

2

m

. . . « m

• 5 <- 2 · 5 <- 2

2

• 3 . . . <— · 3 . . . <-

. . . 2 - 9 < - 2 - 7 < - 2 - 5 < - 2 - 3 . . . « ... 9 ^ 7

^ 5

« - 3 .

Nonlinear Politics

133

(As before, "<—" means "precedes.") I f / ( x ) has period-3, then it must include all periods n when n is an arbitrary integer. This inclusion forf(x) must also contain a chaotic region. B y definition, the emergence o f period-3 implies three f r e q u e n c i e s : / , / , a n d / , wheref = f // (Falconer 1990). Period-3 is one example where chaos was predicted from a mathematical theorem but could not be seen in earlier computer simulations (Mayer-Kress and Haken 1981). 1

2

3

3

2

1

2

Richards restates the Li-Yorke Theorem, sketches a proof, and links it to the three-cycle for the majority-rule function. Written without the work o f Browne, James, and M i l l e r (1991) at hand, Richards clearly delimits and urges the analysis they pursue. That is, based on the presence o f a strange attractor, Richards suggests that within the dynamical decision process, a new solution set is likely to emerge. In a sequel, Richards (1992) extends the logic o f the analysis considerably by linking the presence o f period-3 cycles to multidimensional social choice situations i n which the core is empty. Without testing numerically for the presence o f chaos, Richards indicates with a series o f bifurcation diagrams the fractal quality o f the complex decision behavior over increasingly smaller scales. Unraveling the dynamics o f axiomatic theories under conditions o f ma­ j o r i t y rule has presented some interesting analytic problems. In their highly innovative w o r k , James, Browne, and M i l l e r (1993) are reconsidering the dynamics o f both individual decision makers and resultant system response. Policy space is allowed to automatically initiate system preserving reactions as a result o f changes in a decision maker's behavior. Both proposals, to move the status quo and acceptance o f the change, are allowed and indeed expected to have an effect on a policy space surface. When a decision maker proposes or changes a policy position, the system responds in an effort to automatically reestablish the previous alignment. This systemic action may cause local disturbances, and the resulting ripple through the adjacent regions o f policy space is allowed to produce a changed surface. The decision maker, as a consequence, faces an altered policy environment at the next iteration. These innovations allow for what we expect to be an "institutional response o f entrenched interests against significant alterations o f the system status quo. The expected consequence o f such resistance would be a reordering o f positions in the vicinity o f a changed status quo that, at the same time, preserves the general stability o f existing relationships" (Brown et al. 1992, 13). The inability to preserve such stability could lead to greater oscillations in support and to behaviors that may cause the system to go extinct. Because systems do respond, such events are likely to be rare. Ideal point dynamics are comparably developed. Choice theory assumes that once a position is taken, it doesn't change. This assumption limits result­ ing cyclical behavior to a subregion o f policy space. James et al. (1993) relax this constraint and allow the ideal points to move within policy space condi-

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tional to the iterated status quo locations. As the status quo iterates in policy space, the change i n the position o f the status quo indicates the degree o f stress that must exist on the surface o f the policy space. These forces consti­ tute the sum o f the moments impacting a subregion in policy space; their limits symbolize the critical values for phase transition. A t the micro-level, James et al. (1993) provide a decision maker w i t h a changed context w i t h i n w h i c h to reassess what might maximize his or her utility. Hence, the movement o f the ideal point is not random in issue space but dependent upon the change in the immediate local field o f forces, a very old electoral behavior idea brought to bear on behavior in multidimensional choice environments. A t the onset o f cycling behavior, t w o things must occur i f the system is in a chaotic state: (1) there is a rapid diminishing o f information on where the next status quo is likely to be and (2) there is a dilemma as to what to do next. A g a i n , Browne, James, and M i l l e r state: " . . . continuous cycling is likely to produce strain in the region o f the policy space, and rather than accept this state, actors (especially those whose ideal points are most distant from the cycling location) have an incentive to break the cycle by relocating their ideal points" (Browne et al. 1991b, 13). Implicitly, they are shifting the meaning o f ideal points away from the strict rational-choice definition to a relative pre­ ferred policy position based on what it is currently possible to realize. In any phase transition, for instance, any smoothed approximation, or envelope, o f the volume occupied by area such as a status quo w i l l grow in volume as it is moved. Unperturbed, the séparatrices divide oscillatory and rotational trajectories o f the various contours in utility space. Utility space, o f course, exists as a projection o f indifference functions. As the utility space is perturbed, a stochastic layer could form in its vicinity. What Browne, James, and M i l l e r show is that the region occupied by the stochastic layer contains chaotic motion and holes that a stochastic trajectory cannot enter. This stochastic layer, moreover, is not uniformly thick, and the topology o f the augmented space is therefore assumed to be coarse-grained or irregular. Chaos as Allegory

Maybe the revolution caused by quantum theory during the first half o f this century swamped P o i n c a r é ' s w o r k . Maybe the nature o f nonlinearities would have been impossible to see without the development o f high-end computers and the graphical ability to visualize complex mathematics. But it probably took serious applications or even potential applications to move developments like chaos theory o f f the pile o f intellectual toys. Clearly, the characterization of complex dynamics and chaos w i t h low degrees o f freedom have opened up a scientific reality in chemistry, physics, and biology. But what is reasonable

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for chemistry, physics, or neurobiology is more difficult to accept in the social sciences. It is difficult to find crisp illustrations o f chaos in politics. They are rare precisely because politics is characteristically defined and affected by inter­ actions among individuals, between individuals and groups, among groups, and so o n . Political reality is often shaped by larger metapolitical environ­ ments. Thus, the dynamics and interactions that are the source o f political chaos may obscure its presence. A science o f politics, like other quantitative sciences, may one day soon begin to reconcile itself to the fact that not everything important to under­ standing politics is accessible to traditional methods o f data collection and statistical analysis. Some might say it is unfair to expect prediction and accurate forecasting because politics has been systematically investigated for too brief a time. Compared to astronomy, in which observations have taken place for thousands o f years, or even demography, for hundreds o f years, the politics o f mass publics, elites, or even the behavior o f political business cycles, have been studied only for decades. Some would suggest we have been too quick to attempt to generalize based on data over a short period o f time and a limited range o f political situations and contexts. Maybe. Others might say that while a time series would be useful, what we really need is a better understanding o f the formal nature o f political decision making. That is, we have not reduced the individual and the logic o f the collective decision to a small enough package o f variables to really understand the most basic political motives and actions. But as we have already suggested, even formal areas o f political theory such as collective choice are now under investigation by nonlinear science. Systems w i t h strong nonlinearities, which we showed above to exist in relatively simple polities, and the possible chaos that results do present a practical problem for continuing conventional techniques o f data analysis. I n potentially chaotic systems, as several o f the chapters in this book show, the observed variation is a mixture o f real noise or stochastic error and determin­ istic chaos. M a y (1989) says a problem o f nonlinear relations is separating density-dependent noise from density-independent noise. I n some patches in a fluctuating environment, nonlinear relations are sufficient to produce chaos. The odiousness o f the w o r l d is also able to generate noise by environmental stochasticity. When or where chaos lurks in politics probably can only be k n o w n by statistical prediction. The beauty o f recent developments in the theory o f chaotic behavior is that they offer a wealth o f predictions, given that we know the dynamics o f the system (equations o f motion). Research i n areas such as cellular automata may begin to give us such equations for politics. Develop­ ments also provide powerful methods to extract the characteristics o f chaotic

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dynamics (attractors, invariant measures, characteristic exponents) from data when we do not k n o w the underlying dynamics. A t a m i n i m u m , the theory o f chaos warns us o f the dangers o f extrapola­ tion from a political time series. But the questions raised (and yet un­ answered) are profound: Can we predict that a political system w i l l display deterministic chaos? Can we link our current statistical methods and data to the existence o f deterministic chaos? A n d does the existence o f deterministic chaos imply that predictability in politics is impossible, or can the analyst learn from it and refine predictive models to ultimately reflect control over political processes? Implicit and explicit assumptions exist about the political link between the micro- and macro-levels. For instance, some believe that a direct, simple, additive relationship exists between micro traits and macro manifestations. I n the earlier work in electoral behavior, the level o f an individual's partisan strength was assumed, when aggregated, to indicate the strength o f the party system; the summation o f individual identification was assumed to give a clear indication o f the standing vote o f the citizenry, and so on. Such simple micro-macro links were thought to hold for a variety o f measurable variables: ideology, trust in government, and partisanship. These measures indicated the state o f the political system and gave the analyst a handle on the immediate nature o f the electoral system, as well as an ability to assess the long-term nature o f the entire political system. Nonlinear theory involves an approach quite different from conventional methods i n quantitative political science. Conceptually, it shifts the emphasis in formal theory to investigating dynamical microscopic interactive behavior. A t the macroscopic level, the emphasis is shifted from short-term forecasting based on specific, complete initial data to long-term predictions based on generic, incomplete data. Explicit theory relies upon elusive contextual vari­ ables like political climate or mood, level o f political stimulation, instability, hegemony, or volatility. These variables are ill-defined as aggregation o f individual attitudes but may be well defined in terms o f certain statistical properties o f the system. Beyond Chaos

The purpose o f theory is to make nature stand still when our backs are turned, Einstein reportedly said. For political scientists, nature often laughs and dances around behind us. Is the dance chaotic? In other disciplines, scientists are finding that chaos is a normal part o f any process. Standing still—order, stability, and continuity—is merely a special case w i t h i n a more complex process. Some political mechanics govern electoral realignment, equilibrium be-

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tween parties, the collapse o f the Soviet Union or the return to order in Lebanon, the occurrence o f a coup d'etat, and so on. While politics is a complicated process, its complexity may even be understandable and manage­ able once political behavior can be precisely described. Yet despite the wealth o f contemporary data and information, we really do not know how to pre­ cisely describe political behavior. We do not know the exact degrees o f free­ d o m or the equation(s) o f motion that describe (and maybe predict and con­ trol) political dynamics. We don't know which political phenomena are linear and hence i f they can be Fourier transformed. Do equations that describe the cyclical nature o f electoral politics also describe other political dynamics? D o political dynamical equations contain universal features similar to other social, economic, or biological dynamics? To borrow and to twist a line from Feynman's Lectures on Physics, political scientists cannot say whether something beyond these equations, like free w i l l or personal liberty, is even needed. Recent discoveries in nonlinear dynamics may help us to eventually discover political laws and their qualitative content.

N O T E S

1. Mutual cooperators therefore score 1; mutual defectors score 0; D scores b against C , who receives 0. The reported results are not altered when P = e, when 0 < e « 0. T > R > P > S remains strictly satisfied. 3

2. Period-3 means that f (x ) 0

x

0

< f(x ) 0

<

2

f (x )0

CHAPTER 7

The Prediction of Unpredictability: Applications of t h e N e w Paradigm o f Chaos i n Dynamical Systems t o t h e Old Problem of t h e Stability of a S y s t e m o f Hostile N a t i o n s Alvin M. Saperstein

The social sciences have long tried to emulate the procedures and results o f the physical sciences—the laws o f economic determinism o f Marx come immediately to m i n d . To a large extent, the science o f international relations has not been successful in this emulation; its capability o f predicting the outcome o f international competition (war or peace?) has been very limited. Perhaps this is because too many variables seem to be required, intuitively, to describe the interactions between nations. Also, nations are composed o f multitudes o f individually complex people, making it hard to believe that the internation variables, representing their collective behaviors, can satisfy rela­ tively simple functional relationships. A n y easily expressible theory, relating such variables, is bound to be incomplete and thus suspect, whether the theory is expressed verbally or mathematically. The physical sciences owe much o f their long-term successful hold on the public imagination and pocketbook to their long record o f successful predic­ tions o f physical events and processes, predictions that have given rise to our powerful modern technological society. In the process, the sciences have learned that incomplete descriptions o f the systems being studied are still very useful. O f the many variables evidently needed to describe a physical system, some are major, more are minor, and often the latter can be ignored while still producing very useful predictions about the system. For example, celestial mechanics can give a very good predictive description o f planetary motion in our solar system without including the effects o f the motions o f the various moons about their respective planets. The minor variables only lead to impor­ tant effects in those regions o f the system where the incomplete model, consisting only o f the major variables, is itself unstable. Successful detailed predictions have usually been obtained by avoiding these unstable regions, 139

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regions i n which small changes i n the (major) variables included in the model, mirroring the effect o f the excluded (minor) variables, lead to major changes in the system. Such instability, in a deterministic system, is now commonly referred to as chaos. Physics also deals w i t h multitudes o f complex entities, such as mole­ cules. When appropriate—that is, when the interest is in the collection, for example, a gas, rather than in its individual constituent molecules—physics successfully uses very few variables. These are often the result o f averaging over the constituent multitudes, satisfying relatively simple functional rela­ tionships, such as the gas laws and the fluid flow equations (e.g., Kinetic Theory, Fluid Physics; see Lerner and Trigg 1991). These laws are necessarily incomplete and hence sometimes incapable o f leading to detailed predictions. But we have learned how to predict their unpredictability and have found that this incomplete knowledge is still useful as the model is extended to include further variables, that is, to more completeness. Following the lead o f physics, it w i l l be assumed in this chapter that international relations may be productively modeled by relatively few vari­ ables, simply related; the choice o f variables depends upon the system o f interest and the questions to be answered. It w i l l further be assumed that the breakdown o f these incomplete models into chaos may be more useful than the models themselves, that the qualitative prediction o f model instability may be more meaningful than the quantitative predictions o f specific system varia­ tion. Given these assumptions, a procedure is developed for exploring a num­ ber o f questions that are o f major concern to the understanding o f international relations and the practical application o f this understanding to the creation o f international security policy. The procedure is the creation o f simple mathe­ matical models for the interaction o f competing states and the examination of the numerical output o f these models for regions o f stability and chaos (Saperstein 1990). The general questions to be explored in this chapter are: " W h i c h is more war-prone—a bipolar or a tripolar world?"; "Are democracies more or less prone to war than autocracies?"; and " W h i c h is more warprone—a system o f shifting alliances or a collection o f go-it-alone states?" (Further questions suitable for exploration with this procedure should become apparent to the reader.) For example, it w i l l be shown that the region o f instability in a tripolar w o r l d is larger than that o f the corresponding bipolar w o r l d . From this it w i l l be concluded that a tripolar w o r l d is more prone to war than a bipolar one. Before applying these assumptions to the development o f models and an analytical technique for getting useful information from these models, I w i l l first try to justify them by attempting to formulate a general definition o f science and prediction and then draw from this requirements for the more

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specific science o f international relations. A n important part o f this definition w i l l be the relation between science and technology, a relation that has been fundamental to the growth o f the natural sciences and their related technolo­ gies. The technology corresponding to the science o f international relations should be the making o f successful international security policy. Following the general discussion o f prediction and nonpredictability, a tool for recogniz­ ing chaotic regions is developed. The chapter concludes with the development of a simple model for each question and the application o f the chaos tool to the answering o f that question. Science ( U n d e r s t a n d i n g ) and Technology (Control) Many students in introductory courses in the sciences have great difficulty coming to grips w i t h the notion o f science as a system o f understanding, not a compilation o f facts or ideas about the system to be understood. Exacerbating the problem is the difficulty o f compactly setting down what is universally meant by "understanding" in the physical or social sciences. B y "understand­ i n g , " I mean the development o f a relationship between the phenomenon to be understood and a previous set o f concepts already understood (even i f only innately). These prior concepts, axioms for this latest level o f understanding, are similarly related, in turn, to previously understood axioms, which, in turn, are related to still more primitive understandings. Thus, science is a tower o f relationships constantly growing at the top and the bottom. By "relationship" between two levels o f concepts or phenomena, I mean that one level can be derived from (caused by) the other level acting as axioms. " D e r i v e d , " in turn, means that the one follows logically from the other, arising by necessity from the structure o f the language being used; there is a "functional relationship" between the t w o levels. For example, i f the prior level consists o f the two statements: " a l l men are mortal" and "Socrates is a man," then the derived statement is "Socrates is a mortal." More formally, i f the system in question can be described by a variable (or set o f variables), x, and i f x and x represent t w o neighboring levels, then we might be able to write n

*n+\

n + l

=/(*„.

A

)>

(!)

where / designates the functional relation that may depend upon a set o f parameters A. I n physical science this might be the relationship between the present configuration o f planets in the solar system and a previous configura­ tion. I n biology, it might be the relation between the behavior o f a cell in a salt solution to that o f the same or similar cell in plain water. I f there is a time sequence between the two l e v e l s — i f one occurs at a later real time than the

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other—then the implied later level is determined by the former. The resulting system o f understanding—henceforth also referred to as "the theory"—is then a deterministic system. ( A n illustration o f a deterministic system w o u l d be the sequence o f positions along the orbit o f an artillery shell in the absence of air resistance. A nondeterministic system would be the sequence o f posi­ tions o f an electron in an atom, though the probabilities o f these positions are well determined by quantum physics.) For example, suppose the theory deter­ mines the value o f some variable, x, at all future times, t, given the starting value o f the variable, x (this could be the result o f iterating the relationship o f equation 1 for all positive values o f n starting w i t h n = 0 ) ; the result o f the theory is the curve x(t) shown (see fig. 7 . 1 , part a); x{t) is determined by JC and the theory. 0

0

A check on the adequacy o f the understanding system (the theory) is provided by observation o f the system being understood. Are the axioms always observed to be accompanied by the derived concepts or phenomena (logical equivalence)? Or are they always followed by the next level? In this case, the consequences are predicted by the axioms. Such an empirical check requires confirmation o f the validity o f the axioms in the system under consid­ eration, as w e l l as the observation o f their outcome. Hence, the prior l e v e l — the initial state o f the system to be understood—must also be carefully observed. Actual measurements o f the real world—either physical or social— are always accompanied by noise, random fluctuations in the values that would be obtained in a series o f identical measurements upon identical sys­ tems (e.g., "error analysis" in Lerner and Trigg 1991). Hence there is always some uncertainty (also commonly called experimental error) i n the observed system parameters or starting values, and hence some lack o f certainty in the adequacy o f the system o f understanding. The result, therefore, is a range o f possible starting values, 8x , and a consequent range in determined outcomes, 8x(t) (fig. 7 . 1 , part b). 0

The phenomena being investigated may not refer to definite events but to the probability o f occurrence o f the events. For example, we may be investi­ gating the probability o f survival o f a bacterium as an antibiotic is added to its growth medium. The events themselves may be random, nondeterministic, even though their probability is determined by the relations o f the theory. I n this case, single observations o f the system, no matter how precise, are not sufficient. Repeated observations on an ensemble o f identical systems (theo­ retically an infinite number o f identical observations) are required to deter­ mine a probability (see, e.g., Feller 1950). For example, a single toss o f a coin w i l l not tell you the probability o f a head for that coin. N o r w i l l the observation o f a single human life span give any actuarial probability informa­ tion. Thus a scientific theory relates some initial state (definite or a probability,

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Fig. 7 . 1 . T i m e e v o l u t i o n o f m o d e l s o l u t i o n s under v a r y i n g d y n a m i c s : (a) s i n g l e s o l u t i o n o f m a t h e m a t i c a l m o d e l ( d e t e r m i n i s m ) ; (b) f l o w o f s o l u t i o n s ( p r e d i c t a b i l i t y ) ; (c) f l o w o f s o l u t i o n s (chaos).

but observed somehow) o f the system to be understood to some other state(s) o f the same system. Since the initial state is uncertain, to the extent o f the observational noise encountered in empirically determining i t , knowledge o f the related states may also be uncertain. I n linear theories, small changes in the parameters describing the initial state lead to correspondingly small changes in the related states. For example, i f the theory is described by equation 1, then the small changes are similarly related: x

n + l

+ 8x„

+l

=f(x„

+ 8x„, A ) .

(2)

144

Chaos Theory in the Social Sciences

Hence, given the theory, the parameters o f the determined states are as w e l l k n o w n as those o f the initially given states (see fig. 7 . 1 , part b). It follows that such a deterministic theory is also a predictive theory. The future state o f the system is as well known as its present initial state. The ability o f the physical sciences to make predictions, subsequently verified, has given them a strong hold on the human imagination. A popularly observed eclipse bolsters the public support o f astronomy, as well as o f all the other sciences, presumably applying the same methods. (Naturally enough, other intellectual and pseudointellectual disciplines thus strive to at least appear to be using the same scientific approaches.) In nonlinear theories, differences between initial states may not evolve into allowed differences o f final states; that is, equation 2 is not valid. Hence, small initial uncertainties about the system may lead to large final uncertain­ ties; little or nothing may be k n o w n about the outcome, no matter how precise the knowledge o f the initial state. For example, i n figure 7 . 1 , part c the range in outcomes determined by the theory is equal to the full range o f values possible to the system; prediction may be impossible. I f a small change i n the initial configuration o f a deterministic theory leads to large indeterminate changes in the output configurations, the theory is said to be chaotic (Schuster 1988); it precludes precise prediction. A given theory may be chaotic for some ranges o f input or system parameters and nonchaotic and predictive for other possible conditions. For example, the theory obtained by applying the usual Newtonian dy­ namical laws to the motion o f physical fluids is often referred to as the NavierStokes equations. Given the initial state of, say, a body o f water—the velocity of every particle o f the water at some given starting time—the solutions o f the equations w i l l give the velocity o f each o f these particles at every subsequent time. O f course, noise and measurement errors are associated with the deter­ mination o f the initial and system parameters (e.g., the configurations o f the rocks i n the creek bed over which the water flows; the random fluctuations o f the molecules making up the fluid around their mean, fluid, motion). Hence, there w i l l be fluctuations in the results produced by the equations. I f the output fluctuations are small, the motion o f the fluid is well predicted: k n o w l ­ edge o f the behavior o f the water at one place, at one time, gives knowledge of the fluid flow at neighboring places at subsequent times. Such a nonchaotic flow is termed laminar flow and is characteristic o f liquids flowing slowly in smoothly varying channels. The same theory w i l l describe the motion o f the same water flowing rapidly in a boulder-strewn river, a flow that we usually describe as turbulent (see, e.g., Lerner and Trigg 1991). I n this chaotic regime, knowledge o f what the water is doing at one time and place provides almost no insight into what it w i l l be doing later at the same and neighboring positions.

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The difference between the laminar and turbulent regimes o f the theory is determined by the value o f a parameter, the Reynolds number, calculated from the fluid speed and the size o f the obstructions to the fluid flow. For values o f the Reynolds number less than a critical value, determined by the theory, the flow is laminar; exceeding the critical Reynolds number implies turbulence. The turbulent regime can further be divided into regimes of soft chaos and hard chaos. I n the soft regime, the output fluctuations, though large w i t h respect to the input parameter fluctuations, are small compared to the extent o f the system. For example, there may be turbulent ripples on the surface o f a body o f water even though the motion o f the bulk o f the fluid is well predicted. I n hard chaos the fluctuations dominate the entire system, as does the flow uncertainty i n a fully turbulent, chaotically tumbling and s w i r l ­ ing river rapids. Although the flow o f water in a turbulent stream may not be predictable, the onset o f turbulence itself may be predicted. The transition from laminar flow to turbulent flow depends upon the Reynolds number, a parameter that changes as the stream is followed. K n o w i n g how it changes, from a k n o w l ­ edge o f the configuration o f the stream bed and its sources, allows a prediction as to where and when the stream w i l l change from laminar to turbulent flow and vice versa. Similarly, the transition to and from turbulent air flow over an aircraft's wings can be predicted as the shape o f the aircraft and its aerial maneuvers are varied. Or, since a large storm may be construed as a turbulent part o f the earth's atmosphere, the possibility o f such a storm may be pre­ dicted even i f the air flow w i t h i n such a storm remains chaotic and unpredict­ able. Hence, a theory o f a system that manifests considerable chaotic, and therefore unknowable, regimes, but that allows the prediction o f where and when these chaotic regimes w i l l start, still represents considerable under­ standing o f that system. Such theories may not have the presumably a l l powerful predictability o f celestial mechanics, which has caught the public eye. But all natural systems may not be predictable (as we have learned in this century), and one o f the characteristics o f successful science is to do the best you can. The study o f such unknowable systems has become very popular recently in the physical sciences (see, e.g., chaos in Lerner and Trigg 1991). The possibility o f prediction implies the possibility o f deliberate control. Control o f a process means that known intended consequences w i l l follow from deliberate acts: prior conditions are modified to ensure desired outcomes or to avoid undesired outcomes. For example, i f the outcome x„ in equation 1 is not desired, either the prior condition, x„, or the system parameters, A, can be changed appropriately. ( I f a car is going in a dangerous direction, the parameter A—the steering wheel position—can be changed or the system can be started w i t h the car on a different highway.) I f prediction is not possible, there is no way o f k n o w i n g the outcome o f a given act or policy, which is +l

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synonymous w i t h saying control doesn't exist. Hence, technology, which is the controlled manipulation o f some part o f the human environment, requires predictability. A successful technology, one that produces anticipated ends in a system from given input policies and means, is thus a confirmation (though not a proof!) o f the understanding o f that system. Conversely, an unsuccessful technology, one i n which anticipation is thwarted, is a definitive proof o f the failure o f the underlying theoretical understanding. The failure o f a bridge or an agricultural policy is an indication that the appropriate understanding o f the behavior o f physical materials and environments or o f farmers is lacking. The success o f the bridge or policy, however, does not guarantee the correctness o f the theoretical understanding. (There may be alternative theories to the one that maintains that guardian angels hold up the bridge and protect the crops.) It follows that there is a symbiotic relationship between the physical and biological sciences and engineering and medical technologies, a relationship that has existed ever since these sciences have existed. Scientific advances have led to new technologies that, in turn, "proved" the correctness o f the scientific understanding and provided the tools for further measurements and theory. This symbiotic relationship has led to the success o f both natural science and technology and has served as a model for all other sciences. The Science o f I n t e r n a t i o n a l Relations Turning now to international relations, whose understanding is part o f the political and social sciences, the corresponding technology is international policy m a k i n g , w i t h its subset, international security policies. I f we truly understand the international system, we should be able to make successful policy, that is, policy whose desired outcome, given the existing system and starting configurations, actually occurs (see, e.g., Saperstein 1990). I n the past, a successful international security policy meant that a nation obtained what it wanted, either without having to resort to war or by winning its wars without unreasonable costs to itself. That is, the international system, o f which the nation is a part, either doesn't change to the detriment o f the nation or, at the very least, the component o f the system encompassing the nation thrives even i f the rest o f the system is fundamentally changed. In the present nuclear age, a war between major nuclear powers is unlikely to leave either the w o r l d system or any o f its subcomponents intact. Hence, a major goal o f any international policy has to be the avoidance o f such a war. Thus, the symbiotic science must be able to predict the outbreak o f war. I f that science could also predict the outcome o f a war, there would be no need for the war. Nations often strive for overwhelming military strength in the hope o f ensuring (predicting) their success in the event o f war. They have often been unsuccessful in this prediction because o f the large uncertainties in

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the reactions o f their potential opponents. Considering war as a chaotic pro­ cess (see, e.g., Clausewitz 1982), like a turbulent flow over boulders, it may be possible to predict its outbreak but not its outcome—that is, which side o f the boulder gets a given portion o f churning water or which side wins the war. A science o f international relations that only dealt w i t h probabilities could lead to no policy technology and hence would be useless. Testing o f probabilistic arguments requires an ensemble o f identical systems, whereas we only have one such system—our real w o r l d . Thus, a scientific conclusion that there is a likelihood o f nuclear war w i l l be treated as a prediction o f certainty; the corresponding policy w i l l almost certainly be avoided. I n such a situation, probabilistic predictions such as " l i k e l y " or "unlikely" (where is the separation?) w i l l be treated by policymakers as "yes" or " n o , " which are the results o f a definitive or deterministic science. Hence, we seek a science o f international security with at least some deterministic aspects. Can we postulate a system o f competing, hostile states to be determinis­ tic? Experience indicates that nations respond to one another's actions in reasonably determined ways—at least for major actions and responses. I f these patterns o f action and response didn't exist, there would be no political science, and perhaps no history, o f international relations—just readings o f tea leaves or goat's entrails! But why should we expect deterministic behavior o f a collection o f nations, given that each one is composed o f many, many individuals, each o f w h o m often acts in an unpredictable, chaotic manner? A n answer—that deterministic behavior o f entities exists precisely because o f the random nature o f large numbers o f their subentities—is obtained by analogy to the physicist's description o f a gas. Such a gas is a collection o f an enormous number o f randomly moving molecules whose precise description requires a similarly enormous number o f stochastic variables. Yet its overall physical behavior is deterministically described by a few gas variables, such as pressure and temperature, each a complex average over the many molecular variables. O n a gross scale o f observation these few gas variables usually change smoothly, though on a very fine scale they exhibit random minor fluctuations, mirroring the underlying random molecular variables. So we might expect the behavior o f modern nations to be governed by a relatively few deterministic variables, each a complex average over the behaviors o f each o f its multitudinous population. The usual goal is to find rules governing these variables, admitting that there w i l l be occasional minor fluctuations—a love affair or assassination at the higher levels o f government or society—not subject to these rules. M y additional goal is to see how a system, evolving according to such rules, responds to such fluctuations. The core o f international relations can be described as follows: Nation A responds to the actions o f nation B in the context o f the other nations o f the w o r l d system and its o w n present and past internal situations. Nation B then

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responds to nation A's response in a similar context. Each response is deter­ mined by the system's previous responses (Richardson 1960a,b). Hence, the situation is recursively deterministic. I f A and B designate the appropriate sets o f variables describing the two nations, the situation might be represented in a manner similar to that o f equation 1 : n+l

B,

= a(A„, B „ , A) = b(A , B„, A),

(3)

n

where a and b are functional relationships representing the responses o f the two nations; the parameters À may also represent some average over the other nations in the w o r l d system. For example, A„ might represent the hostility (or the exports) o f A to B in year n, w i t h B similarly defined. Then equation 3 describes how next year's hostility (or exports) o f A to B w i l l be determined by this year's hostility (or exports) o f A to B and o f B to A . The role o f the political scientists (and the need o f the policymakers) is to attempt to predict the outcome o f the intertwining chain o f responses on the basis o f their understanding o f the system and its history. Usually this understanding is verbal, expressed in the words o f historical and policy treatises. ( I n equation 3, the relationships a and b and the parameters À would be verbally ex­ pressed.) n

Verbal understanding possesses the vividness, vagueness, redundancy, and contradictions o f ordinary, daily, spoken and written language. It usually contains many implicit assumptions and innate biases. The logic taking it from premise to conclusions is often incomplete and/or fuzzy. It is often difficult to know what the premises are and how well they are verified, what the conclu­ sions are, and how we got from one to the other (e.g., the functions a and b in equation 3 are not explicit). Hence, how do you test the model against reality? In contrast, mathematical formalisms tend to be explicit in their premises, transparent i n their logic, and concrete in their conclusions. There is thus an often-expressed desire to describe all aspects o f the macroscopic interactive behavior o f nations via explicit mathematical recursion relations between numerical variables representing those behaviors. Equation 3 becomes a set o f mathematical relations between numerically defined variables: A, B, A, and so on. I f the resultant equations could be solved, given all the necessary input representing system parameters and starting conditions, they would produce concrete predictions that could be tested against observed behavior to confirm or negate the mathematical model. However, given the complexity o f the real-world system, we are unlikely to ever have all the necessary input information, or sufficient understanding, to create a model system o f equations complete enough to really represent the w o r l d in all o f its manifestations. A n y practical model would have fewer

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variables than are really required. The resultant equations, though far short of the complexity o f the w o r l d , w o u l d still be complicated and nonlinear i f they are to represent some o f the essential attributes o f our strongly interacting real w o r l d , ruling out the possibility o f simple, explicit solutions (see, e.g., Prigogine and Stenger 1984). These obvious shortcomings o f any mathemati­ cal model are no different than the difficulties o f any reasonable verbal under­ standing o f the w o r l d . The difference is that the incompleteness o f the mathe­ matical model is usually immediately obvious, whereas that o f verbal models is concealed in their fuzzy verbosity. In both cases, the resultant predictions are necessarily incomplete, making theoretical testing and practical policy formulation difficult. The more relevant factors left out o f a model or incompletely described w i t h i n the model, the more gross and incomplete its consequent predictions w i l l be. Overall system variations may be correctly predicted though the detailed outcomes may not be trustworthy. Such incomplete predictions may or may not be useful, depending upon the circumstance o f the policy making or theory building. It becomes important to determine what kind o f gross predictions, from necessarily incomplete theoretical models, can be useful in a given arena o f human experience. I t is the contention o f this chapter that the prediction o f unpredictability can be very useful i n attempts to understand and control the advent o f war in the w o r l d system o f nations (Saperstein 1984). The Prediction o f Chaos a n d t h e O u t b r e a k of War Crisis instability in the international system usually implies a configuration i n which small insults can lead to major changes—the loss o f a nail leads to the loss o f a shoe . . . to the loss o f a kingdom; an assassination o f a minor duke can lead to the deaths o f millions and the profound transformation o f the w o r l d system. Such instability represents the loss o f control and the great potential o f war. This parallels the definition o f chaos (Schuster 1988): small disturbances o f a deterministic mathematical system lead to dispropor­ tionately large changes in the system and the consequent loss o f control. Prediction o f unpredictability i n a system is a prediction o f the onset o f chaos—soft or hard. The range between soft and hard chaos among nations is the range between minor and major loss o f control i n international relations. I t is the range between the possibility o f loss o f an unspecified soldier i n a given battle, the w i n n i n g or losing o f that battle, o f the war, and o f the existent w o r l d system. We postulate that the presence o f hard chaos, in a theoretical system modeling the international system o f competing nations, is a represen­ tation o f major crisis instability and o f the extreme likelihood o f the outbreak of major war in the real w o r l d being modeled. I f the prediction o f hard chaos is believable, then the international security policies associated with that part

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o f the model leading to the onset o f chaos should be avoided. A useful analogy is to the testing o f a new aircraft to determine its behavior under a wide variety o f desirable and undesirable circumstances. I f the theoretical model (either purely mathematical or based upon w i n d tunnel modeling) indicates that certain maneuvers are likely to lead to loss o f control and possible loss o f plane and crew, the pilots w i l l be instructed to avoid those maneuvers—even i f they w o u l d lead to otherwise desirable results. Even i f the model is not sufficiently complete to indicate how the aircraft w i l l behave in the danger zone but is believable in predicting the existence o f the loss-ofcontrol zone, that w o u l d be sufficient to mandate changing policy so as to stay away from that zone. Thus, i f the prediction o f unpredictability i n an inter­ national system is believable, the ability to make such predictions, even i f incomplete i n details, can be used to answer political science questions as w e l l as to help determine practical national security policy. Thus we must face the question: is the prediction o f unpredictability believable in an incomplete model? I f the onset o f chaos portends the outbreak of war, can we believe a model's prediction o f war even though the extent and outcome o f the war are not predicted and may not even be describable by the model? Is the prediction o f instability (or stability) o f a model itself stable i n the face o f expansions o f the model to include more relevant aspects o f the system? W i l l a model's prediction o f chaos be softened or disappear as more variables and/or more complicated interactions are added to the model? A formal answer to the above question requires a working knowledge o f the complete model, knowledge that doesn't exist. ( I f it d i d , the question would be irrelevant!) In its place, we must rely on analogy with other systems—usually from mathematics or physics—where complete models, and their incomplete component models, exist and are used. Mathematical experience w i t h dynamical systems indicates that "chaos first appears in the neighborhood o f non-linear resonances" (Reichl 1992, 14), where new vari­ ables first make their impact upon the system. There is no experience o f such chaotic regions disappearing as new variables are introduced. Empirical expe­ rience with fluids indicates that chaos (turbulence) appears earlier and stron­ ger when new variables, such as temperature differences and heat flows, become important in the system. Theoretical experience with specific mathe­ matical models o f real phenomena (the Navier-Stokes equation for fluid flows; the recursive equations' modeling o f the evolution o f tripolar systems from bipolar ones [Saperstein 1991], developed later i n this chapter) suggests that the regions o f stability (areas o f absence o f chaos) decrease in extent when additional variables come into play. We thus presume that qualitative (gross) predictions o f the loss o f model stability are much more believable (more stable) than are the quantitative (detailed) predictions from the same model. Hence, the prediction o f hard

The Prediction of Unpredictability

151

chaos in a model reasonably representing the international system o f compet­ ing nations is a fair warning to policymakers: embarkation upon the modeled course o f action is to be done with extreme dread and care. However, a contrary prediction o f model stability is not an absolute assurance o f system stability; complacent acceptance o f the corresponding policy is not warranted since a more faithful, more complete model o f the system in which the policy is being applied may very well allow chaotic breakdown. Similarly, when exploring theoretical questions about the international system, the appearance o f chaos in an appropriate model portends the breakdown o f stability in the system; however, stability o f the model does not necessarily imply a corre­ sponding stability i n the system. The stability o f a mathematical system can be tested directly i f sufficient computing power is available. For example, a large number o f possible differ­ ent but neighboring initial configurations o f the system o f interest may be specified. Each one o f these starting configurations gives rise, via the mathe­ matical model, to a final configuration (the system at some specified future time). I f these predicted final states are as similar to each other as the initial states, the system is stable. I f they are w i l d l y divergent, as compared to the starting values, the system is chaotic. As an illustration, the S D I system proposed by President Reagan can be described by variables representing the number o f offensive missiles available to the opponents, the number o f defen­ sive antimissile weapons, and the number o f anti-anti-missile weapons (also offensive) (Saperstein and Mayer-Kress 1988). The model consists o f rela­ tions between these variables specifying how each o f the opponent parties procures weapons in response to the analogous procurements o f the other nations in the system—a typical arms race. The initial configuration is spe­ cified by the initial numbers o f the different weapons stocks and some numeri­ cal estimates o f their capabilities: neighboring starting configurations imply slightly different values for these initial numbers. (In the real w o r l d , these numbers and parameters would not be precisely known to any o f the compet­ ing parties.) The computer then tracks the evolution with time o f the system emanating from each starting configuration and displays the variations in the possible outcomes—the numbers o f the different types o f weapons i n the stocks o f each o f the opponents. I f the final variations are comparable to the corresponding initial variations, the system is stable. I f the final variations are large compared to the initial ones, but still small enough that system d o m i ­ nance (superior and inferior weapons stocks) can be distinguished, the system displays soft chaos. I f the final outcomes vary so widely that they encompass the entire variation possible to the system and obscure all differences between the variables corresponding to the different nations, then no prediction o f the outcome o f the arms race is possible—we have hard chaos. A more modest approach, suitable for systems with fewer variables, is

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the calculation o f Lyapunov coefficients for the model in question. These coefficients are direct measures o f the rate at which initially neighboring configurations drift apart as the model system evolves. Each possible starting point traces out a path toward the future by means o f the model mechanism, as is illustrated in figure 7 . 1 . I f the paths from closely neighboring starting points remain close (fig. 7.1b), prediction is possible; i f the paths separate exponen­ tially (fig. 7.1c), so that the final outcomes cover all possibilities allowed in the system, everything is possible in the future, meaning nothing is k n o w n about the future, and prediction is impossible—the situation is chaotic. The Lyapunov coefficient (see, e.g., Schuster 1988) is a useful measure o f such path separation that is easily calculated given a computer able to follow the evolution o f neighboring paths. Let X and XQ be any two possible neighbor­ ing starting configurations, separated by a small distance 8 . I n the course o f time n these evolve respectively (e.g., via equation 1) to the t w o configura­ tions X and X ' , separated by the distance 8„, where 0

0

N

N

=

8

0

*"«*o>.

(4)

This defines the Lyapunov coefficient Ç(X ), which depends upon the starting configuration as w e l l as the dynamics o f the system. Since the interest is in the long-time evolution o f the system, it follows that 0

limit S&o)

=

limit

°° fir,

1 0

n

log

x:

(5)

I f £ < 0, two configurations starting close to each other w i l l remain close to each other. Thus, predictability is possible (fig. 7.1b), so such a system w i l l be defined as stable. I f £ > 0, the initially close configurations w i l l drift ever further apart, making prediction impossible. Thus £ > 0 is the signature o f chaos or instability. In this chapter, several aspects o f international arms competitions, nor­ mally modeled qualitatively and verbally, w i l l be put into equivalent algebraic forms. Only algebra is used! The variable expressing the arms level or pro­ curement level o f one country in one year is expressed in terms o f similar variables describing the status o f that country and o f its competitor nations in the prior year (e.g., equation 3). These expressions only involve a small number o f additions, subtractions, multiplications, and divisions. Hence, they can be put into standard spreadsheets on a desktop computer and iterated from year to year to generate the time evolution o f the armaments levels o f the competing nations. Thus, the future o f these levels, and o f the arms competi­ tion among these states, is determined. Similarly, the corresponding Lyapunov coefficients, describing how the uncertainty associated w i t h these levels, given that there are initial uncertainties, evolves with time, are also

The Prediction of Unpredictability

153

computed. O f course, one doesn't reach limits o f infinity or zero, as is required by the definition in equation 5, on a desktop computer. Instead, the iterations are run up to large values o f n ( i . e . , 100) for varying small values of 8 . 0

The prediction by a mathematical model is meaningful only i f the results are stable, that is, i f small changes in the input and/or model parameters lead to corresponding small changes in the predicted time evolutions. Otherwise, the model's output is chaotic and indicative that the system being modeled is crisis-unstable and likely to undergo a phase transition from a competitive, but nonshooting, cold war to a hot war. Since the models are necessarily incomplete, their detailed predictions are not really believable, even i f mean­ ingful within the context o f the model itself. But, as we have presumed before, the prediction o f chaos is itself meaningful and believable. Using the desktop computer, the models are explored over different ranges o f parame­ ters, or different algebraic forms, to see which regions lead to stable solutions, which to chaos. Since different parameter ranges or algebraic forms represent different policy choices, we are thus able to predict which policies w i l l be dangerous, in that they may lead to crisis-unstable situations. We have briefly described one approach that can lead to specific policy choices (e.g., S . D . I , should not be developed i f it w i l l lead to a decrease i n the range o f stability o f the international offensive nuclear warhead missile system). I n the remainder o f this chapter the chaos approach w i l l be applied to three political theory questions: Are bipolar international systems more or less stable than corresponding tripolar systems? Is a system o f democratic nations more or less stable than corresponding systems o f autocratic states? Is a system o f nations that strives for a balance o f power via shifting coalitions o f states more or less stable than one in which each nation individually seeks to balance the power o f the others? There is a great body o f literature exploring the answers to these ques­ tions via qualitative verbal analysis, and no claims are made for new or surprising answers. However, since all o f these theoretical approaches, quali­ tative and quantitative, are incomplete and hence not completely convincing, a juxtaposition o f different approaches that lead to similar answers should add to our understanding o f these questions and their applicability. A disagree­ ment between the results o f the different approaches should be a signal that understanding is still lacking. W h i c h Is M o r e W a r - P r o n e — A Bipolar or a Tripolar W o r l d ? Is a bipolar w o r l d more or less stable than a tripolar world? As we leave a world configuration dangerously dominated by two nuclear superpowers for a w o r l d perhaps dominated by none, one, or many, it is important to gain

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insight into whether w e must be even more cautious or can be a little less t i m i d in addressing the many nonmilitary security problems o f a twenty-firstcentury w o r l d . One approach to this question is to compare the range o f stability o f a two-power system w i t h that o f a corresponding three-power w o r l d . The variable chosen to describe national behavior in the interactive model o f equation 3 is the devotion o f a given nation to war preparation—the ratio o f arms procurement ( i n the most general sense) i n a given year to the gross national product o f that nation i n the same year. O f necessity, the numerical value o f this variable must lie between zero and one. The bipolar w o r l d is then heuristically modeled by the simple nonlinear Richardson-type model represented by the coupled equations below X Y

n+l

= 4ar„(l - Y ) = 4bX (\ - X„).

(6)

n

n+l

n

X is the devotion o f nation X to war in year n. The procurement o f arms by X in the year n + 1 is in response to, and assumed to be proportional to, the previous year's devotion o f its opponent, nation Y. The nonlinear term (1 — Y„) expresses the assumption that i f the opponent's resources are stretched to the breaking point ( i . e . , his previous year's procurements were so great that they almost preempted his entire G N P , making i t impossible for h i m to procure any more), there is no need to stretch any further this year, so that this year's procurements may be correspondingly diminished. This model is so simple that its region o f stability may be analytically computed (Saperstein 1984) as a function o f the model's proportionality parameters, a and b, which must also lie between zero and one so that the X , Y„ remain between these bounds. The resultant curve represents the critical relation between a and b; the region above the curve, in which the two Lyapunov coefficients, £(X ) and £(Y ), each defined as in equation 5, are positive, is the model's chaotic region. n

n

0

0

In equation 7 the model is extended to three nations (when e = 0, equation 7 reduces to the previous two-nation model, equation 6): X

n+1

Y

n+l

Z„

+ 1

= 4aY„(l

- Y ) + 4 e Z „ ( l - Z„)

= 4bX„(l

- X„) + 4eCZ„(l

= 4e[X„(l

- X„) + CY„(l

n

- Z„)

(7)

- /„)].

As e increases from zero, the coupling between X and Y and the third nation, Z , becomes increasingly significant. Numerical computations o f the three Lyapunov coefficients pertinent to this model, £(X ), £(Y ), and £(Z ), made using simple spreadsheets on a small desktop computer (Saperstein 0

0

0

The Prediction of Unpredictability

155

1991), indicate that the stability region decreases in area as e increases, that is, as the third nation becomes more significant in the w o r l d system. Hence, the model suggests, in accord w i t h a great deal o f other scholarly evidence, that a tripolar w o r l d is more dangerous than a bipolar one. A r e Democracies M o r e or Less Prone t o War? Given the propensity o f many o f the world's major democracies to aid and abet less-than-democratic governments elsewhere, it would be useful to know i f such policies hinder or further the w o r l d peace that these democracies presumably seek and guard. Required first is a definition o f democracy that is easy to quantify and fit into a model o f competitive arms procurement, such as that o f equation 6, suitably generalized. It seems easiest to define democracy in terms o f diffuseness o f decision making. The larger the fraction o f the population that has significant input into matters o f peace and war, the more democratic the nation w i l l be considered to be. W i t h this definition, the numerical difference between democracy and autocracy w i l l be based upon population-sampling theory. I assume that every citizen o f nation 1 has some intrinsic feeling for the totality o f its opponent nation 2, and conversely, ranging from complete confidence to extreme fear and loathing. Let a, be the "fear and loathing" coefficient for any one citizen of nation 1. Then the policy o f nation 1 toward its opponent w i l l be deter­ mined by (a,), the mean o f a, over the class o f citizens o f 1 who are significant decision makers; (a ) is similarly defined. The fraction o f the citizens belong­ ing to this class w i l l be small in an autocracy, large in a democracy. 2

Assume that there is a natural spread o f values o f the fear and loathing coefficient whose distribution is characteristic o f humanity, not nationality. Each large nation w i l l contain many people manifesting small values o f the coefficient, intermediate values, and so on. Thus the mean, a, o f this coeffi­ cient, taken over an entire large population, is a human characteristic, inde­ pendent o f nation: d , = a = a. (Note that the known examples o f extreme interethnic hostility usually occur in comparatively small groups.) 2

The decision makers o f a nation are a sample o f the entire nation. I f it is a very large sample, means over this sample w i l l be close in value to means over the entire population. Hence, for large democracies, (a) = d. Means over small samples may vary significantly from the means over the entire popula­ tion. Thus, given a random set o f autocratic nations, there w i l l be some for which (a) greatly exceeds a and some for which (a) is much less than d . I n a corresponding set o f democratic nations, all w i l l have intermediate values o f {a) = d. Therefore, i f is the effective fear and loathing coefficient o f nation X for nation Y , i f a l l nations i n the set containing X and Y are large democra­ cies, = d , whereas i f the set contains autocracies, can range from very small values to very large values.

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Chaos Theory in the Social Sciences

The effective fear and loathing coefficient is taken to be the propor­ tionality constant i n a nonlinear Richardson-type arms procurement recursion relation. I n a competitive situation between nations X and Y, the annual procurement o f arms by X , A X , w i l l be proportional to the amount by which X's arms stocks were exceeded by Y in the previous year: Y — X . (The proportionality constant w i l l be taken to be a^; the similar coefficient for Y's annual arms buildup is a .) Procurement w i l l also increase proportionally to the total strength o f Y : Y„; i f Y is very weak, i t w o n ' t matter i f Y is stronger than X ; the more powerful Y is, the more damaging any discrepancy in power w i l l be i n any potential conflict. However, arms buildups cannot continue indefinitely. There w i l l be economic constraints; nations can't procure more than the total economy w i l l allow. I f C is the m a x i m u m annual expenditure for arms possible by X , assume a smoothed economic cutoff function n

n

yx

x

0 ( 1 -

AX/C )

0 ( 1 -

= (1 - AX/C )

X

X

AX/CJ,

(8)

where 0 is the unit step function (0(TJ) = 1 for 17 > 1, 0 = 0 otherwise). I f X ' s arms stocks exceed those o f Y , then X can decrease his arma­ ments. Assume that the arms build-down in any one year (AX negative) w i l l be proportional to the amount by which X exceeds Y ' s strength in the previous year and also proportional to the total power o f X : the more confident a nation is in its strength, the more it can afford to build d o w n . The proportionality parameter for build-down is the confidence coefficient, w h i c h , for simplicity, is taken to be the inverse o f the build-up fear and loathing coefficient. Finally, there are no economic constraints for build-down. (It is assumed that recon­ version o f the military-industrial complex works!) The result o f combining the coupled arms buildups and build-downs is the following set o f recursion relations, relating the total arms stocks o f X and Y i n any year to the corresponding stocks in the previous year:

x„

+i

=x„

+ Y„(Y„

~^-X (X n

u

Y

n+l

=Y

n

x ) 0 (1 - [x

-

axy

n

-

n

n+l

- x yc ) 0 n

x

(Y

n

Y„)@(X -Y ) n

n

(9)

xy

+ a X (X yx

-

n

Y {Y n

n

n

- X„)

Y„) 0

(1 -

0 (Y„

-

[Y„+l

-

Y ]IC ) n

y

0

(X

n

X ). n

"xy

These relations may be iterated, given arbitrary starting values X and Y at n = 0, again using standard spreadsheet routines on a desktop computer. Going on to large n (n = 100), the t w o Lyapunov coefficients can be calcu­ lated from equation 5. Typical results (Saperstein 1992a) are given in figure 0

0

The Prediction of Unpredictability

157

x<0) = y(0) = 0.3 epsilon = 0.001

b

20

V 0

10

20

30

40

a Fig. 7.2. T r a n s i t i o n t o e s c a l a t i n g a r m s race

7.2, where the region above the curve is the region o f instability. Note that small values for both a = a^ and b = a imply stability whereas instability results when both a^ and a get large or when either gets very large. Since large values o f the effective fear and loathing coefficient imply chaos and war, and since any collection o f autocratic states (or large mixed collection o f autocratic and democratic states) is more likely to have some pairs o f nations w i t h large values o f this coefficient than a corresponding collection o f democratic states, it follows that an outbreak o f war is more likely i n a collection o f autocratic states than in a similar collection o f demo­ cratic states. Even though it is often easier, in foreign affairs, to deal with autocratic governments, it is evident that, in addition to the usual human desire to nurture similar systems elsewhere, democracies are more likely to ensure w o r l d peace by supporting the democratization o f other members o f the international system. yx

yx

W h i c h is M o r e War-Prone: A S y s t e m of S h i f t i n g Alliances o r a Collection o f G o - l t - A l o n e States? As a last example, consider the question: which is a more stable system o f competing states: one in which each nation individually procures arms so as to

158

Chaos Theory in the Social Sciences

match any other state in the system or one i n which weaker nations j o i n coalitions to collectively procure arms against the strong and in which they shift alliances when a coalition member becomes too strong—independence security or balance-of-power security? To model the t w o cases simply, con­ sider three competing states, either independent o f each other or in which any two w i l l be allied against the stronger third. Considering any pair o f the three, arms buildup or build-down is pre­ sumed to occur just as in the previous case (equation 9), w i t h the effective fear and loathing coefficient o f X for Y and l/a^ the corresponding confi­ dence coefficient. However, now X may pair against Y or Z , i f all three are independent, so the resultant arms procurement or build-down is the sum o f the corresponding pair terms: AX = X

= [ Y„(Y

-X„

N+L

AXY

-

N

X„) 0 (Y

- X„) + a Z (Z

N

XZ

N

N

-

X) N

l

e (z„ - x )] © ( - = ^ ) n

- Y ) 0 (x -y„)~

- ~- x (x n

AY

= y„

+ 1

YX

Y ) 0 (X N

- x„) © (Y -x„)-^-

N

- Z„ = [a X„(X„ ZX

N

-

Z„)] 0

- -L z (z n

YZ

N

N

-

Y) N

Y„(Y„ - Z ) 0 (Y - Z„)

N

N

u

0 (Y

Y„) + a Z (Z

(10)

yx

+ 1

-

N

N

n

N

N

= Z„

-

N

n

Y )} 0 ( ^ - ^ )

- - L Y (Y

AZ

N

- Z„) 0 (X - Z„)

~ x (x

n

= [a X {X

-Y„

0 (Z„ -

u

N

n

(

X

- Z„) 0 (X„ -=^

Z„) + a Y (Y ZY

N

N

- Z„)

)

- x ) © (z„ - x ) - -L z (z

n

N

yz

n

n

n

n

- Y„) © (z„ - Y„).

In the alliance model, X w i l l j o i n its procurement w i t h Y to match a superior Z , or j o i n w i t h Z to match a superior Y , and similarly for build-down (here the appropriate coefficients are written b^, b , and so on): x z

AX = [b^Y,,

- X „ - Z„) 0 (Y

0 (Z„ - X

N

n

-

Y )} 0 N

(

~ X „ - Z„) + b Z (Z XZ

)

N

N

- X

N

-

Y) N

The Prediction of Unpredictability

TT~ V« + z )(X n

+ Z

n

-

n

Y ) 0 (X n

n

159

Y)

+ Z„ -

n

xy

(X

AY

n

n

n

+ Y - Z„) 0 (X„ + Y n

n

n

0 (z„ - y„ - x„>] © r - (Y + z„)(Y

-

n

n

Z„)

n

- Y „ - Z„) 0 (X

= [b X {X„ yx

+ Y )(X

- Y

n

- Z„) + b Z (Z„ yz

- Y „ -

n

X) n

1 - AY

+ z„ -

© (r„ + z„ -

x) n

(ID - ^ - (y„ + x„)(Y

n

AZ =

[b X (X ZX

N

N

+ x„~

- Z „ - Y„)

z„) 0 (K„ + x„ -

0 (X„ -

K„) + fo /„(K„ - Z„ - X„)

Z„ -

0 (K„ - Z„ - X„)] 0 (

z„)

z

)

- j - (z„ + r„)(z„ + Y„ - x ) 0 (z„ + Y„n

x) n

zx

-T~(Z

n

+ X )(Z„ n

+ X „ -

Y ) 0 (Z„ + X N

N

-

Y ). N

The t w o models each consist o f three coupled recursion relations (eqs. 10 and 11). The answer to the question posed is obtained by seeing which o f the t w o models is more stable. The analysis is done by assuming that the initial values, X , Y , Z , in each o f the three nations are almost the same and asking whether these small differences grow as the recursion progresses. A spreadsheet iteration o f the recursion relations is again carried out, starting off w i t h XQ, Y' , Z' , represent­ ing small random differences about an arbitrarily chosen starting point, X . Lyapunov coefficients are computed (again by iterating out to n = 100 w i t h small 5 ) , varying the starting value X and the single model parameter a (representing a common value for a^, b^, a , and so on). 0

0

0

0

0

0

0

0

xz

Four possibilities become evident, as the parameters a and X are varied, for the independent nation model as a result o f the numerical computations (Saperstein 1992b). I n the first case, strong stability, all sequence limits exist and all Lyapunov coefficients are negative, which implies that all sequence limits exist and are the same. There is complete predictability and hence no war. I n the second case, weak stability, all sequences have limits that are 0

160

Chaos Theory in the Social Sciences

equal to each other, but the Lyapunov coefficients are positive. This occurs because all limits are less than the starting value, X , except for the singular case when all start off exactly equal to X , in which case the limiting value is X . Hence, the numerator in the definition o f the Lyapunov coefficient doesn't become less than the denominator, implying a positive coefficient. This situa­ tion is an illustration o f bifurcation, in that one point leads to results com­ pletely different from all other points. Since the limits are still well defined, the sequences are predictable, not chaotic, and so again there is no war. I n the third case, weak chaos, none o f the sequence limits exist and so there are no Lyapunov coefficients. However, the variation within any one o f the se­ quences, X' , Y' , Z' , and the differences between the sequences (X' — X ) remain small compared to the arbitrary j o i n t starting value, X . The situation resembles case b i n figure 7 . 1 , rather than case c; hence, the unpredictability is minor and so, i n the paradigm o f this chapter, does not represent war. The last possibility is one in which the sequences do not have limits and the variations w i t h i n each sequence, as w e l l as the differences between them, are large compared to the starting value. In this case the range o f unpredicta­ bility is the entire range possible, there is no way o f knowing—even approximately—the outcome o f any policy or action, and hence major fluctu­ ations may result from minor perturbations. This is the criterion for crisis instability and war. 0

0

0

n

n

n

n

n

0

The numerical results for the independent nation model are displayed in figure 7.3. N o sharp boundary was evident between the regimes o f weak stability and weak chaos and so the two regimes are considered to be a single realm i n w h i c h crisis stability is still valid. Note that there is no upper l i m i t to the weak chaos regime, no breakdown o f crisis instability, i f the fear and loathing coefficient is small enough (a ^ 1). I f there is enough confidence in the system, perturbations to the independent nation international system may lead to continued fluctuations and uncertainty, but not to war, no matter how large the initial, equal, stocks o f arms are. On the other hand, with sufficient fear and loathing (a > I), war w i l l always break out i f the initial arms stocks are large enough. The numerical predictions o f the alliance model are quite different. For large enough symmetric starting values, X = Y = Z , the system imme­ diately and completely disarms, no matter what value is chosen for the fear and loathing coefficient. This is region I o f figure 7.4; since all sequences tend immediately to a zero l i m i t i n g value, it is a strong stability region. I n region I I o f the figure, a l i m i t has not yet been reached numerically at n = 100; nonetheless, the fluctuations in the monotonically decreasing sequences are small and hence this region is appropriately characterized as weak chaos: the course o f events is clear and the eventual complete disarmament is completely predictable. 0

0

0

a Fig. 7.4. N u m e r i c a l p r e d i c t i o n f o r t h e alliance m o d e l

162

Chaos Theory in the Social Sciences

Hence, using the possibility o f strong chaos in the model as an indicator o f the possibility o f war in the corresponding international system, it follows that a policy o f building (perhaps shifting) alliances is guaranteed to keep an initially symmetric system peaceful. A policy o f seeking national security by "going i t alone" allows the possibility o f war in such a system i f the contend­ ing parties start out w i t h large enough levels o f armed force. I know o f no real-world situation in which a system o f comparable competing states have balanced themselves d o w n to a complete disarmament as is predicted by the alliance model o f this chapter. Perhaps this is because a shifting o f alliances has costs i n the real w o r l d , costs not included in this model. Also possible is that there are no real balance-of-power situations; real nations may resort to balance-of-threat instead, a much harder situation to model (Walt 1987). Or, the presumed anarchy may not have always been present. Conclusion

Humans have always developed theories (often mythologies) to explain their w o r l d . W i t h the advent o f science, theories had to have practical as well as literary or philosophical implications. A successful theory o f international relations should be an important component o f competent policy making in the arena o f international security in a competitive world. Conversely, a careful analysis o f practical policy—successful or unsuccessful—must be the foundation for any scientific theory. Given the complexity o f the real international system, any theory o f it is bound to be incomplete, describing some aspects o f it more thoroughly than others. A truly massive modeling enterprise—such as is done i n economics w i t h national input/output models or in meteorology w i t h large-scale computer models o f the atmosphere—may at best lead to very rough, shortterm, quantitative agreements between predicted and observed experience. The quality o f the results is often vastly disproportionate to the effort required to secure them. It thus seems useful to look for simple models and methods o f model analysis that can lead to qualitative robust predictions. One such approach is the prediction o f unpredictability in simple non­ linear models o f the competitive interactions between rival states. It can be an important approach to policy and theoretical questions, given the fundamental assumption that such unpredictability may represent crisis instability and war in the international system. Given that policy questions are usually more specific and complex than more theoretical questions, we expect to require more complex models for the former, although they would still be consider­ ably simpler than any intended to address quantitative questions. For example, very simple theoretical models, easily analyzed w i t h spread sheets on a desk-

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top computer, have allowed us to conclude that a tripolar world is less stable than a corresponding bipolar w o r l d , that a set o f democratic nations is more likely to be stable than a similar set that includes autocracies, and that a group o f nations attempting to guard their security via balance-of-power alliance formation is more stable than a group in which each member makes individual attempts to cope w i t h other potentially hostile members. A bigger and more complex computer model indicates that a policy o f introducing S D I into a situation similar to the recent cold war nuclear confrontation between the United States and the Soviet Union would likely dangerously destabilize such a world. None o f these theoretical or policy results is new. Similar conclusions have been reached before by many others using conventional verbal analyses. But the alternative route presented in this chapter—the qualitative analysis o f simple quantitative models—leads very quickly to the answers without any hidden implicit assumptions, biases, or faulty logic. Finally, agreement o f results obtained via traditional means and the methods developed here brings added credence to both approaches. Science is a fabric: its ability to cover the w o r l d depends upon the existence o f many different fibers acting together to give it structure and strength.

CHAPTER 8

C o m p l e x i t y in t h e E v o l u t i o n o f Public O p i n i o n Michael McBurnett

Most political phenomena are not static. Presidential campaigns and the elec­ torates' response to them; the emergence and outbreak o f international hostili­ ties; the formation o f racial attitudes—all these and more are inherently dynamic processes. Be that as it may, we often study such phenomena as though they were static. Cross-sectional analyses continue to dominate disciplinary research, for two reasons. First, we typically choose to analyze cross-sectional data. Exemplary are the many analyses o f year-specific surveys o f the electorate during presidential and off-year elections (see Huckfeldt and Sprague 1987; Cohen et al. 1991; and examples based on specific National Election Studies), w h i c h , it might be noted, represent some o f the most sophisticated empirical work in political science. Second, our methodological training has largely concentrated on techniques well suited to the analysis o f cross-sectional data; ordinary least squares and its cousins continue to hold sway. Although scholars increasingly are turning to the analysis o f phenomena over time (Tuma and Hannan 1984; Huckfeldt 1990; B r o w n 1991), the study o f dynamic processes i n a truly dynamic way remains in its infancy. One o f the most foreboding challenges is to identify the complexity that inevitably characterizes dynamic phenomena (Ford 1986). In the pages that follow, three methods are used to analyze complex dynamical time series. First I discuss the phase portrait. Second, the correlation integral is introduced and applied to a variety o f time series. Finally, Lyapunov exponents are used to illustrate the divergent properties inherent in complex dynamics. I discuss those methods w i t h i n the context o f the 1984 race for the Democratic nomina­ tion. When candidates do w e l l , estimates o f their chances o f winning increase. When the candidates fare poorly, citizens report a corresponding decrease in their estimates o f the candidates' chances o f winning the nomina­ tion. This pattern o f probability estimates does not follow an orderly cycle, however; the pattern is considerably more complex—so complex that it is best described as chaos. Chaos implies a deterministic structure in a nonlinear dynamic process. 165

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I first demonstrate that cycles can emerge naturally in competitive nomi­ nation races. I then discuss the phase portrait, a technique designed to reveal systematic patterns in over-time data. A p p l y i n g this technique to a 1984 NES study identifies a very complex, yet regular structure. The heart o f the anal­ ysis is a demonstration that this time series is chaotic. After showing that application o f an alternative linear model fails to account for the complicated dynamics in the time series, I conclude w i t h some suggestions for extended application o f this technology to other data series. The Emergence o f O p i n i o n Cycles i n Competitive Nominations The empirical pattern o f voter support for candidates as the presidential nomi­ nation race unfolds provides direct evidence o f how a cyclical pattern in public opinion may develop during the campaign. Figure 8.1 shows the vote percentages received by Hart and Mondale in the 1984 Democratic race for each election date. Where multiple races were held on a single day, as on Super Tuesday, the results have been averaged over all races held on that day. The graph presents three pieces o f evidence about the dynamics o f candi-

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date competition in presidential nomination races. First, w i n n o w i n g , which occurs as a result o f competition (Aldrich 1980; Mayer 1987), appears to take place early in the campaign, as those candidates who are not viable are eliminated. This is inferred by summing the percentage o f the vote captured by Hart and Mondale. Second, the graph illustrates the concept o f momen­ t u m . For the three races following the Iowa caucuses (the first point on the graph) Hart's percentage o f the vote climbs. Following Hart's crushing defeat o f Mondale i n Vermont's nonbinding primary, the percentage o f the vote Hart wins begins a three election decline. As Hart's electoral fortunes go into decline, Mondale's reverse. T h i r d , as the nomination race progresses, Mondale and Hart trade the lead a number o f times. The predominant pattern in the race is one o f oscillation, indicative o f the ebb and flow o f electoral fortunes in the race. This demonstrates that momentum works in two ways: it can be positive or negative. The oscillatory pattern can be demonstrated, first, w i t h a simple model o f the process o f winning votes (see Sprague 1976). Expectations o f future candidate performance in the nomination race are based on the past perfor­ mance o f the candidate i n electoral contests, weighted by the number o f contests or the number o f delegates that remain to be won (Aldrich 1980; Bartels 1988, 3 5 - 3 6 ) . Two versions o f this model are used to illustrate these dynamics. The first is unweighted and the second uses the percentage o f the contests that remain as a weight. This model (for Mondale) has the form Mondale Percent o f the Vote, = a + (3 • (Mondale Percent o f the V o t e , . , ) .

(

1

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The parameters are estimated by O L S . The results establish that the percent­ age o f the vote Mondale wins does oscillate about a limiting value. I n the unweighted model, the estimate for the lagged variable is —.389 ± .22. The constant term in this model is .49. The estimate in the weighted model is — .409 ± . 19, w i t h an intercept o f .52. Both parameter estimates are negative and the absolute value o f each estimate exceeds zero but is less than one. The time path for both versions o f this model is an oscillatory trace that converges rapidly to an equilibrium. Both demonstrate that oscillation in the percentage o f the vote a candidate wins can occur during the nomination race. However, the pattern produced by the parameters is short-term, period-two oscillations, a pattern not found in the original data. A more elaborate model demonstrates the long-term oscillatory nature o f the election outcomes more convincingly. The traces for Hart and Mondale in figure 8-1 show long-term oscillations rather than period-two oscillations or an oscillatory convergence over time. Competition is an integral part o f the

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campaign, so a model should account for the competition for votes between candidates. Using only Hart and Mondale, a dynamic model o f competition is given by AMondale, = j 8 -

j8, Mondale, + j8 Hart,

0

2

(2) AHart, = B' -

j3',Hart, + /3 Mondale,

Q

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where AMondale, refers to the difference in the vote percentage Mondale received between time t + 1 and time t, B is a constant, j8, reflects the level o f difficulty campaigning plays in attempting to w i n votes by Mondale, and B measures the threat Mondale feels due to Hart's level o f success in repeated elections. B' , B\, and B' in the second equation are interpreted similarly for Hart. This model closely resembles the Richardson model o f arms races. 0

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One equation can be eliminated by substitution (see Huckfeldt et al. 1982). This produces a second-order linear difference equation in one candi­ date in which the interdependent structure o f competition is implicit. The procedure is symmetric for the other candidate. The result for Mondale is Mondale,

+2

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+1

(3)

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where b represents (B — B B — B B[B ), b represents (1 + B — B — B B[), and b represents (B B[B - /3 /3, - B B\ - B ), so that equation 3 is isomorphic to a regression equation with three parameters. The parameters are estimated using O L S , as before. The results for Mondale and Hart are shown in table 8 . 1 . After estimation, the equation can be put in the homogeneous form: 0

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TABLE 8 . 1 . E s t i m a t e s f o r t h e S e c o n d - O r d e r Dynamic Model Candidate

b

Mondale Hart

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are the only alternatives (Goldberg 1958; Huckfeldt et al. 1982). Substituting the estimates from Mondale's equation into the quadratic equation gives _ r

ii

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- ( . 1 0 2 0 ) ± V ( . 1 0 2 0 ) - 4 · (.4188) 2 '

' '

It is obvious that the sum under the square root radical is less than zero. Hence, the roots are complex conjugates and the time path described by this model contains long-term oscillations. The pattern is similar for Hart. For particular values o f these roots, the time path w i l l show stable long-term oscillatory motion (Baumol 1958). This implies that the competition between candidates persists over time during the nomination race, an implication that can lead to the media's portrayal o f competition between candidates as a horse race (Brady and Johnston 1987, 184; Popkin 1991, 117-18). Campaign man­ agers for the candidates in the 1984 race agree that this race was one in which the competition persisted and one in which the outcome was i n doubt for some time (Moore 1986, 56). The outcome from this model resembles the time paths shown in figure 8 . 1 , where neither candidate shows a simple oscillatory motion and the lead changes several times during the primary season. This simple model may provide some insight into the problem survey respondents have when asked to estimate the chance each candidate in a large pool o f candidates has o f winning the party nomination. The problem o f accurately assessing the chances o f a number o f candidates at a particular point in time is liable to be quite difficult when the candidates are locked in a race in which the lead changes a number o f times (Battels 1988, 3 2 0 - 2 3 ) . ' The simple models explored in this section demonstrate that competition between candidates can lead to fairly complex long-term behavior in the nomination race. I n this changing environment, where members o f the elec­ torate are liable to switch their candidate preference ordering subject to the results o f recent electoral outcomes, forecasting the eventual outcome be­ comes extremely difficult. Preferences i n one state are driven by recent elec­ toral results in another state. Issues become less important and electability increases in importance (Brady and Johnston 1987). The Mondale campaign discovered that even what is considered to be unshakable support can be swayed in this fashion. Fifty percent o f Mondale's lead in Maine disappeared between the New Hampshire primary and the Maine caucuses and this is attributed to Hart's early successes (Moore 1986, 76). The forecasting prob­ lem for campaigners is difficult, but it is precisely this difficulty that makes the problem interesting, for campaigners and scholars alike. I turn now to a method designed to analyze time series graphically and to discover hidden patterns in the data that may not be revealed by other methods.

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Revealing S y s t e m a t i c Patterns i n Time-Series Data The visual inspection o f time-series data rarely reveals an underlying regu­ larity in the data. For one thing, the time series may be too short: patterns that w o u l d be evident in very long series w i l l not appear. For another, the level o f noise i n the data may be large relative to the systematic movement in the observations o f the time series. In this section, I offer a simple example to motivate a strategy that allows one to discover patterns in time-series data that might not otherwise be visible due to the length o f the series and to noise in the data. After demonstrating how the technique works—and that this technique works when noise is present—I use it to investigate a time series constructed from the 1984 NES Rolling Cross Section. A n a l y z i n g D e t e r m i n i s t i c Data Assume that a candidate's mean thermometer rating among survey respon­ dents follows a regular long-term cycle during the course o f the nomination race. I f such a cycle were present, it might appear as shown in figure 8.2. This

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figure was produced by graphing a two-dimensional linear harmonic function that is centered on the value 50 percent and that ranges between thermometer scores o f 60 and 40 percent (see Hale and Kocak 1991). It is deterministic, clearly periodic, and contains long-term cycles (the oscillations are greater than period t w o ) . For the purposes o f illustration, the function is graphed over an interval o f one year. The equation for the function is given by AY, = aY, + bX

t

(6) A X , = cY, + dX„ where a = 0, b = 1.0, c = —1.0, a n d d = 0, and the time interval is 1, 2, 3, . . . , 366. To demonstrate that a cycle is present, a phase portrait is constructed. The phase portrait is a graph that shows the long-term behavior o f the time series w i t h time removed. Put another way, a phase portrait displays the asymptotic behavior o f a dynamic system. It is constructed by plotting the values generated from equation 6 o f X, against Y . t

This figure contains an ellipse and this ellipse is known as an oscillator. A n oscillator is an equilibrium solution that arises when a dynamic system undergoes " w e l l defined cyclic changes in t i m e " (May 1974, 16). A similar pattern o f survey responses can emerge in a contest like the 1984 Democratic race, where candidates swap the lead several times. I f the system is perturbed, as time evolution continues, the trajectory w i l l simply orbit at a new level o f equilibrium. This is an explicitly oscillatory phenomenon, and the example here is based on a linear harmonic oscillator. The simple second-order models devel­ oped in the prior section have nonlinear solutions that produce oscillatory time paths. The roots o f the dynamic equations describing the percentage o f the vote Mondale and Hart won in the sequence o f elections were complex conju­ gates. These roots have representations in the real plane as trigonometric functions that mimic dynamical behavior similar to that shown in figure 8.3. A n a l y z i n g Noisy Data What happens when the system contains noise? Noise may enter the data through measurement error, random response variation, sampling error, or by unknown other means. Figure 8.4 shows the simulated time series o f mean thermometer scores w i t h random variation about each point. The initial func­ tion is identical to the first part o f equation 6, but random error has been added to each observation o f the time series. The random component o f this series is normal (/it = 0, a = 0.025). I f the procedure o f examining the phase portrait is adopted to study the

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dynamical behavior o f this series, the technique is to plot Y against Y and view the result (see fig. 8.5). Though it is k n o w n that an equilibrium is present, simple random variation in the data masks i t . The result is a hope­ lessly tangled mess. t

t+l

A solution to the problem is a variation o f a technique suggested by Packard et a l . (1980). They demonstrate a way to recover a strange attractor from a time series. A strange attractor is an attractor that is not a cycle or equilibrium point, but is stable in the sense that the trajectories o f a dynamical system remain nearby as the system evolves in time ( M o o n 1987, 267). These attractors do not have ordinary shapes; they are not elliptical or circular i n the phase space. I n part it is their odd shapes that earned them the name strange. Packard et al. observe that any time series contains all the information that went into creating i t , even a series consisting o f several variables. Thus, one can use lagged values o f one variable in the series to construct a phase portrait and study the long-term dynamics o f a multivariate time series. This enables a sufficiently low-dimensional attractor to be revealed i f one is pre­ sent. I f a strange attractor can be revealed, then an equilibrium cycle can also be revealed.

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Mean Thermometer Rating at Time f for x Fig. 8.3. Phase p o r t r a i t o f average c a n d i d a t e race

thermometer

scores i n a t w o -

Complexity in the Evolution of Public Opinion

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Technically, this operation amounts to inserting a two-dimensional slice into a higher-dimensional space and mapping all the points down onto the plane. The result is the phase portrait. Some orientations o f the t w o dimensional plane show more interesting views o f the dynamics than others. The problem w i t h revealing the attractor is found in the orientation o f the graph w i t h respect to the viewer. The data cloud in figure 8.5 is oriented from lower left to upper right, similar to a person holding his or her hand a distance of one foot in front o f his or her eyes (as i f the person were preparing to issue a left-handed salute). From this angle, one can see the thumb and first finger, but little else o f the structure o f the left hand. B y rotating the hand so that the thumb turns toward the floor, the entire structure o f the back o f the hand is revealed. H o w can this idea be applied to the tangled structure that emerges from a plot o f Y, against Y w i t h error? Note that this implies a threedimensional attractor, for obviously a hand is not a two-dimensional object. t+l

Nicolis and Prigogine (1989, 277) display an example o f a technique that uncovers a strange attractor by using the appropriate lag structure in a threedimensional representation. The key is that the lag structure used is that which is appropriate. Experimentation is required to reveal the appropriate lag. The

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Chaos Theory in the Social Sciences J

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identification o f the lag structure proceeds as follows: use successively higher shifts i n the lag, defined as integer multiples o f a fixed lag, T ( T = mAt), where m is an integer and A f is the difference interval between successive samplings (Nicolis and Prigogine 1989, 276). Ar is always 1 for maps such as are used here (it differs from 1 only when the dynamical system is based on differential equations and describes a step size for the numerical integrator in those cases). m is an increasing index, 1 , 2 , . . . , n. This process leads to a set o f discrete variables Y„ Y , · • • , Y,+ t + l

n

Nicolis and Prigogine point out that "for a proper choice o f T these variables are expected to be linearly independent, and this is all we need to define a phase space" (1989, 277). A l l o f these variables are derived from a single time series, such as is shown in figure 8.4, and are provided by the data. This procedure yields an experimental time series and it provides the data to transcend the one-dimensional space o f the original (univariate) time series and to "unfold the systems' dynamics in multidimensional phase space" (Nicolis and Prigogine 1989, 277). Figure 8.6 provides a graphic revelation o f the structure o f the attractor k n o w n to be present in the noisy data representing the simulated thermometer scores. Here the value o f the thermometer score at time t is plotted against the thermometer score at time t + 10. The tenth lag o f the series has been used to

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Mean Thermometer Score at Time t Fig. 8.6. Phase p o r t r a i t r e c o n s t r u c t i o n o f noisy t h e r m o m e t e r scores, r e v e a l i n g t h e u n d e r l y i n g l i m i t cycle

reveal the equilibrium cycle. It retains its elliptical shape. This is strong evidence that this technique produces an insight that might otherwise be masked by random variation or noise in a time series. Clearly, in order to illuminate fine structure in a dynamical system, 2

empirical methods may be required. What happens when these techniques are applied to a time series o f public opinion data? I n the next section, the time series used in the later analysis is described. C o n s t r u c t i o n of t h e T i m e Series f r o m Public O p i n i o n Data In 1984, the Center for Political Studies conducted a telephone poll o f the national electorate using a probability sample over a very long period. This survey is k n o w n as the Rolling Cross Section. Although it has some draw­ backs, such as a fairly large sampling error (Allsop and Weisberg 1988), it is the only national sample o f its type. This analysis uses that portion o f the survey containing those respondents interviewed after January 11, 1984, and before June 12, 1984. Those respondents were interviewed during the period in which all primary and caucus activity took place. The survey asked respondents to rate the chances o f the various Demo-

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cratic candidates winning their party's nomination. As Bartels points out, most respondents failed to rate candidates' chances as i f they had probability theory i n m i n d (1988, Appendix A ) . Though Bartels used this survey for a quite different purpose, I make use o f his coding scheme, which measures each individual's perception o f each candidate's chance o f winning the Demo­ cratic nomination (1988, 3 2 2 - 2 3 ) . The individual responses were averaged according to the date o f the responses. This results in a measure o f the perceived mean daily probability o f each candidate's chance o f winning the Democratic nomination. In order to see i f any interesting dynamical patterns were evident, I smoothed the data using a five-day moving average. Smoothing a time series that contains noise may allow a cyclical or otherwise interesting pattern to be identified visually, i f one is present. Smoothing reduces, but does not e l i m i ­ nate, the level o f noise in the time series. This time series is shown in figure 8.7. The graph shows the smoothed mean probabilities that Mondale, Hart, and Jackson w i l l w i n the 1984 Demo­ cratic nomination over the span o f the nomination p e r i o d . 3

The phase portraits suggest that the dynamics o f these time series are not regular or oscillatory. Instead, the phase portraits reveal what may be a strange attractor resembling the Lorenz attractor from climate dynamics (Lorenz 1963; Schuster 1988). Strange attractors have so far been identified only in chaotic dynamical systems (Schuster 1988). The identification o f this type o f structure underlying the dynamics o f public opinion is important because o f the implications this has for forecasting the outcome o f nomination races as well as the study o f change in public opinion over time generally (Huckfeldt 1990). The Lorenz attractor is a strange attractor and the Lorenz equations have all the identifying characteristics o f chaotic behavior: the motion is highly erratic, the attractor is contained in a bounded region in the phase space, and the dynamics display sensitive dependence on initial conditions (Lorenz 1963). H i g h l y erratic motion means it is difficult to tell, a p r i o r i , exactly what value the next observation in the series w i l l take on, even when the equations governing the behavior o f the system are k n o w n . The dynamic behavior o f the series is contained in a restricted region o f the phase space. Being bounded means that the series does not diverge toward ± ° ° over time. Sensitive depen­ dence on initial conditions is a property o f nonlinear systems. It is defined as "separating initially close trajectories exponentially fast in a bounded region of phase space" (Schuster 1988). I n practice, this means that two nearby points selected as starting values for a dynamic system w i l l separate exponen­ tially quickly over time as the system is iterated. The dynamics illustrated i n the phase space (chap. 2, figs. 2.10, 2 . 1 1 , and 2.12) meet t w o o f these requirements: the attractor is contained in a

Complexity in the Evolution of Public Opinion

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Days Since January 1,1984 Fig. 8.7. P r o b a b i l i t y o f w i n n i n g t h e 1984 D e m o c r a t i c n o m i n a t i o n (all respondents)

bounded region in phase space and the motion is erratic. The governing equations for the system used in this analysis are not k n o w n , so it is impossi­ ble to tell, given the information presented thus far, i f this series displays sensitive dependence on initial conditions. However, i f this system can be shown to be sensitively dependent on initial conditions, then it is established that public opinion measures o f the mean likelihood o f candidates winning their parties' nomination are very probably unpredictable far in advance. This is disconcerting, not because forecasts may be slightly off from time to time, but because forecasts may be completely wrong between two adjacent polls only narrowly separated i n time. This means that pollsters may predict the wrong candidate winning elections. This has occurred in recent nomination races. James Johnson, Mondale's campaign manager in the 1984 race, relates that we d i d a poll [one week prior to the New Hampshire primary] o f our o w n people in the straw p o l l , and it was 50-50 with Hart. These were people who had signed in blood on October 1 that they were for Mondale until death, and we were 50-50. (Moore 1986, 73)

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Peter Hart was Mondale's pollster in the 1984 race. His forecast o f Mondale's support i n New Hampshire was off by 50 percent. The electoral result i n the New Hampshire primary was a boost for Hart and tighter competition for the Democratic nomination. The 1976 Democratic race also provides examples. Morris Udall's p o l l ­ sters reported three weeks prior to the Wisconsin primaries that the vote split was Carter 34 percent, (Henry) Jackson 24 percent, Udall 17 percent, Wallace 15 percent, and Undecided 10 percent. Udall desperately needed to w i n in Wisconsin in order to stay in the race. This p o l l nearly forced h i m from the race. Ten days prior to the election, another poll revealed a new mean prefer­ ence ordering between candidates among the public: Carter 34 percent, Udall 30 percent, Henry Jackson 15 percent. This p o l l , as well as the previous one, was conducted by Peter Hart (Schram 1977, 126). The Wisconsin primary results (the first primary in the 1976 Democratic race) were Carter 37 percent and Udall 36 percent, w i t h only five thousand votes separating these two candidates (Schram 1977, 132). On the Thursday prior to the Wisconsin primary, Carter's pollster, Pat Caddell, reported that "the weighted results stood at Carter 38%, and Udall 2 7 % , " w i t h 5 percent no-preference respon­ dents, who were expected to vote for Udall (Schram 1977, 133). This p o l l , conducted only days prior to the election, is close to the true outcome. The forecast error increases as the distance from the actual election date i n ­ creases. This problem has obvious implications for theoretical political science, as well as for practical politics. Can this series, or other similar series, be shown to be chaotic? The answer turns on the ability to determine i f the phase portrait is a t w o dimensional slice o f a higher-dimensional strange attractor with a positive Lyapunov exponent. This attractor must have a fractional, that is, noninteger dimension; no strange attractors are k n o w n w i t h integer dimensions (Schuster 1988, 106). I f it can be shown that the phase portrait is a snapshot o f a strange attractor w i t h a positive Lyapunov exponent, then the dynamics o f the un­ k n o w n equations o f motion must display sensitive dependence on initial con­ ditions and the time series is chaotic. 4

A demonstration o f sensitive dependence on initial conditions in a time series o f opinion data has ramifications that extend far beyond this essay. Nonlinear processes define the underlying structure o f opinion formation i f the series can be shown to be chaotic. Forecasts o f election results based on opinion data w i l l be unreliable unless the initial levels o f public opinion are k n o w n quite precisely. This condition is never met in practice. Linear models are functionally misspecified and results from these models are suspect. We must probe the dynamics w i t h other tools that provide deeper insights into the dynamics. One tool in common use is smoothing, which can help reduce the variation in the observed series. Smoothing has problems o f its

Complexity in the Evolution of Public Opinion

179

own (see Schaffer et al. 1990). Another tool is spectral analysis, which can uncover regularities in dynamical time series in the presence o f considerable noise. As this is the subject o f a separate chapter in this volume I refer the reader to it (see chap. 2). The time series in this chapter are analyzed i n much greater detail there. Smoothing and spectral analysis are both designed to probe for periodicity in the time series. What is needed is to attack the problem from another angle. Two techniques are available to probe the geom­ etry o f the time series: the correlation integral and Lyapunov exponents. In the next section I explain the construction and use o f the correlation integral. I also posit an alternative (linear, oscillatory) model and recompute the correlation integral. I follow w i t h an examination o f the largest Lyapunov exponent for the Mondale time series. Measuring t h e Dimension of a Strange Attractor This analysis uses the Mondale time series rather than data from the entire three-candidate series. It is known that the shape o f attractors in lowdimensional phase space can be recovered using the phase space representa­ tion from an appropriately lagged univariate time series (Packard et al. 1980; Nicolis and Prigogine 1989). This result allows the use o f one series to analyze the dynamic properties o f the entire three-candidate series. 5

Some vector notation is required for the analytic procedure that follows: Xj represents a point i n phase space with the coordinates for the point given by the ordered n-tuple { X , X - , X ,- , • • • , Xot-JTypically, one begins w i t h pairs o f points, then moves into a three-space, and then continues into higher-dimensional phase space. The analysis takes place in this subspace o f the N-dimensional space in which the original attractor lies (Packard et al. 1980, 712). I t is a consequence o f Takens's theorem (1980) that this technique works. The dimension o f the attractor can be measured i n a number o f ways. A measure that is computationally efficient for a discrete time series such as this one is called the correlation dimension. 0 /

0t {

0

2

The correlation dimension is useful when one does not know the exact dimension i n which the attractor lies. It is an experimental measure that w i l l determine the dimension o f the attractor numerically. Based on the ordered n-tuple constructed above, a reference point x is chosen and all distances |JC, — jt-| from the N — I remaining points in the series are computed (Nicolis and Prigogine 1989, 278). These distances are arrayed against various values o f a radius r from the point X in the phase space. This process is repeated for all values o f i. This computation calculates a correla­ tion integral. The correlation integral is denned as t

N

C(r)

=

2

l i m (l/N )

.^

6(r -

\ x,•. -

X j

|),

(7)

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Chaos Theory in the Social Sciences 6

where 6 is the Heaviside function, 6(x) = 0 i f x < 0 and 6(x) = 1 i f x > 0 . The Heaviside function acts as a dummy variable, allowing one to count points i f they lie inside the radius, r. C(r) varies over the range 1 IN to 1. This measure can be constructed using Euclidean distances or a non-Euclidean measure such as the sum o f absolute values o f the component vectors ( M o o n 1987, 216). The former measure is used here. The fact that C ( r ) fails to vanish measures the extent to which any one data point, X , , affects the positions o f the other points (Nicolis and Prigogine 1989, 278). Hence the reference to C(r) as the correlation integral. This method is also known as the ball-counting method (Barnsley 1988). Each point has a ball superimposed over it w i t h radius r. A l l the other points in the phase space inside this radius are counted and stored. The radius is incremented until all points lie inside the boundary. This radius defines the upper l i m i t . The lower l i m i t for a particular dimension is UN, for individual points must occupy a unique address in phase space. This process is repeated for a number o f dimensions based on the construction o f the ordered n-tuple 2

Despite the similarity i n names, the correlation dimension is unrelated to the statistical correlation coefficient commonly used in the social sciences. The correlation dimension provides a quantitative description o f the rate o f divergence o f nearby points i n the phase space, as opposed to the direction and strength o f association between two separate variables in a linear relation­ ship. For many attractors, the correlation dimension is known to scale as a power law function. That is, C(r)

~ r

d

(8)

or, i n other words, the dimension o f the attractor is determined by the slope o f In C ( r ) versus In r in a particular range o f r (Nicolis and Prigogine 1989; Grassberger 1986). This relation is In C ( r ) = d · In r.

(9)

The procedure then is to calculate C ( r ) for radii varying from zero to that sufficient to cover all points in the phase space for successively higher dimen­ sions. The relation between In C(r) and In r can be seen graphically in figure 8.8, where the correlation integrals for dimensions four through ten are plot­ ted for the phase portrait o f public opinion data related to Mondale's chances o f winning the 1984 Democratic presidential nomination. The slope, d, from equation 9 is estimated using bivariate regression from the various dimensions over the regions in figure 8.8 that are approx-

Complexity in the Evolution of Public Opinion J

I

|

I

I

1

I

I

I

181 L

Log Radius Fig. 8.8. C o r r e l a t i o n i n t e g r a l s f o r t h e M o n d a l e t i m e - s e r i e s d i m e n s i o n s four to ten

imately linear. It is the range o f linearity that allows the inference o f the dimension o f the attractor. The slope is calculated for each dimension. I f the slopes reach a saturation point, that is, become parallel as the dimension increases, then the saturation point is interpreted to be the dimension o f the attractor (Nicolis and Prigogine 1989, 279; Grassberger and Proccacia 1983, 196; M o o n 1987, 233). Where the slopes fail to saturate and the correlation integral scales according to the relation In C(r) = d • In n,

(10)

where n is the dimension in which the correlation integral is constructed, the dynamical process is understood to be Gaussian white noise. This relationship is represented by a 4 5 ° line in the plane. This clearly differentiates the two processes that might be expected. Evidence o f the presence o f a strange attractor is given by a series o f estimates for d that lie under a 45° line. Figure 8.9 graphs the slopes for dimensions t w o through ten for the phase portrait for Walter Mondale. The error bars show the range o f standard errors for the slope estimates. The slopes saturate or become parallel after dimension five. The horizontal line drawn through the saturated values to the

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vertical axis is the mean o f the slopes for dimensions six through ten. This is the dimension o f the attractor. The value is approximately 3.58. Since this value is noninteger, we make two inferences: the dynamics are governed by a nonlinear process and the attractor is a strange attractor. This time series has a fractal structure and it may be chaotic. The dimension o f the attractor is between three and four and is noninteger. The next-largest integer value greater than the dimension o f the attractor—four—provides us w i t h knowledge o f the maximum number o f dynamical variables needed to model the behavior o f the system (Schuster 1988, 128; Nicolis and Prigogine 1989, 279). The number o f variables re­ quired to completely explain the dynamics o f the entire system is less than the number o f candidates that ran for the Democratic nomination. This insight provides another perspective on candidate viability i n presidential nomination contests. Unfortunately, no one has as yet devised a way to deduce which variables are required for modeling the dynamical behavior, though some imaginative ad hoc forecasting techniques have been applied w i t h limited success (Crutchfield and McNamara 1987; Casdagli 1989). These efforts have thus far failed to recover the functional form o f the original equations, even where it is k n o w n a p r i o r i .

Complexity in the Evolution of Public Opinion

183

The Correlation D i m e n s i o n and t h e Effects o f Noise Noise can be the bane o f statistical analysis. The survey used in this analysis, the 1984 Continuous Monitoring Study, is known to contain more sampling error than other National Election Studies (Allsop and Weisberg 1988). The original survey contains about 11 percent error. I n the construction o f the time series used in this analysis, the sampling error increases. The increase in the error is approximately equal to the square root o f the sampling interval m u l t i ­ plied by the original sampling error. The original time series used a sampling interval o f seventeen-day "weeks" in which the goal was to obtain two-thirds of the interviews in the first actual week and use the remaining time to complete the interview schedule (ICPSR 1984). The error in the collapsed time series I use here is slightly more than . 4 5 . This is a large amount o f error for any analysis and the question naturally arises as to whether or not the results in the prior section can be trusted to identify a deterministic component in the system dynamics. 7

In this section I report the results o f two Monte Carlo simulations on a time series w i t h k n o w n dimension, where progressively larger error has been added to the system dynamics in an attempt to destroy the resolving power o f the correlation integral. The results show that the method is robust. The Monte Carlo simulation uses the model constructed in equation 6 as the baseline, w i t h slight modifications. That model is a linear harmonic os­ cillator in t w o variables. It oscillates in the phase space, tracing a circle. A circle is a closed line and a line is a one-dimensional geometric figure. Hence, the dimension o f the experimental time series is k n o w n . Equation 6 was iterated 1,500 times to obtain a series o f long length and the output was scaled into the unit interval so as to be able to easily calculate the size o f the error component. Random normal error with k n o w n variance and mean zero was added to each observation. Ten computations o f the correlation integral from the X-Y phase space were conducted for each level o f added error. Summary results o f this experiment are shown in table 8.2. 8

The results i n the table show that this measure is robust in the presence o f a normal error component. The subset o f the points used to estimate the correlation dimension were selected using the method described by Chen (1988). The linear region o f the correlation integral is identified by plotting the log o f the radius, r, against the ratio formed by dividing the log o f the radius into the change i n adjacent points in the correlation integral, that is, plot [ A C ( r ) / A l o g ( r ) ] · log ( r ) . The linear region is identified as a plateau or flat segment in the modified plot (see Chen 1988; Abraham et al. 1986; M o o n 1987). I n the simulations run with 50 percent error added to each observa­ tion, the estimate for the correlation dimension is, on average, only 2.3 percent larger than the true dimension. This result is consistent with Schaffer 9

184

Chaos Theory in the Social Sciences TABLE 8.2. M o n t e Carlo S i m u l a t i o n of t h e Effect o f Error on Estimate of the Correlation Dimension Beta

Std. Error

Error Variance

Percent Error

.142 .222 .258 .112 .171 .247 .269

0.02 0.03 0.04 0.05 0.06 0.07 0.08

7% 14% 18% 25% 30% 42% 50%

.991 .9944 1.004 1.005 1.016 1.016 1.023

Source: Author's computations. Note: N = !,500 for all cases, all entries are means, statistics computed over ten runs.

(1988). The results i n the table show that the standard error o f the estimates increases as the amount o f error increases. The simulation above is based on a time series o f length 1,500. Next I report the results o f a second simulation where the error components remain constant and the sample sizes increase. This shows the effect o f noise in small samples. Two runs are reported. The first has 15 percent error and the second has 50 percent error. The series lengths are 200, 300, 400, . . . , 1,000, for each run. Table 8.3 contains the results. This simulation is based on one run per series. The results show that this technique allows one to recover the k n o w n dimension o f the linear harmonic oscillator quite accurately w i t h small sample sizes. These results are similar to

TABLE 8.3. M o n t e Carlo S i m u l a t i o n of t h e Effect o f Error o n E s t i m a t e of t h e C o r r e l a t i o n D i m e n s i o n w i t h F l u c t u a t i o n i n S a m p l e Sizes

N

15% Error

200 300 400 500 600 700 800 900 1,000

1.00 1.01 1.02 1.01 1.01 1.01 1.004 1.02 1.01

± ± ± ± ± ± ± ± ±

.011 .02 .022 .023 .023 .024 .025 .029 .027

Source: Author's computations.

50% Error .994 1.00 1.03 1.02 1.01 1.01 1.01 1.01 1.01

± ± ± ± ± ± ± ± ±

.038 .033 .024 .030 .027 .027 .024 .023 .020

Complexity in the Evolution of Public Opinion

185

a simulation reported by Schaffer (1988) for the logistic equation with varying sample sizes (see also Albano et al. 1987). I n both cases the standard errors o f the estimates are larger w i t h small sample sizes. The larger the sample, as always, the more accurate the estimate. The results o f this simulation give added weight to the results from the aggregate time series. The ability to recover the true dimension is not com­ pletely destroyed by having a short or noisy time series. Can the movement in opinion in the primary period be explained by a simpler model? I test this hypothesis i n the next section. A Test of an A l t e r n a t i v e M o d e l It can be argued that the results derived in the previous section are the result o f some unknown nonlinear yet regular process similar to the example used in equation 6, where the deterministic relation was based on a linear harmonic equation. A n even stronger argument might begin by positing that the ob­ served relationship is actually explained by a linear model. A test o f an alternative model is used to demonstrate that the empirical results detailed above are not due to noise that happens to follow an unusual cyclical pattern. When coupled w i t h the Monte Carlo results reported above, this provides further evidence for a low-dimensional deterministic process at work. Experimental work on physical (Ben-Mizrachi et al. 1984; Ott et al. 1985) and economic time series (Scheinkman and LeBaron 1989) containing an error component has revealed that the correlation dimension can be recov­ ered when noise is present in the data. Ott et al. describe how an attractor's volume changes with the addition o f an error component to a time series. I f e, represents the error component o f the series, the presence o f noise in a dynamical system "fattens up the attractor by an amount proportional to e" (1985, 65). This can be seen by referring back to figures 8.3 and 8.6. The equilibrium in figure 8.3 is "fattened" in figure 8.6 by the addition o f the random error component. This has the additional effect o f further separating nearby points from one another. I n the context o f calculating the correlation integral, this addi­ tional separation results in fewer points falling inside small radii, r , which means that larger and larger radii w i l l be required to discover the upper l i m i t required to cover the entire set o f points in the phase space. Noise has the effect o f increasing slightly the calculated dimension o f the attractor using the correlation integral (Abraham et al. 1986). The estimated slopes over the correlation integral w i l l , i f the system is dominated by noise, fail to saturate. Thus, in the presence o f noise, the point at which the slopes o f the various correlation integrals saturate w i l l be marginally greater than the saturation point in a system without noise.

186

Chaos Theory in the Social Sciences

However, i f the level o f noise fails to swamp the deterministic structure in the data, the correlation integral w i l l reach a saturation point and it is expected that this point, measuring the dimension o f the attractor, w i l l exceed the original dimension due to the increased proportion o f the series that is noise. To illustrate this, an A R 2 model was estimated for the Walter Mondale series. A n A R 2 model is used because the time path o f such a model can contain an oscillatory component (Brock 1986). The residuals were computed and used in a calculation o f a second correlation integral (Brock 1986, 182— 84). The A R 2 model explained approximately 45 percent o f the variance. I f the amount o f noise in the data is at all large and the deterministic component small, the expectation is that only noise remains to be used in calculating the correlation integral. 10

It is k n o w n that the level o f noise in the Rolling Cross Section is large compared to other NES/SRC surveys o f the electorate. Therefore, it is ex­ pected that saturation, i f it occurs at a l l , should require testing across many dimensions o f the phase space. The correlation dimensions were calculated for dimensions two through

Fig. 8.10. S l o p e s a t u r a t i o n i n l o w d i m e n s i o n f o r t h e M o n d a l e series based o n t h e residuals f r o m an AR2 m o d e l

Complexity in the Evolution of Public Opinion

187

ten, as before. Estimates for the slopes were derived as before, and are shown in figure 8.10. The results clearly show that the deterministic component in the data remains after use o f the A R 2 model. The correlation integral saturates at approximately 3.70, the mean slope over the last five dimensions, and the standard errors o f these estimates overlap the standard errors o f the slopes based on the original Mondale time series. The fact that the linear model fails to account for the deterministic component in the model coupled w i t h the repeated result o f saturation in the slopes over the scaling interval i n the correlation dimensions based on the residuals from the A R 2 model is strong evidence that the time series contains a nonlinear deterministic component. It remains to be shown that this series is chaotic. Chaos is confirmed by the presence o f a positive ( > 0 ) Lyapunov exponent. The next section confronts this problem. L y a p u n o v Exponents and t h e Presence of Chaotic D y n a m i c s The evidence provided to this point indicates that the aggregate time series constructed from the 1984 Rolling Cross Section may be chaotic. Confirma­ tion o f chaos requires a positive Lyapunov exponent. Lyapunov exponents measure the rate o f divergence o f nearby points during the evolution o f a dynamical system. This section discusses the technique o f estimating Lyapunov exponents from an experimental time series, shows that the tech­ nique is robust in the presence o f large amounts o f noise, and estimates a positive Lyapunov exponent for the constructed time series. Estimating Lyapunov Exponents from Experimental Time Series The Lyapunov exponent measures the rate o f orbital convergence or d i ­ vergence i n a dynamical system. Positive exponents indicate divergence and chaos while negative exponents indicate convergence or stability (Wolf 1985, 1986). I f the governing equations for a dynamical system are k n o w n , the exponents may be derived explicitly and easily; hence, when the equations are k n o w n , it is simple to determine i f the structure o f the dynamical motion is chaotic or not. When an observed time series is all that one has, the problem is much more difficult. The logistic equation is a useful vehicle for illustrating the properties o f a positive Lyapunov exponent. The logistic equation w i t h driving parameter r = 3.7 is known to be chaotic. I f we write the equation in the form n+\

X

r

' n X

'

0

n)

X

(11)

188

Chaos Theory in the Social Sciences

and investigate error propagation using linear stability analysis, it can be shown that nearby initial conditions diverge after only a few iterations. Fol­ l o w i n g W o l f (1986), define the error in measuring x to be dx„, so that n

x„

+i

+ dx

= f[x„ + dx ]

n+l

n

~ f[x ] n

+ dx„ •

f'lxj

= dx„- f [x ]

dx i n+

n

= dx

n

· r(l -

2x„).

I f one uses the initial uncertainty, which amounts to a small measurement error, one can write n-l

\dx„\ = \dx \ • in= 0 !/'(*,·)! 0

(13) \dx \ • 0

fi

|K1 -

2x . i

1= 0

For any value o f r, the uncertainty grows i n time i f the product o f the local stretching, | r ( l — 2x,)|, exceeds one. I f equation 13 is consistent w i t h dx„ = dx

0

• 2*n

(14)

the uncertainty grows exponentially fast. Here A is the Lyapunov exponent. From this equation, one can easily see how convergence and divergence occur. I f A < 0 then nearby initial conditions converge exponentially fast. This is the situation i n a stable l i m i t cycle w i t h a perturbation. I f A = 0 then the dynamics are neutrally stable and nearby initial conditions remain sepa­ rated by the identical distance throughout the evolution o f the dynamics. This occurs i n equilibrium cycles. I f A > 0 then nearby initial conditions diverge exponentially quickly and the system w i l l display chaotic motion. Equation 14 requires that n- 1

A =

lim 1 2

log |/'(*,)!· 2

(15)

Note that logarithms are reported in base t w o . I n practice, this means that the Lyapunov exponent, A, is reported in bits per iteration.

Complexity in the Evolution of Public Opinion

189

Generally it is practical to estimate only the largest Lyapunov exponent from an experimental time series (see Bryant et al. 1990 for an algorithm for simultaneously estimating all Lyapunov exponents from a time series). Since the Lyapunov exponents are simply generalized eigenvalues that give average rates o f expansion or contraction on an entire attractor, one begins the estima­ tion process by reconstructing the attractor as described above. After recon­ structing the attractor, one searches for points that are close to each other in the phase space, but separated by at least one orbit. The t w o trajectories, the fiducial and the test trajectories, are followed for a predetermined time inter­ val or until the distance between them exceeds some specific value (Schaffer 1988; W o l f 1986). Then one searches for another point close to the fiducial trajectory (a new test trajectory) and repeats the procedure. I n this way the search process continues until the entire time series is exhausted. Some o f the paired trajectories w i l l diverge and some w i l l converge toward each other. The average o f the stretchings and contractions is taken at the end o f the process. This procedure describes W o l f ' s Fixed Evolution T i m e algorithm and it has been used effectively to estimate the Lyapunov exponents for a number of dynamical systems, including the logistic equation, the Rossler equations, the Lorenz equations, and the Henon map (Wolf et al. 1985; W o l f 1986). As shown in equation 15 above, A is based theoretically on an infinitely long time series. N o experimental time series approaches this length, so a question naturally arises regarding the relationship between the length o f the experimental series and the accuracy o f the estimated Lyapunov exponent. In table 8.4 I show the result o f an experiment based on the Henon m a p , ' where the length o f the series from which the Lyapunov exponent is estimated decreases from 500 to 50 observations. Wolf et al. (1985) report that as few as 1

TABLE 8.4. C o m p u t a t i o n of t h e Largest L y a p u n o v E x p o n e n t f o r t h e Henon M a p N

500 450 400 350 300 250 200 150 100 50

Exponent

Time Delay

Evolution Time

.4661 .4021 .3622 .4092 .4004 .4460 .4737 .3692 .4511 .3852

1 1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2 2

Source: Author's computations.

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Chaos Theory in the Social Sciences

128 points were needed to calculate the largest (positive) exponent for this attractor to w i t h i n 5 percent o f its value. Hence, the data requirements appear to be less rigorous here than for the computation o f the correlation integral. Note that the estimates for the Lyapunov exponents are consistently greater than zero. I n this simulation I used one time series and drew increas­ ingly short samples from it in order to compute the Lyapunov exponent. W o l f et a l . (1985) report on similar experiments for a number o f other attractors: the Rossler attractor, the Lorenz attractor, hyperchaos (a 4 - D variant o f Rossler's system), the delay differential equation, and others. Their results indicate that the data requirements for difference equations are far shorter than for systems of differential equations. The time delay is the lag used to construct the phase portrait. Evolution time refers to the number o f iterations o f the fiducial and test trajectories at the end o f which the paired trajectories are measured. Noise i n the data poses no problem here. W h i l e adding noise to every observation fattens up the attractor, noise o f the standard normal variety moves points about in random, equiprobable directions. When computing the Lyapunov exponents, these variations tend to cancel out and the resulting estimates remain positive and far from zero. This is demonstrated i n another simulation by constructing a new time series based on the Henon map but in which a large random error component is added to each observation. The error amounts to 50 percent. The results are shown in table 8.5. These results indicate that error does not interfere greatly with the struc-

TABLE 8.5. C o m p u t a t i o n of t h e Largest Lyapunov Exponent for the Henon Map w i t h 50 Percent Error N

500 450 400 350 300 250 200 150 100 50

Exponent

Time Delay

.4057 .4944 .3760 .4236 .3762 .3555 .3280 .4640 .4354 .3371

1 1 1 1 1 1 1 1 1 1

3

Evolution Time

b

2 2 2 2 2 2 2 2 2 2

Source: Author's computations. a

The time delay is the lag used to construct the phase portrait.

b

Evolution time refers to the number of iterations of the fiducial

and test trajectories at the end of which the paired trajectories are measured.

Complexity in the Evolution of Public Opinion

191

ture o f the attractor and i f chaos is present it can be detected. Table 8.5 is based on an error component that exceeds the error component in the aggre­ gate time series based on the 1984 Rolling Cross Section. It seems reasonable to proceed on the assumption that i f a positive Lyapunov exponent exists in the time series o f candidate chances, it w i l l be detected. A p p l i c a t i o n t o the Constructed T i m e Series The correlation integral informs us that the dimension o f the attractor for the Mondale (and the Jackson) time series is between three and four. We make use of this information in the computation o f the Lyapunov exponent by comput­ ing the magnitude o f the exponent for a range o f evolution times and time delay reconstructions o f the attractor for dimensions three and four. I n this case a small amount o f information can dramatically reduce the amount o f w o r k necessary. The time series I have constructed is based on daily data. It is a mapping, like the Henon map used above. Hence the appropriate time delay for recon­ structing the attractor is one, just as for the Henon map or the logistic map. What must vary is the evolution time for each choice o f dimension. I f the time series is chaotic, we must expect that all o f the Lyapunov exponents computed w i t h the fixed evolution time algorithm w i l l be positive. The results are shown in table 8.6. A l l estimates exceed zero, hence this time series is chaotic ( M o o n 1987, 191-92; W o l f et a l . 1985; Ruelle 1989; Baker and Gollub 1 9 9 0 ) . This evidence also is an indication that the evolution o f public opinion during the 1984 presidential primary season is sensitively dependent on initial conditions. 12

TABLE 8.6. L y a p u n o v E x p o n e n t s f o r t h e M o n d a l e T i m e Series, D i m e n s i o n s T h r e e a n d Four Evolution Time 2 3 4 5 6 7 8 9 10

Lyapunov Exponent(3)

Lyapunov Exponent(4)

0.1739 0.1722 0.1837 0.1150 0.1096 0.0633 0.0549 0.0806 0.0683

0.1255 0.1350 0.1202 0.0871 0.0833 0.0826 0.0493 0.0445 0.0862

Source: Author's computations.

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Chaos Theory in the Social Sciences

In this result we find the reason that shifts in public opinion can occur in dramatic fashion. The Lyapunov exponent is interpreted as a loss in predictive power measured in bits per iteration. Large exponents, such as for the logistic equation w i t h driving parameter equal to four, where the Lyapunov exponent is one, mean that one bit is lost each iteration. I f one measures w i t h eight-bit accuracy, all predictive power is lost after eight iterations. By choosing the exponent for dimension four and a time evolution o f two, we see that one bit is lost after thirteen iterations (days). Extend this to six months and the result reported by James Johnson (in Moore 1986, 73), Mondale's pollster in 1984, is clear: the predictive power has been lost as people switch their candidate preference ordering. Effectively, such a long forecast may as well be a roll o f a probability die. Because the exponents in the Mondale series are relatively small, only a small amount o f information is lost per iteration. The effective result is to allow longer forecasts with greater accuracy and this effect is reflected in the polling done by the Udall campaign in 1976 (Schram 1976, 126, 132). We know that forecasts made close to an election date are more accurate than those made long before the election. The presence o f positive Lyapunov exponents explains w h y that result occurs. Conclusions

Radical opinion shifts can occur in a short period during primary election campaigns. This has been demonstrated with the Mondale time series con­ structed from the 1984 Rolling Cross Section NES survey. These shifts in opinion can be induced by a response to the competition between candidates for convention delegates. The concept o f momentum is important here. M o m e n t u m can be positive or negative (or zero) and the full range o f possibilities is shown in the time series used here. Mondale w o n in Iowa, but Hart did better than expected and reaped the electoral benefits in later elections. I n New Hampshire, Maine, and Vermont, Hart captured ever-greater shares o f the vote. Mondale began to recover on Super Tuesday and he then captured the higher percentage o f the vote in a sequence o f elections. This long-term oscillatory pattern persists throughout the campaign and the electoral results affect public opinion forma­ tion, perhaps by altering individual candidate preferences reported in the survey. The survey design unfortunately did not ask respondents whether they had altered their perceptions o f candidates' chances o f winning the nomina­ tion and each individual was only sampled once, so this hypothesis cannot be confirmed. However, it seems very likely that respondents in this survey reacted to electoral events in the same way that Mondale's deserters did—as a reaction to Hart's ability to w i n votes when he was not supposed to.

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What process is at work here? I show several linear models based on simple dynamic models o f the primary electoral process (see A l d r i c h 1980). The first t w o illustrate that simple models cannot capture the rich dynamics observed i n electoral outcomes during the 1984 primaries. The last was posed as an alternative to the hypothesis that the dynamics are governed by non­ linear equations. Each o f these failed to correctly characterize the observed dynamics. The analysis eliminates the linear model as an option when model­ ing the dynamics. A nonlinear process is at work here. The presence o f a positive Lyapunov exponent is convincing evidence for chaos in this and any other time series. As chaotic dynamics are associated w i t h deterministic processes, I am led to conclude that such a process is at work here. This should not be surprising. M a n y time series, most notably economic time series, are thought to consist o f t w o components: a deterministic component and a stochastic component. What this analysis shows is that the stochastic component may be much less influential than previously thought. The combination o f the several tools in this analysis simplifies the model-building process. We know how many dimensions are required to model the process. The correlation dimension gives the upper l i m i t on the number o f equations required to model the dynamics—four. The Lyapunov exponent confirms that the equations must be nonlinear—no linear process can produce chaotic behavior. The remaining problem—and it looms large— involves deducing a plausible mathematical model o f candidate competition that reproduces the observed dynamics to within the error envelope. Chaos has been confirmed by research in a number o f disciplines outside physics (see Holden 1986; B a i - L i n 1990; for a thorough review o f the physical evidence, see Jackson 1989, 1991). Biologists studying epidemics (Degn et al. 1987; Schaffer et al. 1986; Schaffer 1987; Schaffer et al. 1990) and population biology ( M a y 1974, 1983, 1987) have developed theoretical models that can display chaos and have discovered empirical evidence that supports theory. Economists show a keen interest i n chaos. Studies o f stock market dy­ namics are intriguing for the simple reason that i f the stock market can be shown to have a deterministic element, then the hope is that it can be con­ trolled or manipulated (see Brock 1986; Anderson et al. 1988; Boldrin 1988; Brock 1988; Chen 1988; Eckmann et al. 1988; Casti 1989; Scheinkman 1989; Scheinkman and LeBaron 1989; Brock et al. 1991). The article by Eckmann et a l . (1988) is particularly interesting, as it estimates a positive Lyapunov exponent for stock market data on gold futures prices. The magnitude o f their estimate is comparable to the one reported here for public opinion over time. Where should political scientists search for chaos? Whenever one has a time series the possibility exists that the underlying governing equations are nonlinear. For example, the political business cycle is a good candidate for a

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nonlinear dynamical process precisely because no one can find a cycle. I n ­ stead, what is found is a pattern o f aperiodicity very similar to what is found in economic, biologic, and physical systems exhibiting chaos. The political business cycle links the president's popularity w i t h a number o f dynamic variables, each o f which can be described by a separate equation (Hibbs 1987). Presidential performance, as measured by the number o f bills sent to and passed by Congress, provides a discrete time series that may be analyzed w i t h these methods (Hammond and Fraser 1984). Additional work can be done on the nomination process (see McBurnett 1990, 1991). We need not confine our efforts to presidential nominations; Congressional as well as lower-level races should provide fertile ground for research. Perhaps a systematic set o f govern­ ing equations may be discovered that operate across many electoral levels. The search for the underlying equations o f motion that govern the behav­ ior o f the nonlinear systems is an area o f intense research activity (Crutchfield and McNamara 1987; Casdagli 1989). Only recently have researchers discov­ ered how to recover the structure o f the governing equations from real data (Mees 1992). This topic is beyond the scope o f this paper but clearly it is the next step i n the analysis. We may need to adjust our perception o f the viability of the linear model as a research tool. Linearity is rare in nature; the norm is the nonlinear model (Devaney 1986; Casti 1988). The techniques described and used in this paper are simple to apply, require little subjective interpretation o f the results, and use no a priori k n o w l ­ edge o f the structure o f the dynamic to be investigated. The results from one analysis can be used to narrow the range o f approaches taken i n later analysis o f the same time series. The results are conclusive, whether the result is a confirmation o f chaos or not. I suspect that many models w i l l be replaced w i t h a deeper understanding o f the dynamical structure that exists in much o f political behavior that changes over time.

NOTES The present version of this chapter is much improved thanks to the insightful comments and critique of Courtney Brown, Thad Brown, Jim Kuklinski, Mike M a c K u e n , Steve Seitz, Seymour Sudman, and Dina Zinnes. Remaining errors should be attributed to the author. The data used in this paper were made available by the I C P S R . Neither the original collectors of the data nor the I C P S R are responsible for the interpretations made. 1. This is also a serious problem for campaign managers and candidates. The candidates rely on accurate professional polls that correctly identify the issues voters believe are important. These data aid the candidates in the formation of policy statements in their attempt to sway voters into their camps. When, unexpectedly,

one

Complexity in the Evolution of Public Opinion

195

candidate begins to do "better than expected" according to election results, voters in future elections may change their allegiance in favor of a candidate who is overtaking the front-runner. This occurred in the 1984 Democratic nomination contest, and Mondale's and Hart's pollsters documented Hart's surge in popularity after the Iowa caucuses as well as his later decline (Moore 1986, 6 1 - 9 6 ) . 2. Though the attractor is clearly visible using the tenth lag of the time series, it appears with lower order lags. Experiments indicate, with this simulation, that an identifiable limit cycle emerges using as few as three lagged observations. The tenth lag is used here so as to make the structure obvious. In practice, a sequence of graphs might be used in which the lag structure is advanced by one unit of time per graph. Thus the attractor, if it is present and of sufficiently low dimension, will quickly become recognizable. The elliptical shape of the attractor is partly a consequence of geometry and the aspect ratio of the graphic display. If one can determine the period of the cycle, use it (or the closest integer value) to set the lag, and plot K, against K ,

t p e r J o d

, the result will

be a circle. Elliptical distortion, if any, will be due to a problem in the aspect ratio in the output device. This also suggests a one-step method to identification of lowdimensional attractors if the governing equations are known a priori. 3. In the analyses to follow, the dynamics of these time series are probed in the phase space. To see how the phase space provides information on the dynamics of these particular series, I refer the reader to chapter two of this volume, where I discuss the spectrum for the time series, and the accompanying figures contained therein (figs. 2.10, 2.11, and 2.12). 4. According to Schuster (1988), "all strange attractors that have been found up to now in dissipative systems have fractal HausdorfT dimensions." A dissipative system is one in which the volume of the attractor shrinks, but not necessarily uniformly in all directions. Further, "the property which makes the attractor strange is the sensitive dependence on initial conditions," which means that the dynamics have a positive Lyapunov exponent and this is a quantitative measure of the stretching and shrinking of the volume of the attractor as the motion evolves on the attractor. In short, Schuster's claim is that no strange attractors exist that do not also display chaotic dynamics by virtue of the fact that fractal dimension and positive Lyapunov exponent are inextricably intertwined. However, a pair of mathematical counterexamples have been constructed of strange attractors that are not chaotic. These examples are continuous time models (Grebogi et al. 1984). No counterexamples are found in discrete time. 5. It does not matter which of the series is used; the results are the same. This entire analysis was first conducted with the Jackson series and then replicated with the Mondale series, on the suggestion of a reviewer. The results are the same. 6. The correlation integral is only defined in the limit. A s no experimental time series approaches this length, it is customary to assume that the available data are sufficient to support the necessary computation. 1 make this assumption here. Some guidance as to the necessary length is available (see Abraham et al. 1986). 7. I thank Seymour Sudman for his help in deriving the nature of the increase in the error in my construction of the time series. 8. The use of both variables in the computation of the correlation integral speeds up

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the process considerably by reducing to one (from six or more) the number of dimensions over which the calculation of the correlation integral takes place. This technique (see Ben Mizrachi et al. 1984) is exactly equivalent to the procedure described in this paper and the dimension estimates are the same. For example, using the technique described in this paper, I estimated the dimension of the linear harmonic oscillator using the time-delay method in a series of two hundred data points with no error and the result was a set of six slope estimates of 1.00 with error less than 8.9e - 9 for each. With 20 percent error added to the original series, the estimate for the correlation integral converged to 1.008

± 0.031 at the reconstructed five-dimensional phase

space. Because the multidimensional correlation integral is easier to interpret in the context of the simulation, its use and the results are reported here. There is no difference between the two measures. The results reported here are available from the author. 9. A l l techniques presently available that allow a determination of the linear region of the correlation dimension integral involve some intuition on the part of the researcher. The main question, and it has yet to be resolved, revolves about identification of the linear region. This technique tends to maximize the fit of the regression as measured by

2

R.

10. Solution of the resulting difference equation reveals no oscillatory motion. This does not detract from the results that follow. 11. The Henon map is given by the coupled difference equations

X

n+I

Y

n+l

= 1 - aX

2

+ Y

n

= bX , m

where a = 1.4 and b — 0.3, and X

0

= Y = 0.5. This map is the two-equation analog 0

to the logistic map. 12. A l l exponents remain positive when this table is extended to include evolution times up to thirty.

Part 3 Chaos Theory and Economics

CHAPTER 9

Chaos Theory and Rationality in Economics J. Barkley Rosser, Jr.

A central assumption in economic theory is that o f rational behavior by economic agents, although this has been widely criticized by many noneconomists. Since the work o f M u t h (1961), this assumption has increasingly taken the form o f rational expectations, that agents over time on average accurately predict the future. This somewhat simplistic formulation o f the assumption depends on agents knowing the underlying "true" model o f reality and fully utilizing information that, in turn, is assumed to arrive in a random pattern that reflects a normal distribution. Agents may make errors from time to time, but these errors tend to cancel each other out over t i m e . 1

M u t h (1961) originally applied the theory to explain behavior in agri­ cultural markets. Its use has since spread throughout economics at both the microeconomic and macroeconomic levels, as well as into financial theory. A m o n g the most controversial applications have been in macroeconomics (Lucas and Sargent 1981), where models using the assumption have been used to support a "New Classical" view that systematic stabilization policies by governments w i l l be ineffective and can only cause inflation in the long run. The essential argument is that economic agents w i l l rationally forecast the effects o f government policies and w i l l act in ways that nullify those effects. The assumption has acquired the status o f a Lakatosian hard core axiom of the economic research program (Rosser 1993), despite some empirical findings that it does not hold for a variety o f economic agents (Lovell 1986). The argument for this hard core axiom view has been articulated by Sargent (1982, 382), who has asserted that it is a hypothesis not amenable to empirical testing because o f "the logical structure o f rational expectations as a modeling strategy, the questions that it invites researchers to face, and the standards that it imposes for acceptable answers to those questions." A significant complication for the use o f this assumption has arisen in the form o f the increasing awareness o f the widespread reality o f nonlinearity i n many dynamical economic systems. Such nonlinearity can lead to chaotic dynamics in economic systems. A number o f theoretical models have been 2

199

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Chaos Theory in the Social Sciences

developed in different areas o f economics that assume rational expectations but can generate chaotic dynamics for certain parameter values. This is ironic because in a situation o f chaotic dynamics, the condition o f sensitive dependence on initial conditions holds. This implies that even a minute error in estimation can lead to serious errors in long-run forecasting. This raises very serious doubts about the realistic applicability o f the rational expectations assumption under such circumstances. It is this contradictory situation that is the central theme o f this chapter. Chaotic M i c r o e c o n o m i c M o d e l s w i t h Rational Expectations Chaotic Preference Cycles Generally, in microeconomic analysis it is assumed that individual prefer­ ences are constant and that a person's behavior reflects their responses to changes in relative prices or income. However, it has been recognized since at least Pareto (1909) that this is a highly unrealistic assumption, and some economists have occasionally attempted to develop models that dynamicize preferences. Such an effort that is both consistent with rational expectations and also allows for chaotic patterns o f individual consumption behavior is due to Benhabib and Day (1981). They propose that a person's preferences reflect past consumption pat­ terns and that such relationships may generate cyclical patterns o f preferences and consumption as between two goods. Thus people may alternate between ski and beach holidays. Letting the goods be x and y , w i t h m the person's money, they assume that the person's utility from consuming the goods can be represented by the standard Cobb-Douglas utility function,

U(x,y) = xayx— a

(i)

They hypothesize t w o different dynamic processes that both operate through changing the value o f a over time. The first one, which is analogous to the Lorenz (1963) climate model, is given by (2) w h i c h , in turn, implies a dynamic demand curve x

t + i

= oanx,(jn — x,).

(3) 2

The control parameter for this process is am . A t values slightly greater than one there is a stationary demand curve. As its value increases, period-

Chaos Theory and Rationality in Economics

201

doubling bifurcations emerge w i t h the resulting emergence o f cyclical con­ sumption behavior. Chaotic dynamics appear for values greater than 3.57. The second process is given by (4) which implies a dynamic demand function

x

t+l

=

ffu e f

a<1

(5)

*>>

I f m = 1, this is analogous to a population dynamics model due to M a y and Oster (1976) for which a is the control parameter. Chaos appears when it approximately equals 2.6924. Both o f these processes imply that as a declines, higher values o f m are required for chaotic dynamics to occur. Given that a is the experience dependent parameter and m is income, Benhabib and Day (1981, 463) conclude that "the t w o models then characterize experience dependent demand as con­ verging to a stable long run pattern for relatively low incomes, but exhibit increasing instability and eventually become completely erratic as income grows reflecting the whimsical, seemingly arbitrary behavior o f the compla­ cent, or the very r i c h ! " Interpersonal Utility Effects and Chaotic Dynamics Another realistic but infrequently allowed aspect o f individual preferences in microeconomic analysis is the sociological observation that a person's prefer­ ences may depend on the behavior o f others, an observation originally made by Veblen (1899) and later studied by Leibenstein (1950). I n particular the latter distinguished between bandwagon effects, wherein one wants to buy something when others are buying i t , and snob (or reverse bandwagon) effects wherein one does not want to buy something when others are buying i t . Granovetter and Soong (1986) postulate that a person could exhibit both effects w i t h respect to a particular good for different levels o f market satura­ tion. Thus someone might be inclined to buy a push-button phone when 20 percent o f their friends do so, for bandwagon reasons, but may incline toward an old rotary phone when 90 percent o f their friends have push-button ones, for snob reasons. Such a situation implies a nonlinear dynamic demand pro­ cess and i f it is sufficiently nonlinear the dynamics can be chaotic. Figure 9 . 1 , taken from Granovetter and Soong 1986, shows the demand curve in such a situation. As price, P, declines below />,, period-doubling bifurcations appear. P* is the threshold o f chaotic dynamics. 3

Iannacone (1989) has criticized this model for having consumer prefer­ ences depend only on the behavior o f others during the most recent period. He

202

Chaos Theory in the Social Sciences

Quantity Fig. 9 . 1 . C h a o t i c d y n a m i c d e m a n d process. (From S o o n g 1986. R e p r i n t e d w i t h permission.)

Granovetter a n d

shows that an adaptive expectations strategy drawn from Heiner 1989 wherein consumers respond to a weighted average o f past behavior by others reduces the likelihood o f chaotic dynamics. This is an argument we shall consider in more detail later. Chaotic C o b w e b s Originally conceived by Cheysson (1887), cobwebs are oscillatory and possi­ bly explosive dynamics in markets that can arise when there are lags in the production process, the classic example being i n agriculture. Ironically, the original motivation behind Muth's (1961) development o f the rational expec­ tations assumption was to rule out such outcomes for agricultural markets. He argued that farmers would rationally forecast the true long-run equilibrium and w o u l d not foolishly indulge in alternating increases and decreases in output in response to short-run price fluctuations.

Chaos Theory and Rationality in Economics

203

However, it has since been shown that oscillatory and even chaotic dynamics can emerge because o f production lags i f supply or demand curves are nonlinear and nonmonotonic, even w i t h rational expectations. Artstein (1983) posited partially downward-sloping supply curves for agricultural commodities i f farmers (rationally) forecast that very low prices in one year w i l l trigger government price supports in the next year. Artstein shows that such a case can generate a three-period cycle, thus implying chaotic dy­ namics. Jensen and Urban (1984) derived similar results for models w i t h backward-bending supply curves (not uncommon in labor markets and fish­ eries) and double-valued demand curves (much rarer, perhaps Irish potatoes in the mid-nineteenth century). Chiarella (1988) showed the possibility o f chaotic cobwebs even with normally shaped supply and demand curves. However, his results are not consistent w i t h rational expectations because they depend on the existence o f a lag in expectations formation as well as a lag in production, realistic as that may b e . This model o f Chiarella's underlies the empirical studies o f the pork and m i l k cycles by Chavas and Holt (1991, 1993). 4

Chaotic M a c r o e c o n o m i c M o d e l s w i t h Rational Expectations Overlapping Generations Models The first macroeconomic models to exhibit chaotic dynamics that are consis­ tent w i t h rational expectations (Benhabib and Day 1980, 1982), labeled "Strong New Keynesian" by Rosser (1990), depend on the use o f the over­ lapping generations concept developed by Allais (1947) and Samuelson (1958). In each period o f time let there be two generations alive, w i t h the young possessing endowments o f w and the old possessing endowments o f w and the o l d possessing fiat money at the beginning o f time equal to M. Let p(t), c (t), and c {t) be, respectively, prices, consumption by the young, and con­ sumption by the old in time t. 5

a

y

a

The young seek to maximize utility max U[c (t), y

(6)

c (t + 1)], a

subject to their intertemporal budget constraint p(t)c (t) y

+ p(t + \)c {t a

+ l)=p(t+

l)w . 0

(7)

These generate an intergenerational offer curve, O, by the young as p{t) and p(t + 1) vary. The old in time 1 face the constraint

204

Chaos Theory in the Social Sciences

p(l)c (l)

= p(l)

0

+ M.

Wo

(8)

Given a Ricardian intertemporal production possibilities frontier, R = [(c , c ) : c y

a

y

+ c

a

= w

y

+ wj,

(9)

a set o f perfect foresight (hence rational expectations) equilibria are given by a sequence o f prices and consumption levels satisfying equations 6, 7, and 8. The pattern generated by this sequence w i l l depend on the "humpiness" of the intergenerational offer curve, O. Sufficient humpiness can lead not merely to endogenous cycles, but to chaotic cycles such as that shown in figure 9.2. Benhabib and Day (1982) show that this humpiness w i l l depend positively on the variability o f the constrained marginal rate o f substitution between individuals' present consumption and future consumption. The pre­ cise sufficiency conditions for chaos also depend on the endowments o f the young and on the rate o f population growth. They also show that these chaotic trajectories are efficient, that is, Pareto-optimal. 6

Cy Fig. 9.2. Chaotic i n t e r g e n e r a t i o n a l offer curve

Chaos Theory and Rationality in Economics

205

A somewhat different overlapping generations perfect foresight model that can generate endogenous chaotic dynamics is due to Grandmont (1985). His model focuses on interest rate changes initiating conflicts between inter­ temporal wealth effects and intertemporal substitution effects. I f older agents have a marginal propensity to consume leisure sufficiently greater than that o f young agents, then chaotic dynamics can emerge. More specifically, let a be real wealth and V the indirect utility function o f either generation. Then, the Arrow-Pratt relative degree o f risk aversion, which measures the curvature o f V, is R(a)

= -V"(a)a/V'(a).

(10) 7

Chaotic cycles w i l l occur i f R (a ) > R (a ). Figure 9.3 illustrates such a case derived from Grandmont's (1985) model. The axes are real money balances in time t, fx, on the horizontal and the perfect foresight dynamics o f real money balances in time t + 1, x(fi), on 0

0

y

y

Fig. 9.3. Perfect f o r e s i g h t m o d e l i n o v e r l a p p i n g g e n e r a t i o n s e n d o g e n o u s c h a o t i c d y n a m i c s . (From G r a n d m o n t 1985.)

with

206

Chaos Theory in the Social Sciences

the vertical. The humpiness o f x(ix) increases w i t h R (a ). Grandmont argues that Keynesian-style governmental policies o f proportional intergenerational money transfers can pin down expectations about the interest rate and e l i m i ­ nate any cyclical outcomes. However, this depends on both the government being able to make accurate forecasts under conditions o f chaotic dynamics w i t h the attendant problem o f sensitive dependence on initial conditions and people having complete confidence in government policies. 0

0

Brock (1988a) has dismissed the realism o f such assumptions as amount­ ing to a "nirvana fallacy." Indeed, D w y e r (1992) has developed a model wherein stabilization policy generates chaotic dynamics in a situation when otherwise such would not occur. This echoes monetarist critiques o f discre­ tionary Keynesian policies as generating business cycles when otherwise they w o u l d not occur. Such an argument has been reinforced by DeCoster and M i t c h e l l (1992), w h o show that a chaotic policy w i l l be magnified into even more chaotic dynamics, even when the economy is basically nonchaotic and reflects linear rational expectations. Discounting and Chaos in Infinite Horizon Models Strong advocates o f the rational expectations approach object to the kinds o f models presented in the previous section on the grounds that they either do not involve optimization over an infinite time horizon (Benhabib and Day 1982; Grandmont 1985) or involve some kind o f incompleteness o f markets (Woodford 1989). Thus it is argued that they do not represent truly complete general equilibrium solutions. Defenders o f the above approaches may cite their apparently greater realism, but the critics note that agents may well have time horizons beyond personal lifetimes, either for family bequest motives (Barro 1974) or because important agents may be transpersonal entities such as corporations or governments w i t h presumably very long time horizons. Thus the question arises as to whether or not chaotic dynamics can arise in such models w i t h completely general infinite horizon equilibria. The answer is yes. One way that chaos can appear in such models is i f there is a failure to converge to a steady-state equilibrium, even though one exists. For optimal, infinite horizon, multisector models, Sutherland (1970) initially showed that a low discount factor (high real interest rate) could lead to such nonconvergence. Boldrin and Montrucchio (1986) established that in such situations "every possible" kind o f behavior, including chaotic dynamics, is possible. B o l d r i n and Woodford (1990) noted that in the simplest case the discount rates involved imply 10,000 percent real interest rates per period, clearly ridiculous. However, alterations to the model can allow for the phenomenon w i t h i n more realistic ranges (Neuman, O ' B r i e n , Hoag, and K i m 1988). I n particular, Boldrin (1989) has shown that increasing the number o f substi-

Chaos Theory and Rationality in Economics

207

tutable factors and the ranges o f their respective substitutabilities can allow for chaotic dynamics within realistic discount ranges. Deneckere and Pelikan (1986) have considered further factors beyond the discounting issue that can also bring about chaotic dynamics in one-good, two-sector, optimal infinite horizon, complete markets models. These include the presence o f strongly decreasing returns to scale and the possibility that the relative capital intensities o f the two sectors reverse order as the level o f production rises. This latter condition is key to the arguments o f Boldrin (1989) noted above. Deneckere and Pelikan take the strong position that i f optimal behavior is truly chaotic it w i l l never be observed. This is because the learning processes necessary for perfect foresight (and the use o f rational expectations more generally) can only operate in relatively regular environments. Sensitive de­ pendence on initial conditions w i l l destroy this possibility and w i l l also render impossible any Grandmont-type optimal governmental policy intervention as well. Nonseparable Utility and Chaos in Infinite Horizon Models Yet another case i n which chaotic dynamics can arise i n optimal infinite horizon models w i t h complete markets is when real money balances enter directly into agents' utility functions in a manner nonseparable from consump­ tion goods. Matsuyama (1991) has examined this case. Let p be prices, c be consumption, and m be real money balances, which grow at a rate u,. Let B be the discount factor and let 8 = /JL/(B - 1) > 0. Let 8 be the elasticity o f intertemporal substitution o f real balances, w i t h 17 being a parameter 3: —1 such that 8 = (17 4- 2 ) . Then Matsuyama assumes a utility function o f the form _ 1

uic, ) m

=

\; ^\ rr [

m

1(1 +1)1

l In g(c) + In m,

(i i) i f 17 = - 1 ,

w i t h g > 0 and g' > 0. This implies an optimal time sequence o f prices given by (p, )n +l

= (1 +

- p,).

(12)

Matsuyama proceeds to demonstrate that the nature o f these sequences w i l l vary according to combinations o f 8 and 17. A n important variable deter­ mining the nature o f these sequences is A(TJ) = TJ-^I

+ i7)

1+1

i -

1.

(13)

208

Chaos Theory in the Social Sciences

In particular, he shows that for any TJ > 0, there exists a value A*(-n) satisfy­ ing 2 r] < A*(T/) < A(TJ) such that a period o f cycle three and hence chaotic dynamics exists i f 8 > A*(TJ). These results and Matsuyama's more general categorization o f possible outcomes are summarized in figure 9.4. Matsuyama argues that this model suggests the possibility that even for comparative statics analysis the set o f possible equilibrium prices may be "topologically complex." This result is probably even more destructive o f standard views o f economic theory than is the associated undermining o f the usefulness o f the rational expectations hypothesis. Empirical Evidence M u c h empirical evidence has been gathered that is suggestive o f the possible existence o f deterministic chaos in various economic time series. However, there has yet to be a definitive demonstration o f such existence in any case. W h y is this? Brock and Sayers (1988) have presented a methodology for searching empirically for chaos. The first step is to test for dimensionality and associ-

Fig. 9.4. M a t s u y a m a ' s prices

t o p o l o g i c a l c o m p l e x i t y i n possible e q u i l i b r i u m

Chaos Theory and Rationality in Economics

209

ated nonlinearity. Brock, Dechert, and Scheinkman (1987) have developed a test called the B D S statistic that is based upon the use o f the Grassberger and Procaccia (1983) correlation dimension. This test has been used on a variety o f economic time series and has convincingly shown the presence o f serious nonlinearities in many o f them. Nonlinear dependence has been found in monetary aggregates (Barnettand Chen 1987), U . S . stockreturns (Scheinkman and LeBaron 1989), gold and silver markets (Frank and Stengos 1989), U . S . Treasury b i l l rates (Brock 1988b), work stoppages (Sayers 1988), employment, unemployment, and pig iron production (Brock and Sayers 1988; Sayers 1990), industrial production (Brock and Sayers 1988; Ashley and Patterson 1989), Japanese G N P (Frank, Gencay, and Stengos 1988), foreign exchange rates (Hsieh 1989; Papell and Sayers 1989), pork pro­ duction (Chavas and Holt 1991), and m i l k production (Chavas and Holt 1993). 8

The second step is to estimate Lyapunov exponents to determine i f the greatest exponent has a positive real part, thus indicating the presence o f sensitive dependence on initial conditions. Brock and Sayers (1988) argue that although algorithms have been developed for estimating Lyapunov expo­ nents (Wolf et al. 1985; Eckmann et al. 1986; McCaffrey et ai. 1992), no statistical inference theory is k n o w n , although this is disputed by Barnett and H i n i c h (1993). Nevertheless, tentative indications o f the existence o f positive real parts o f Lyapunov exponents have been found by Barnett and Chen (1987) for monetary aggregates, by Brock and Sayers (1988) for employment data, and by Eckmann et al. (1988) for stock returns. W h y then the continuing empirical skepticism? The argument has been made that i f a series is really driven by determin­ istic chaos then it should be possible to make short-period forecasts that beat a random w a l k . A l s o , the dimensionality should be maintained, even as the intervals o f measurement are shortened. Efforts to make such tests on some o f the most promising series have not been favorable. Brock and Sayers (1988) could not support the chaos result for any o f the series they studied. Efforts to forecast using nearest neighbor techniques failed to beat a random walk for foreign exchange rates (Diebold and Nason 1990) and stock returns (Hsieh 1991). Mayfield and Mizrach (1992) found changing the time interval changed the dimension estimates for stock returns. However, none o f these tests can be viewed as definitive and the question remains o p e n . 9

10

Indeed, the discussion o f the estimation issues involved has been much longer and more complicated than we have indicated here. Let us note one argument that may be o f relevance, namely that o f aggregation. Almost all these tests have been performed on fairly highly aggregated data sets. How­ ever, there is reason to believe that even i f chaos exists at the specific or micro-level, it may disappear as aggregation to the macro occurs. Sugihara,

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Grenfell, and M a y (1990) found evidence o f chaotic dynamics in city-by-city measles epidemics. However, upon aggregating to the national level the dy­ namics reduced to a noisy two-year cycle. A similar argument has been made w i t h respect to the dynamics o f spatial economic systems: that the more lo­ cal they are the more likely they are to exhibit unstable cyclical dynamics (Dendrinos w i t h M u l a l l y 1985). Thus it may well be that the future search for economic chaos should focus more on highly disaggregated microeconomic series. The Problem for Rational Expectations Even i f full-blown chaos has not been confirmed for any major macroeconomic time series, many o f the problematic aspects o f chaotic dynamics still adhere to series that are highly nonlinear while not fully chaotic. We return to our central contradiction. O n the one hand, the possible coincidence of chaos and rational expectations could mean the possibility o f improved forecasting. O n the other hand, the problem o f the sensitive dependence on initial conditions means that introducing any stochasticity at all into such a system probably ruins any hope o f improving forecasting ability and renders meaningless the very idea o f rational expectations. DeCoster and Mitchell (1992) have suggested a possible response to this problem in the form o f running many simulations and using the expected value, although this is very time-consuming and expensive. 11

This problem relates to an o l d issue that has plagued rational expecta­ tions models almost from their inception, namely, the possibility o f inde­ terminacy due to an infinity o f possible equilibria (Rosser 1991b). Such a possibility was first shown by Gale (1973) and numerous efforts by New Classical economists have been made since to demonstrate the primacy o f the golden rule steady state that the economy is supposed to tend to in such models. Generally the non-golden rule steady state equilibria have the qual­ ity o f being self-fulfilling prophecies, also known as "sunspot equilibria" (Shell 1977). A recent line o f arguments that has sought to rule out such extra equi­ libria has been to argue that they are not dynamically stable. I f one assumes certain kinds o f learning processes then the system w i l l converge on the golden rule steady state. Evans (1985) first presented such arguments and Marcet and Sargent (1989) have further developed them by arguing for the socalled least squares learning process that appears to have this property. But other learning processes lead to different results. Grandmont (1985) has shown learning processes that converge to cycles. Woodford (1990) has shown learning that converges to self-fulfilling prophetic equilibria. Benassy and Blad (1989) argue that learning w i l l not converge on rational expectations

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211

unless one starts w i t h rational expectations. Evans and Honkapohja (1992) have found expectationally stable bubble equilibria, and even in his initial study Evans (1985) noted that for certain parameter values o f his learning process the result is not convergence but chaotic dynamics. Is there a way out o f this indeterminacy? One such way is to abandon rational expectations in general and to adopt a line o f reasoning mentioned above due to Heiner (1989). He argues that when forecasting errors occur, as we would expect i n a chaotic environment impinged upon by stochastic shocks, agents w i l l use an adaptive expectations strategy by stabilizing their behavior i n the face o f heightened uncertainty. He recognizes that for conver­ gence to the optimal trajectory to occur strong assumptions are necessary. But under the right conditions such convergence w i l l proceed slowly at first, then accelerate, and then slow down again as the goal is approached. Rational expectations w i l l only hold in the equilibrium case. Otherwise, simplicity and caution w i l l create order out o f chaos. The ultimate irony here is that the deeper concept o f rationality may be preserved in such circumstances by abandoning any hope o f following rational expectations. 12

Conclusion So what is left o f the edifice o f standard neoclassical theory after the possi­ bility o f nonlinear and chaotic dynamics is recognized? As noted above, M i r o w s k i (1990) has argued that in fact chaos theory actually reinforces standard theory. However, this is certainly a minority view. As we have seen, sensitive dependence on initial conditions is profoundly disruptive o f the ability to develop rational expectations, especially when any stochastic shocks are present. This is so, or perhaps especially so, when rational expectations equilibria are chaotic, as argued above by Deneckere and Pelikan (1986). Nevertheless, it probably remains the case that standard neoclassical theory remains useful for many situations despite the difficulties raised by nonlinear and chaotic dynamics. Let us conclude this discussion with a quota­ tion by Smale, one o f the initial developers o f chaos theory in mathematics, as well as a student o f the structural stability o f Walrasian general equilibria: How d i d Relativity Theory respect classical mechanics? For one thing Einstein worked from a very deep understanding o f the Newtonian the­ ory. Another point to remember is that while Relativity Theory lies in contradiction to Newtonian theory, even after Einstein, classical me­ chanics remains central to physics. I can well imagine that a revolution i n economic theory could take place over the question o f dynamics, which would both restructure the foundations o f Walras and leave the classical theory playing a central role. (Smale 1977, 95)

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NOTES Figure 9.3 is reprinted from J . - M . Grandmont, "On Endogenous Competitive Business C y c l e s , " Econometrica 53 (1985): 9 9 5 - 1 0 4 5 , with permission of The Econometric Society, Department of Economics, Northwestern University, Evanston, I L . 1. The formal definition of rational expectations is that the subjective probability distribution regarding future reality inside the heads of economic actors coincides with the objective probability distribution operating outside their heads in reality. 2. Reviews of applications of chaos theory in economics are in Kelsey 1988; Baumol and Benhabib 1989; Lorenz 1989; Boldrin and Woodford 1990; Rosser 1991a; Brock, Hsieh, and Lebaron 1991; and Bullard and Butler 1993. For formal definitions of chaotic dynamics, see chapters 2, 3, and 4 in this volume. 3. Such a model is potentially applicable to bandwagon phenomena in asset markets such as speculative bubbles. Models of speculative bubbles coinciding with chaotic dynamics have been developed by Day and Hwang (1990); De Grauwe and Vansanten (1990); Ahmed, Ayogu, and Rosser (1990); and De Grauwe, Dewachter, and Embrechts (1993). However, none of these models allow for rational expectations bubbles because of the boundedness conditions associated with chaotic dynamics. 4. Yet another area of microeconomics where chaotic dynamics models have been developed that are inconsistent with rational expectations is in oligopoly theory, especially the case of duopoly dynamics. The first explicit application of chaos theory in economics (Rand 1978) was in such a model. 5. Chaotic macroeconomic models not consistent with rational expectations have been developed depending on nonlinear population dynamics (Stutzer 1980), nonlinear productivity of capital in a Solow-type growth model (Day 1982), high exploitation of labor in the convergence to a Ricardian steady state (Bhadurk and Harris 1987), Lotka-Volterra employment dynamics with markup wages (Pohjola 1981), nonlinear multiplier-accelerator relations (Gabisch 1984), nonlinear investment functions (Dana and Malgrange 1984), Schumpeterian technological long waves (Goodwin 1986), and long-wave socialist investment cycles (Rosser and Rosser 1994), among other approaches. 6. Such outcomes in some economics chaos models have led Mirowski (1990) to criticize them as being sophisticated efforts to preserve neoclassical economics in the face of actual extreme stochastic uncertainty. 7. Woodford (1989) has shown that similar results can be obtained, even in models with infinitely lived agents, if they face short-run borrowing constraints. 8. This nonlinear dependence may take the form of nonlinear dependence in the variance, or conditional heteroskedasticity, rather than in the mean of the series. 9. A very real possibility is a combination of chaos and stochastic shocks. However such a combination will exactly resemble purely random processes because of the sensitive dependence on initial conditions. Every shock will push the system onto a different path. 10. E v e n though chaos has not been confirmed, nonlinearity most certainly has been. A popular approach to analyzing it in stock and foreign exchange markets has been to examine variations in volatility over time following a suggestion of Mandelbrot (1963). Engle (1982) developed the autoregressive conditional heteroskedasticity

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213

( A R C H ) technique to estimate such effects. This has been expanded to generalized A R C H ( G A R C H ) by Bollersley (1986), A R C H in mean ( A R C H - M ) by Engle, Lillien, and Robins (1987), and nonlinear A R C H ( N A R C H ) by Higgins and Bera (1992). 11.

This is not necessarily the case in the long run if there is only one equilibrium.

Intermediate run forecasting may be ruined, but the system may still oscillate around some expected value in the long run, which is consistent with the idea of forecasting errors being random. 12.

Lavoie (1989) has used neo-Austrian theory to argue that out-of-equilibrium

chaotic dynamics will lead to self-organizing order in a manner similar to that described in the nonequilibrium phase transition theory of Prigogine and Stengers (1984).

CHAPTER

10

Long Waves 1790-1990: Intermittency, Chaos, and Control Brian J. L . Berry and Heja Kim

There are t w o contending views o f economic dynamics. Insights into their validity may be provided via concepts o f catastrophe and chaos. The d o m i ­ nant paradigm is that optimizing behavior produces an economy that is inher­ ently equilibrating, tending toward steady-state growth in the absence o f random shocks. Such shocks, in the Slutsky-Frisch-Tinbergen view, are trans­ formed into cyclical oscillations through the filtering properties o f the econ­ omy's growth-propagation mechanisms (Boldrin 1990, 126-27). A con­ tending paradigm associated with Hicks, Kaldor, and Goodwin is that endogenously driven growth is inherently unstable, with deviations that move between upper and lower limits to investment and consumption (Boldrin 1990, 126-27). What we w i l l show in this paper is that long-wave rhythms o f prices and growth (Berry 1991) accompanied by shorter-term oscillations display unpredictable dynamic system order, or deterministic chaos. Phase portraits reveal both intermittent crises at long-wave peaks and quasiperiodic l i m i t cycles that encompass attractor basins between the peaks (for a discus­ sion o f l i m i t cycles see Berge, Pomeau, and Vidal 1984). It is the presence o f these properties that supports the Hicksian view before 1919. We also show that institutional changes since the Great Depression have reduced the limits w i t h i n which prices and growth oscillate, significantly altering long-wave and short-run dynamics by forcing attractor basins toward stabler equilibria. Specifically, we document the following:

1. Slow-moving long-wave rhythms o f prices are the outcome o f the half-century-long logistics by which new techno-economic paradigms move from innovation to market saturation (Berry, K i m , and K i m 1993). The usual definition o f chaos results in instabilities at the top of such logistics (Modis and Debecker 1992), and indeed they leave in their wake the inflationary spirals that define long-wave peaks. 2. Faster-moving annual

fluctuations oscillate around the long-run 215

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Chaos Theory in the Social Sciences

rhythms o f prices, urged toward acceleration on the upwave and deceleration on the down wave. The tension between low-frequency rhythms and high-frequency os­ cillations produces chaotic l i m i t cycles in the periods between long­ wave crises. Instability at the long-wave price peaks gives rise to bifurcations. Upwave l i m i t cycles may be different from those observed during a d o w n wave. Similar bifurcations and l i m i t cycles characterize quarter-centurylong swings and annual oscillations in the rate o f economic growth. Since World War I I , both institutional changes and policies designed to fight inflation have dampened the higher-frequency oscillations o f prices, leaving the long-wave rhythms as the dominant fluctuations: the first return map o f prices has been reduced to the long-wave attractor. But contrariwise, stabilization programs aimed at fluctuations in economic growth have had the opposite effect. It is the long swings o f growth that have been dampened, leaving behind the higherfrequency oscillations o f the business cycle: the first return map o f rates o f economic growth has been simplified to the characteristic cobweb o f a business cycle. The usual notes o f caution are i n order. We have at most two hundred data points: It is not possible to prove statistically "from a data set o f the lengths available in economics . . . whether the data are generated by highdimensional chaos" (Brock, Hsieh, and LeBaron 1991, 3). " I f long cycles . . . exist, we w i l l need another 200 years o f data to determine whether they do exist or are just a statistical figment o f an overactive imagination" (Becker 1988, 7). A t best, therefore, what we have to offer are suggestive graphics. Long W a v e s o fPrices

The idea that there have been half-century-long waves o f prices in the past two centuries begins w i t h the observation o f rhythms in the moving averages o f the annual growth rates o f prices upward and downward on the sides o f inflationary spirals (fig. 10.1). These movements have occurred with a broad cross-national consistency (fig. 10.2). Russian economist N i k o l a i Kondratiev argued that such roughly 55-year trough-to-trough waves (hence, Kondratiev waves) were a fundamental characteristic o f capitalist economies, each wave representing a radical regrouping of, and change i n , society's produc­ tive forces (Berry 1991; also Rosser 1991, 138 f f . , for a discussion o f Schumpeterian discontinuities and historical reswitching). Later investiga­ tions confirm that these regroupings represent the crystallization o f new

3

1810

1830

1850

1870

1890

Ten-year Moving Averages

1910

1930

1950

1970

1990

Annual Rate of Change

Fig. 10.1. A n n u a l a n d t e n - y e a r m o v i n g averages of t h e g r o w t h rates o f U.S. w h o l e s a l e prices 1790-1990. T h e m o v i n g averages have been used as evidence of h a l f - c e n t u r y - l o n g w a v e r h y t h m s , a r o u n d w h i c h t h e annual oscillations fluctuate.

15

* Pri

CO o

10

a

o>

c
5

o

« "5

0

3C C

< ' ' ' 1780

•UK

1930

1960

-US

Fig. 10.2. Ten-year m o v i n g averages o f U.S. a n d B r i t i s h w h o l e s a l e prices, 1 7 9 0 - 1 9 9 0 . N o t e t h e b r o a d c o n s i s t e n c y of t h e l o n g - w a v e r h y t h m s , as w e l l as t h e u p w a r d t r a n s l a t i o n o f t h e level a r o u n d w h i c h prices o s c i l l a t e after W o r l d W a r II. A c o m b i n a t i o n of d e p a r t u r e f r o m t h e d i s c i p l i n e o f g o l d a n d t h e a d o p t i o n of Keynesian m a c r o e c o n o m i c p h i l o s o p h i e s p r o d u c e d a rise of 5 p e rc e n ta g e p o i n t s i n t h e annual i n f l a t i o n rates i n b o t h c o u n t r i e s .

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Chaos Theory in the Social Sciences

techno-economic paradigms i n the aftermath o f inflationary spirals/stagflation crises that define the Kondratiev peaks. Such spirals/crises come in the decade following market saturation by a previous techno-economic paradigm (" . . . the inflationary pressure generated during the final stages o f an up­ swing continues for about 10 years . . . , " Rasmussen and Mosekilde 1989, 279). Thus, the surge o f U . S . growth after the C i v i l War centered on the steel industry as the leading sector and on the expansion and completion o f the railroads, after the canal-building and wind-powered transportation era had peaked i n the late 1850s (fig. 10.3, from Berry, K i m , and K i m 1993). Viewed in this way, inflationary spirals/stagflation crises delineate tran­ sition points beyond which the older techno-economic paradigm declines and a new paradigm takes off, with growth centering on new leading sectors and new supportive infrastructure networks. Initially, new product prices are high, but they decline w i t h achievement o f scale economies and w i t h routinization. Simultaneously, as consumers switch from older to newer products, product prices associated w i t h the now-declining paradigm also collapse, and along w i t h them the values o f their raw materials and fixed assets. O n l y after the new paradigm achieves market dominance in the next Kondratiev trough and then diffuses toward market saturation do demand pressures once again force prices upward, ultimately to be driven by speculation into the next spiraling Kondratiev peak.

Fig. 10.3. R e l a t i o n s h i p b e t w e e n e x p a n s i o n o f a t e c h n o - e c o n o m i c parad i g m t o m a r k e t s a t u r a t i o n a n d t h e l o n g w a v e o f prices. ( F r o m Berry, K i m , a n d K i m 1993.)

Long Waves 1790-1990

219

The regular succession o f inflationary spirals/Kondratiev peaks at 5 0 - 6 0 year intervals (fig. 10.2) suggests a pattern o f transition via intermittency that may generally be confirmed by the existence o f a channel i n the first return map, P = f(P,), as discussed in Berge, Pomeau, and Vidal (1984, 245). Evidence for such channels has been presented elsewhere (Berry 1991, 22ff.); suffice it to say that Kondratiev peaks represent critical bifurcation points beyond which clusters o f innovations can move the economy on to alternative evolutionary logistics (De Greene 1988, 2 9 1 - 9 2 ) . The alternatives are not ordained: there are usually several competitors, o f which only one becomes paradigmatic. t+i

The new growth logistic produced by the successful paradigm may be characterized by the same kind o f equilibrium-seeking behavior as the pre­ vious techno-economic paradigm, or it may result in a different pattern. In either case, just as the success o f a new paradigm may be taken to be stochas­ tic, the equilibria are chaotic, i n that the dynamic processes that are unleashed are captured by attractors o f one kind or another. We seek to document the equilibrium-seeking/shifting behavior o f prices for the successive Kondratiev waves, 1790-1845, 1845-95, 1895-1930, and 1955-present, plus the transition period 1930-55, when, in the aftermath o f the Great Depression and the New D e a l / W W I I / B r e t t o n Woods/Korean War period, the adoption o f macroeconomic policies moved the U . S . economy into an environment o f permanent inflation (Berry 1991, 13, 3 1 - 3 4 ) . Prior to the transition, long waves moved with inflationary/deflationary rhythms; afterward, the successions became inflationary/disinflationary. Shifts on either side o f Kondratiev peaks are o f interest, but o f course so are the peaks themselves. A s Friedman and Schwartz note (1963b, 6 2 - 6 4 ) : The process we have described w i l l tend to overshoot the mark; it w i l l not simply produce a smooth movement to the new path consistent w i t h the new rate o f growth o f the money stock assumed to prevail . . . . (I)n the process o f going from the initial to the new equilibrium, prices must rise at a faster rate than their ultimate rate. Hence the rate o f price rise must overshoot . . . . The tendency to overshoot means that the dynamic process o f transition from one equilibrium path to another involves a cyclical adjustment process. The final picture that might ultimately de­ velop could be o f a partly self-generating cyclical mechanism. Distur­ bances i n the rate o f change in the money stock set in train a cyclical adjustment mechanism including a feedback in the rate o f change in money itself. Additional disturbances from time to time would prevent the fluctuations from dying out. For each o f the time periods we plot the fluctuations in the annual growth rates o f prices (P) in three spaces:

220

Chaos Theory in the Social Sciences

P, = f{t), the conventional time series graph; P, = fiP,), the growth rates o f prices w i t h respect to price levels; and P = fiP,), the Poincare first-return map, o f which t w o versions are given, the scatter diagram and the temporally connected sequence. t+l

We ask how the price rhythms have been reset after each inflationary surge, whether there is evidence o f a catastrophic switch in behavior in the catastrophe theory sense o f the term, and what the underlying conditions producing such a shift might have been. We begin w i t h figure 10.4, covering the period 1790-1845. I n the PIP space (top right-hand graph), oscillations are evident around two price levels. The first and higher level is that o f the upwave leading toward the 1815 Kondratiev peak. After this stagflation crisis, the short-run fluctuations switch to oscillate around a lower downwave level. There is thus evidence o f a bifurcation astride the Kondratiev peak, with oscillation around a higher price level before the main inflation surge, and around a lower level after the surge. The Poincare first-return maps ( t w o bottom graphs in fig. 10.4) reveal something else, however: despite the switching o f annual oscillations after the Kondratiev peak, there is a single underlying chaotic regime in which the oscillations are contained by well-defined limits (see Berge, Pomeau, and V i d a l 1984, 6 6 - 7 1 , 1 0 3 - 4 , 181). The motions are those o f an endogenous chaotic cycle, as evidenced by successive returns to the outer limits in the graph (Rosser 1991, 114), w i t h moderate boom and bustiness revealed by the elongation o f the ellipse o f points along the 45° diagonal (Gleick 1987, 176— 77; Glass and Mackey 1988, 2 6 - 3 4 ; Casti 1989, 2 4 9 - 5 3 ) . Even without the filtering presented by Berry (1991) in the attempt to isolate the attractor describing the longer-run pattern o f boom and bust, the movements point to a strange-like attractor structure that is consistent both w i t h the notion o f a l i m i t cycle (see also Goldberger and Rigney 1990, 3 2 - 3 3 ) and w i t h that o f an attractor basin. Normally, in a limit cycle, the expectation is that sequential movements w i l l be around the edge, w i t h a hollow interior. In the case o f an attractor basin, the interior o f the ellipse is noisy and there is skateboard movement across the short axis that gradually climbs the long axis (the princi­ pal attractor) before descending rapidly from inflationary peak to deflationary depression. The long wave is better defined in the period 1845-95, w i t h a very sharp inflationary surge that peaked in 1865 (top left-hand graph in fig. 10.5). I n the PIP space (top right-hand graph), upwave/downwave levels are discernible, but they are not as sharply differentiated as in the prior long wave. Likewise, while there is a set o f limits in the Poincare first-return maps, they too are less clearly defined, although the oscillations do occur w i t h i n the same limits as in figure 10.4, once the 1865 boom and bust are accounted for. Certainly, the

Long Waves 1790-1990

221

1790 1793 1SO0 1105 1110 ISIS 1S20 1825 1830 1835 1840 1845

- Ten-year Moving Averages

Price (/>,)

- Annual Rate of Change

•20

-10

10

20

30

Annual Rate of Change in Price » 1

Annual Rate of Change in Price [t)

Fig. 10.4. P o r t r a i t s o f price m o v e m e n t s 1 7 9 0 - 1 8 4 5 . T h e t o p l e f t - h a n d g r a p h is t h e c o n v e n t i o n a l t i m e - s e r i e s p l o t a n d s h o w s b o t h a n n u a l rates of c h a n g e a n d t h e t e n - y e a r m o v i n g averages. The t o p r i g h t - h a n d g r a p h p l o t s a n n u a l changes against price levels. The t w o b o t t o m phase port r a i t s p l o t t h e a n n u a l price changes, against t h e i r p r e v i o u s year value, P . T h e l e f t - h a n d g r a p h s i m p l y s h o w s t h e d i s t r i b u t i o n o f p o i n t s , b u t t h e r i g h t - h a n d g r a p h links successive p o i n t s i n t i m e t o reveal t h e year-to-year oscillations. t

sharpness o f the 1865 peak emphasizes the magnitude o f the b o o m and bust, whereas the first-return maps point to endogenous chaos. The 1867-96 downwave phase o f this long wave is characterized thusly in Friedman and Schwartz's Monetary 1960 (1963a):

History

of the United States,

1867¬

40

222

Chaos Theory in the Social Sciences

2o '

' 1M5

1850

!I55

I860

1865

1870

1875

I 1880

1885

1890

1895

"

3 0

20

—•—Ten-year Moving Averages

' 30

1

AO

1

50

Price!/')

—O— Annual Rate of Change

Annual Rate of Change in Price (r)

Annual Rate of Change in Price (f)

Fig. 10.5. P o r t r a i t s o f price m o v e m e n t s , 1 8 4 5 - 9 5 . T h e g r a p h s appear i n t h e s a m e o r d e r as i n f i g u r e 10.4.

The years from 1867 to 1879 were dominated by the financial aftermath o f the C i v i l War. I n 1862, convertibility o f Union currency into specie was suspended as a result o f money creation i n the North to help finance the C i v i l War. From then until the resumption o f specie pay­ ments i n 1879, the U . S . was on a fiduciary (the "greenback") stan­ dard, and the dollar was linked to other currencies via exchange rates that fluctuated freely. Throughout this period, Great Britain main­ tained a gold standard.

' 60

Long Waves 1790-1990

223

In 1879 specie payments were resumed at prewar parity, preceded by a radical decline in the stock o f money 1875-78. After 1879, wide­ spread adoption o f a gold standard, despite new discoveries o f gold and new financial techniques for building a superstructure o f money on it, contributed to a steady downward trend in product prices and mounting political discontent, expressed as "Greenbackism" and "free silver," sharpened by recurrent banking crises (1873, 1884, 1890, 1893). The period ended with the defeat o f W i l l i a m Jennings Bryan in 1896. The resumption o f specie payments in 1879, plus the banking crises o f 1873, 1884, 1890, and 1893 are readily discernible in the greater and lesser inflation spikes in figure 10.5. Yet equally apparent is that each o f these politically significant events falls w i t h i n the limits o f the chaotic regime. N o clear difference is discernible between price behavior in the greenback and gold standard periods, or on the two sides o f the 1865 spiral. Between 1895 and 1930, there is, in contrast, an exceedingly well de­ fined two-phase relationship between price movements and levels, as behav­ ioral switching occurs after the 1920 spiral (fig. 10.6, top right-hand dia­ gram). Preceding the surge to the 1920 peak there is a progressive upwave acceleration o f the rate o f price increase w i t h i n a narrowly oscillating range, a classic example o f intermittency in action (Berge, Pomeau, and V i d a l , 1984, 245). After the Kondratiev peak there is retreat to oscillation around a stable attractor. Yet these are two states o f the same narrowed set o f chaotic limits. As is seen i n the P IP, space, both prior to and after the peak, the oscilla­ tions occur w i t h i n the same restricted range. What could have caused these switches? The Friedman-Schwartz (1963a) characterization o f this period is that t+l

1897 to 1914 combined a long peacetime rise in prices with a relatively high degree o f economic stability. But discontent w i t h the banking structure came to a head in 1907, fol­ lowed by the temporary Aldrich-Vreeland A c t o f 1908 and the perma­ nent step o f the Federal Reserve A c t o f 1913. During and after World War I , weakening o f the international gold standard gave the Federal Reserve System greater freedom o f action than might otherwise have occurred. During W W I , the system served as the engine for inflation­ ary financing o f government expenditures, which continued for eigh­ teen months after the war and then was reversed by abrupt Federal Reserve action. After this reversal, the Federal Reserve assumed a strong directive role, expanding the stock o f money at a steady rate until the Great Depression hit.

224

Chaos Theory in the Social Sciences

40 30 .

1895

_

1900

1905

1910

1915

1920

1925

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,

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2 0

~ 30

—•—Ten-year Moving Averages

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60

70

Price [P )

—o— Annual Rate of Change

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Annual Rate of Change in Price (f)

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30

Annual Rate of Change in Price it)

Fig. 10.6. P o r t r a i t s of price m o v e m e n t s , 1 8 9 5 - 1 9 3 0 . N o t e t h e t w o - p h a s e s t r u c t u r e a t h w a r t t h e 1920 s t a g f l a t i o n crisis ( t o p r i g h t - h a n d graph) b u t t h e c o m m o n phase p o r t r a i t s ( l o w e r graphs).

What was at work were institutional changes resulting from dissatisfaction w i t h the boom and bustiness o f the banking system. The Fed's actions with respect to the stock o f money had the effect o f changing inflationary creep into a decade o f relative price stability (although Weiher (1992) says that this was the inadvertent consequence o f policy designed to help Britain return to the gold standard, rather than purposeful countercyclical policy). But Friedman and Schwartz (1963a) continue:

«0

Long Waves 1790-1990

225

The downturn o f 1929 marked a major transition. There were major bank failures, whose effects were multiplied by tight money policies and by the Fed's reactions to Britain's departure from the gold standard. There began the Great Contraction, during which one-third o f the nation's banks went out o f existence, and there was a catastrophic drop in aggregate demand, a result o f the Fed's failure to stop an enormous decline in the money supply. O n the heels o f the Great Depression, the relative price stability o f 1921-29 vanishes. Rather, in several surges defined by a l i m i t cycle, prices were ratcheted upward (top right-hand graph i n fig. 10.7) as the Fed followed a reflation strategy. Note that the Poincare phase portraits again assume the form o f chaotic l i m i t cycles. The first points are i n the deflationary lower left quadrant during the depression, but the l i m i t cycling is confined thereafter to the upper right quadrant as a permanently inflationary environment is created (bottom graphs in fig. 10.7). Friedman and Schwartz (1963a) say that with the coming o f the New Deal: Like many countries abroad, the United States entered a phase o f cheap money policy, w i t h a network o f controls that only partially and inef­ fectively combatted inflationary pressures. The proponents o f the New Deal were strongly in favor o f easy money: "Easy money was the nearuniform prescription. Inflation was the near-uniform result" (Friedman and Schwarz 1963a, 700). In 1934 came federal deposit insurance as one o f the reactions to the contraction, the dollar was devalued, and subsequently the United States departed from gold. Most importantly, faith in the potency o f the Federal Reserve System collapsed as the United States adopted a fiduciary standard, and the system assumed a passive and reactive role. Federal Reserve policy was subordinated to Treasury policy dur­ ing W W I I . The stock o f money became a function o f governmental taxing and spending and rose 2.5 times between 1939 and 1945, and then once again during the Korean conflict. This inflationary environment, combined w i t h a reactive Federal Re­ serve, provides the context for the final set o f graphs (fig. 10.8). Prices march steadily upward, but annual oscillations are restricted to a limited range along the diagonal in the P /P space. W i t h the compression o f the chaotic l i m i t cycle, what is left are the longer-term booms and the busts, displaced upward to prevent deflation. Only a little more than one-half o f a long wave is t+i

t

226

Chaos Theory in the Social Sciences 40

30

i

A n n u a l R a t e o f C h a n g e i n P r i c e

A

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n

u

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l

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a

t

e

o

f

C

h

a

n

9

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i n

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!

Fig. 10.7. P o r t r a i t s o f price m o v e m e n t s , 1 9 3 0 - 5 5

depicted; there is thus no real opportunity to observe whether any upwave/downwave switching has followed the 1980-81 crisis. There has been much debate on whether fiscal policy has been a potent stabilization tool since the 1950s, given the Fed's inconsistent policies o f money supply growth, slow in the 1950s, higher in the 1960s to keep down interest rates and fuel g r o w t h , a captive o f political forces in the later 1960s and the 1970s (giving rise to runaway inflation), w i t h a sharp reduction in M 2 introduced by Paul Volcker that took hold in 1981 (Weiher 1992). I f the lower panels o f figure 10.8 are compared w i t h the top four graphs in figure 10.9,

Long Waves 1790-1990

Fig. 10.8. P o r t r a i t s o f price m o v e m e n t s ,

227

1955-90

some additional insights into the consequences o f stabilization emerge. Figure 10.9 plots the first-return maps for the long-wave movements o f prices in each of the five time periods. Each traces out a longer-run boom-bust channel along the 45° principal diagonal o f the graphs—a strange attractor that defines the underlying long-wave rhythms. The first-return map i n figure 10.8 (bottom right-hand graph) looks very much like the earlier long-wave movements, showing clearly enough that since World War I I , it is the annual oscillations o f prices that have been suppressed: since 1955 price movements have been constrained to move along the long-run attractor. Thus, whereas short-run

*

a. j

1

A n n u a l Rate o f Price C h a n g e

(D

A n n u a l Rate of Price C h a n g e

{t)

£ TO

20

(f)

20

o

• 20

-10

0

10

20

A n n u a l Rate o f Price Change

30

(t)

40

A n n u a l Rate o f Price Change

(f)

a.

A n n u a l Rate o f Price C h a n g e

(f)

Fig. 10.9. F i r s t - r e t u r n m a p s o f prices (/*,+,//>,) p l o t t e d f o r t h e t e n - y e a r m o v i n g averages, 1 7 9 0 - 1 8 4 5 ( t o p left), 1 8 4 5 - 9 5 ( t o p r i g h t ) , 1 8 9 5 - 1 9 3 0 ( m i d d l e left), 1 9 3 0 - 5 5 ( m i d d l e r i g h t ) , a n d 1 9 5 5 - 9 0 ( b o t t o m ) .

Long Waves 1790-1990

229

stabilization has occurred, the long-run boom-bust pattern o f the long wave remains. L o n g Cycles of Economic G r o w t h We now repeat the foregoing for variations in growth rates o f real per capita G N P for the same time periods. Note (fig. 10.10) that the long swings o f growth appear to diminish after 1950, and that the short-run oscillations also are dampened between 1950 and 1990. Growth behavior from 1790-1845 and 1845-95 is essentially similar (figs. 10.11 and 10.12). Two long cycles o f growth are arrayed on either side o f the 1815 and 1865 inflation peaks/stagflation crises (Berry 1991). These quarter-century-long cycles are named for Simon Kuznets, who first identified them. I n both PIP spaces the first Kuznets cycle involves a surge o f real growth and a cyclical drawback in the stagflation crisis. Likewise, the second Kuznets cycle also reveals growth surges terminated by drawbacks in the deflationary depressions that mark the Kondratiev troughs. The P /P Poincare first-return maps display chaotic limit cycles encircling attractor basins. t+l

t

From 1895 to 1930 the two-cycle Kuznets pattern is not as sharply

Fig. 10.10. A n n u a l a n d e i g h t - y e a r m o v i n g averages o f t h e g r o w t h rates of real a n d p e r c a p i t a GNP i n t h e U n i t e d S t a t e s , 1 7 9 0 - 1 9 9 0 . N o t e t h e c o m p r e s s i o n of t h e a n n u a l f l u c t u a t i o n s after W o r l d W a r II.

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Chaos Theory in the Social Sciences

Fig. 10.11. Phase p o r t r a i t s o f g r o w t h , 1 7 9 0 - 1 8 4 5 . D i a g r a m s o r g a n i z e d in s a m e m a n n e r as f i g s . 1 0 . 4 - 1 0 . 8 .

defined (fig. 10.13), but nonetheless the PIP space reveals that growth pro­ ceeds until the cyclical setback i n W W I and resumes again until the pullback in 1929-30. A g a i n , the P lP, maps show a chaotic l i m i t cycle. t+l

In contrast, the 1930-55 transition involves strongly cyclical behavior (fig. 10.14), but from 1955-90 what is most apparent is the steadiness o f the growth path, marred by occasional business cycle downturns and by the 1980-81 stagflation setback (fig. 10.15). What emerges is an increasingly well defined l i m i t cycle (Berge, Pomeau, and Vidal 1984, 27, 67, 178, 181),

Long Waves J 790-J990

231

an absence o f attractor basin skateboarding, and only limited boom and b u s i ­ ness, surely another outcome o f the reactive federal stabilization policy that has served to suppress sharper swings. Each o f the circuits around the limit cycle is one business cycle in length. I f the lower panels o f figure 10.15 are compared with those o f figure 10.16, the important contrast that has resulted between the stabilization o f growth and prices is brought into focus. Unlike the restriction o f the annual oscillations o f prices after 1950 (producing an annual first-return map for 1950-90 that looks like the long-term first-return pattern before World War

232

Chaos Theory in the Social Sciences

IS

10

-5

0

S

10

A n n u a l Rate o f C h a n g e in G r o w t h ( f )

IS

"

-'"

A n n u a l R a t e o f C h a n g e i n G r o w t h (*)

Fig. 10.13. Phase p o r t r a i t s o f g r o w t h , 1 8 9 5 - 1 9 3 0

I I ) , the long-term first-return maps o f growth (fig. 10.16, top panels) are quite unlike the annual first-return maps after 1950 (bottom panels o f fig. 10.15). The latter display a perfect l i m i t cycle, each 360° circuit o f w h i c h is a business cycle; the former shows the longer-term boom and bustiness o f growth. What thus has been eliminated since 1950 is the longer-term Kuznets cyclicality: the first-return map o f the moving average o f growth since 1950 appears to be a stable attractor (bottom panel, fig. 10.16). What has been produced is a pattern o f business cycles around a steady-state path o f economic growth.

Long Waves 1790-1990

- O — A n n u a l Rate of Change

.15

-10

5

0

5

S

10

Annual Rate of Change in G r o w t h ( f )

15

233

Real Per C a p i t a G N P

15

10

-5

0

5

10

A n n u a l Rate o f Change in G r o w t h ( f )

Fig. 10.14. Phase p o r t r a i t s o f g r o w t h , 1 9 3 0 - 5 5

Discussion We see in the foregoing the advantages o f using multiple phase portraits to analyze dynamic processes. Both prices and growth reveal a tension between longer-run rhythms and short-term variability. The long-run rhythms are un­ stable near their peaks, leading to intermittent shifts in the locus o f shorter-run fluctuations. I n the long run the pattern is one o f Kondratiev waves o f prices and Kuznets cycles o f growth, and with these are associated intermittent

234

Chaos Theory in the Social Sciences

I960

1965

1970

1975

1980

1985

· - Eight-year Moving Averages a— Annual Rate of Change

2

»

5

%

1

ffc

3000

3500

4000

Real Per Capita GNP

*

••t * at

1

Annual Rate of Change in Growth (f)

Annual Rate of Change in Growth (f)

Fig. 10.15. Phase p o r t r a i t s o f g r o w t h , 1 9 5 5 - 9 0

booms and busts. I n the shorter run, fluctuations were chaotic within limits until the Great Depression. The rhythms have not been immutable, however. Institutional changes and policy interventions since the Great Depression have raised the level around which prices fluctuate by five percentage points and, simultaneously, have suppressed short-term oscillations, leaving behind more gradual upward and sharper downward movements along the long-wave attractor. But simul­ taneously, economic growth has been constrained to fluctuate around a longterm steady state o f roughly 2 percent per annum while preserving the shorterterm rhythms o f the seven- to eleven-year business cycle. Thus, since the

ï

î

o

o

o

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IS

<

Annual Rate of Growth Change (f)

Annual Rate of Growth Change it)

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<5

o

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Annual Rate of Growth Change (f)

15

Annual Rate of Growth Change (fî

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<5

Annual Rate of Growth Change (f)

Fig. 10.16. F i r s t - r e t u r n m a p s o f l o n g - t e r m g r o w t h , 1 7 9 0 - 1 8 4 5 , 1 8 4 5 - 9 0 , 1890-1930, 1930-55, and 1955-90

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Chaos Theory in the Social Sciences

Great Depression, a combination o f institutional changes and public poli­ cies has resulted in more optimizing (Slutsky-Frisch-Tinbergen) economic growth behavior, but the interventions have had a different consequence for prices, constraining them to move within a narrower (but still boom-andbusty) l i m i t cycle.

CHAPTER

1 1

C i t i e s as S p a t i a l C h a o t i c A t t r a c t o r s Dimitrios S. Dendrinos

Developments in the mathematical theory o f dynamical stability and bifurca­ tion theory have considerably influenced developments throughout the social sciences. Sources have been either the field o f structural stability emanating from catastrophe theory (see Thorn 1975; Zeeman 1977), mathematical chaos (see Lorenz 1963; M a y 1976; Guckenheimer and Holmes 1983; Thompson and Stewart 1986; Schuster 1988; Devaney 1989), or the theory o f fractals (see Mandelbrot 1977; Peitgen and Richter 1986; Barnsley 1988). The above are only a short list o f basic references among many that have appeared over the past 30 years i n these fields. Almost without exception, social science fields have been influenced by these developments, including areas o f study in archaeology, history, soci­ ology, demography, political science, geography, psychology, and economics. In particular, the fields o f economics (micro- and macroeconomics, including economic growth theory, the theory o f finance, business cycles theory, tech­ nological progress, international trade, and so on; see Anderson, Arrow, and Pines 1988; Barnett, Geweke, and Shall 1989; Barnett and Chen 1990; and Rosser 1991), and spatial analysis (geography, urban and regional analysis, transportation dynamics and traffic flow analysis, and so on), have experi­ enced considerable impact from these mathematical advances. A l o n g w i t h the considerable new (basic as well as marginal) insight gained by such applications, there has also been a revision o f past claims in social theory. To some, a call for reconfiguring the social sciences has emerged from insights gained through these newly developed fields. But, together w i t h this call, the realization has settled in that there are, at least at present, certain insurmountable practical as well as theoretical obstacles to such reconfiguration.

This paper was written during spring, 1992.

237

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Chaos Theory in the Social Sciences

Contributions A t the outset, a few basic points w i l l be made as to the magnitude o f contribu­ tions to and extent o f impact from applications emanating from nonlinear dynamics into social sciences. Then, some remarks w i l l follow, addressing the question o f the considerable drawbacks o f developing a truly dynamic social science at present.

Basic and Marginal Inroads This is not the appropriate forum to survey all the areas where chaos theory has made inroads or to fully assess the impact from or the value o f these contributions. Some o f the papers in this volume do so in their o w n special fields. Whereas in some areas, like those in spatial analysis and geography, the contributions might be basic, in other areas, particularly i n economics, one may find the contributions made to date to be rather marginal. In economics, the most credible o f the attempts to exploit chaos (and catastrophe) theory occurred when existing classical and neoclassical deductionist (causality-based) theories were modified to marginally change preexist­ ing (and widely accepted among disciples o f current orthodoxies) assump­ tions. Such changes were offered as means o f showing how chaotic dynamics can arise from currently prevailing theoretical models, and thus enhance these models' utility. There is another side to this coin, however. Marginal extensions o f currently accepted paradigms in economic analysis point directly to one o f their weaknesses, namely, that under slight but proper modifications in speci­ fication, these models can reproduce almost anything that the analyst wishes to produce through theoretical deduction. A scientist might find such accom­ modations somewhat troublesome.

New Dynamical Features The single most important contribution mathematical chaos has made is to demonstrate the possible presence o f new dynamical features in social sys­ tems that theoreticians had never addressed before. Until very recently, aperiodic movement was an attribute o f social science models to avoid and quasi-periodic motion was never even conceived of. Chaos theory showed that through such dynamics one can detect interest­ ing and novel insights i n the behavior o f social systems. It also supplied a method to classify the various dynamical trajectories obtainable in phase space and to catalog a large number o f possible bifurcations.

Cities as Spatial Chaotic Attractors

239

A New Type of Explanation Mathematical chaos further suggests that periodic, quasi-periodic, and nonperiodic trajectories in the phase portrait o f interdependent socioeconomic variables can be generated by very simple dynamical equations. Chaos occurs in dimensionally very low systems, where it most often has been studied. High-dimensional chaos is still an elusive subject. Efficiently capturing aperiodic motions can be accomplished w i t h simple dynamical specifications. Efficient dynamic specifications can be viewed as a new method o f explanation in the social sciences. Mathematical chaos highlights a topic first addressed about twenty years ago with the advent o f catastrophe theory and the theory o f structural stability. One can label this new type o f explanation reductionist. The example o f this chapter is an exercise in reductionist expla­ nation. It bears little correspondence to the prevailing causal or deductionist explanation, or to the inner structure o f the underlying socio-spatial complex­ ity. A social scientist may find reductionist specifications and inferential ex­ planation to have minimal economic interpretation and to lack any strictly and currently prevailing disciplinarian meaning, for that matter. For a good, but extreme, example o f the issues involved in this argument the reader is referred to the work by Mandelbrot (1977) and Barnsley (1988) on fractals. Simple iterative processes depict in their phase portrait elements that resemble clouds, plants and animals, shorelines, or exotic landscapes. However, the iterative maps responsible for generating such images do not hold explicitly in their formulation either the physics and chemistry o f clouds, the genetics and biology o f plant and animal species, or the geology o f coastal lines and terrestrial (or extraterrestrial) landscapes. These examples are extreme in the sense that the iterative maps respon­ sible for their images are indeed extremely efficient (in terms o f state variables and parameters they contain), but also totally devoid o f any substantive mean­ ing or (deductionist or reductionist) explanation. They can accommodate, however, a wealth o f dynamical features. I n the example to be presented i n this paper, it is demonstrated that an intermediate case exists involving reduc­ tionist explanation coexisting w i t h model efficiency. Some parsimony is traded off for explanation, while the variety o f dynamical behavior is largely preserved. The requirement for efficiency and parsimony, and the need for obtaining a comprehensive list o f dynamical behaviors may pose some questions to currently dominant social science theories. Both criteria were brought to the forefront o f social science discourse with dynamical modeling; the issue is further enhanced w i t h the advent o f mathematical chaos.

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Chaos Theory in the Social Sciences

A need arises for a basic and not simply marginal modification to defin­ ing social systems and for establishing a single, all-encompassing, nondisciplinarian dynamical social science frame o f reference. This is a call for a new and ambitious research agenda w i t h potentially high rewards in descrip­ tive power, but at the cost o f losing explanatory value o f the deductionist type. A theory that offers reductionist explanation, being a different type o f explanation than social scientists have been used to, may be considered a severe shortcoming and may conjure criticisms by those w i t h an incantation for policy analysis. Widespread current dissatisfaction w i t h the multiplicity o f and tenuous base for most policy-sensitive, behaviorally motivated social science fields, where multiple deductive explanations vie for few followers, tempers such criticism. Discounting o f such fears may also be enhanced by the fact that social science orthodoxies are ephemeral and do not commend the recognition o f relatively durable natural science theories. But this, being a very involved and demanding subject, w i l l not be elaborated significantly here, although a bit more w i l l be added later. Difficulties

Next, a few comments w i l l be made touching on the difficulties in setting up and testing for a truly dynamic social science theory, currently perceived as insurmountable. Lack of Data Chaotic paths i n phase portraits o f even the lowest dimensionality, such as those found in the one-dimensional logistical prototype, require an enormous amount o f data to trace, calibrate, and test. Extremely few variables in the social sciences are available at frequent enough time intervals to permit one to do so at any degree beyond a rather rough level o f accuracy. Most o f the variables w i t h extensive time-series data are found i n stock market or mone­ tary tables, in the field o f finance. It is not surprising that the initial attempts to find chaos and statistically test for i t (through Liapunov exponent-based tech­ niques) i n the social sciences occurred in that field. Frequent

Perturbations

In addition to the difficulties surrounding lack o f data, social science variables also suffer from many other ills. One o f these ills has its source in the restriction that social systems do not produce repetitive measurements under experimentally controlled conditions. Perturbations o f social science variables are very frequent, irregular, and unexpected. Further, very rarely, i f ever, are

Cities as Spatial Chaotic Attractors

241

social systems comprised o f identical components with highly homogeneous behavior. These concerns raise issues o f validity i n statistical analyses in social studies. Statistics are based on fundamental assumptions in probability theory that may not be applicable to social science fields. The difficulty is aggravated when dynamics are involved and temporal heterogeneity is added. Specifically, the definition o f and criteria for detecting what is referred to as white noise present an added difficulty to the issue, since to separate white noise from internal dynamics in socio-spatial variables is at present almost impossible, even for dimensionally low-level social systems. O n the side o f chaos, too, there are difficulties at present related to deriving a conclusive test for detecting i t . These difficulties create considerable impediments to deriving a statistical theory o f social systems dynamics, and they render testing almost prohibitive. Inaccuracy When the dynamics o f state variables are extremely sensitive to either initial conditions or the magnitude o f the model parameters, then serious questions due to inaccuracy in measurements o f state variables and in estimating param­ eter values arise in the social sciences. H o w inaccuracy in parameter estima­ tion techniques might, in the presence o f quasi-periodic or chaotic dynamics, hinder statistical methods is a topic still to be fully addressed. Instruments, techniques, and methods o f measuring social science data (state variables) are highly imperfect. Fuzziness in measurement and m u l t i ­ plicity in perception o f social data sets are often necessary conditions for social planning, policy, and action. H i g h resolution or sharpness in data sets is not always feasible or efficient to obtain. It may even be undesirable. Unresolved Issues In the social sciences, theoretical advances in nonlinear dynamics are also severely hampered by binding practical constraints. One o f these is associated with the level o f disaggregation one can employ to analyze socio-spatial systems. Social systems do not lend themselves, for numerous reasons, to very fine levels o f disaggregate analysis. Consequently, one may be forced to deal with rather aggregate data. Coarseness in social data sets might wash away interesting dynamics found at a finer scale, further inhibiting any chances for advancement in empirical, as well as theoretical, work. Finally, questions can be raised regarding the issue o f fractal properties in chaotic attractors associated with socio-spatial forms. Social systems take up physical space and cannot be broken down to an infinitely small selfreplicating scale. Lower and upper bounds in the scale o f human settle-

242

Chaos Theory in the Social Sciences

ment activity, for instance, are due to the size o f buildings and cities. The presence o f these bounds poses certain theoretical questions (unanswered until now) to socio-spatial models producing chaotic outcomes with fractal dimen­ sions. To conclude however, that social science ought to downplay mathemati­ cal chaos theory and its insights, and instead persist in learning from only static, sharp, or stable dynamical models would be erroneous, and even per­ nicious. I t w o u l d deprive the analyst o f an ability to obtain intuitively appeal­ ing qualitative insights, simply because one lacks the comfort o f very many and accurate measurements. Such luxury may never be afforded the social scientist, while the social agent is always confronted with a need to act. The agent w i l l always be either right or wrong in its speculative action; conse­ quently, the agent w i l l be either rewarded or penalized. A mathematical chaos-ignorant social scientist would not be an effective or desirable advisor to the social agent. The Universal M a p o f Discrete Relative Dynamics Nonlinear interdependent dynamics capable o f generating chaotic paths have made salient contributions to the study o f socio-spatial evolution over the past fifteen years or so. They have supplied methods to comprehensively and abstractly model the dynamics o f socio-spatial systems. Even more impor­ tantly, they have provided a platform for some new and at times fundamental ideas and novel insights to penetrate and subsequently permeate socio-spatial analysis. A m o n g the various contributions made during the 1980s a universal map of discrete relative socio-spatial dynamics was developed (Dendrinos and Sonis 1990). It set the stage for a broad framework to emerge in describing, analyzing, and modeling ( i . e . , simulating) abstract spatio-temporal processes, and it is an experiment in reductionist explanation. The framework involved abstract interdependencies among any number o f competing stocks and loca­ tions under a nonlinear interdependent dynamics configuration, based on the comprehensive notion o f spatial relative comparative advantages. So far, the various contributions made through the application o f the universal map o f discrete relative dynamics have been confined to only one type o f formalism built into this map, namely the temporal evolution in the relative size o f interdependent but heterogeneous stocks as distributed among a number o f heterogeneous and interdependent locations. Under this configu­ ration, the phase portrait o f the map was that o f an abstract space depicting dynamics in the relative size o f stocks at different locations. Physical as well as abstract stocks can be modeled in physical or abstract phase space. The universal map can take any number o f configurations, from the simple one-stock, two-location to the multiple-stock, multiple-location

Cities as Spatial Chaotic At tractors

243

specification, and generate their phase portraits. Qualitatively, the dynamical features o f the various phase portraits differ, depending on the model configu­ ration. In what follows, the focus is on topics o f interest to spatial analysts, addressing interurban or intra-urban spatio-temporal allocation processes. Formation o f human settlements or communities is addressed, each w i t h a different temporal vintage. Dynamics o f a variety o f socioeconomic stocks can be accommodated by the various configurations o f the discrete dynamics map used here. The abstract and general presentation that follows can be employed by other social scientists, including economists, sociologists, de­ mographers, political scientists, psychologists, and so on, to simulate the (spatial or nonspatial) dynamics o f interest to their field o f inquiry. It may also be o f interest to ecologists and biologists, in analyzing stocks of animal and plant species moving and spreading in space-time. Further, the method may also be o f potential use to physical geographers and geologists studying the anisotropy and heterogeneity (possibly spatially chaotic varia­ tion) o f landscapes and geological formations. The method may also supply insights into the spatio-temporal distribution o f geological events (for i n ­ stance, the location o f epicenters o f seismic activity occurring o v e r t i m e along fault lines). Next, a few remarks w i l l be made on the basic features o f selected configurations. O n l y a simple statement o f the various configurations and specifications o f the iterative map w i l l be shown, as the detailed analytical exposition is found elsewhere (Dendrinos and Sonis 1990). The Single-Stock Case The central proposition o f the universal map is as follows: at any time period, / (t = 0,1,2, . . . ,T), the relative size o f a stock at some location w i t h i n a prespecified environment (consisting o f a number o f locations) can be simu­ lated (modeled or described) as being directly proportional to a prior level o f locational comparative advantages enjoyed by the stock in question at the site in question. Locational comparative advantage is translated into a likelihood or probability that a particular event w i l l occur at a particular location. A replication (or reproduction) force occurring at regular time intervals is assumed to underlie the iterative procedure. For social stocks (as in the case at hand), the replication procedure may be attributed to internal ( i . e . , among the various locations within the prespecified environment) temporal reallocation (or redistribution) o f the stock(s). In specific, for the case o f a one-stock, multiple-location configuration, the model is given by the following specifications:

Xi

(t

+ 1) = F^jFjit);

i,j = 1,2, . . . , /

(1)

244

Chaos Theory in the Social Sciences

F,it)

= F^itlxM

. . . ,x,(t)} > 0.

(2)

The discrete iterative map describes the dynamics o f the relative distribu­ tion o f the (homogeneous) stock at a location i (0 < *,·(/) < 1) within an environment o f / (heterogeneous) locations, so that at all time periods

2,x,(0 = 1 ; / = 0 , 1 ,

T

(3)

over a time horizon, T, which consists o f time periods (or iterations), t, identical i n length. Quantity x is the probability that an event w i l l occur at any location, i, i n / . Quantity F is the comparative advantage that any location, /', in / has in attracting such an event. The description o f these dynamics is based on the premise that, under a one-time period delay, a reproduction process is i n effect; the relative size o f the stock at location i at the next time period, (t + 1), depends on the normalized current locational comparative advantages given by the function F,{t). This is only one possible delayed response, as any time delay, n (or periodic forcing, t + n) could be incorporated into the above dynamical configuration. Functions F depict nonlinear current locational comparative advantages. A m o n g other factors, these advantages are the result o f interdependencies among the current sizes o f the homogeneous stock at the various locations; more about the specification o f these functions w i l l be supplied later. A distinction is noted i n particular; equations 1-3, through their form, identify interdependencies among stock sizes at the various locations and not inter­ action, that is, flow o f stock(s) among them. Without any loss o f generality, equations 1-3 can be restated and scaled in terms o f a certain numeraire (or reference) location, / , as follows: xff

+ 1) = Fft)l{\

+ 2,-F/r)}

x,{t + 1) = 1/{1 + Fj(t)

= Fj(t)IF,(t);j

(4) = 1,2, . . . , / -

1,

as F,(t) is a strictly positive quantity. The One-Stock, Three-Location Configuration O f special interest is the case o f the iterative map depicting the discrete dynamics o f a single stock as distributed among three locations. It turns out that this is a generic case involving all possible qualitatively different dy­ namics encountered i n the one-stock, multiple-location configuration. De­ tailed specifications o f this model are as follows (see fig. 11.1):

Cities as Spatial Chaotic Attractors x {t + 1) = Ftf/ZjFft);

i,j = 1,2,3; 2 , · * , « = 1,

t

In F (t) = In A, + X^a,-, In A:,-, t

245 (5) (6)

where the vector o f the state variables, *,(**), defines in the three-dimensional space an equilateral triangle w i t h sizes equal to the square root o f 2. The vector o f (strictly positive) parameters, A , depicts in absolute terms and in a descriptive and bundlelike manner the following prevailing condi­ tions at each o f the three locations (among numerous other factors characteriz­ ing each location). First, it identifies the way i n which the environment favors or handicaps the location. Second, it picks up the presence and level (or absence) o f natural resources at the location in question. T h i r d , it records a pattern o f accessibilities to markets (locational access as a place o f origin for trips) and to input factor sources (access as a place o f destination for trips). Fourth, it depicts the technological conditions prevailing there. H o w much each o f these factors contributes to each element in A is unknown, intractable, and indeed irrelevant in this discussion. In combination, all these factors affect production, consumption, market (exchange and inter- as well as intralocational trade), and public decision­ making conditions, which in turn may entail advantages or disadvantages at that location. These factors indirectly affect interaction and flow o f stocks. Whereas the matrix o f real ( i . e . , any negative or positive) coefficients [a] contains entries, each o f which identifies in a bundlelike manner factors directly associated w i t h and affecting (facilitating or impeding) interaction (including interlocational trade, and flow o f stocks), these coefficients identify comprehensive cross-elasticities o f locational comparative advantages w i t h respect to stock sizes, since

{dFJdxjViFtlxj) = a . 0

(7)

W i t h i n these cross-elasticities, effects o f physical and other (ecological) dis­ tances are discounted. Parameters [a] could also be interpreted as comprehen­ sive (political, environmental, social, and not merely economic) prices o f interaction among locations i and j , where transportation (and other spatial transaction) costs are also included. Elasticities exponents weigh the current relative size o f the stock(s) at different locations in the F function. Obviously, the magnitude o f each o f these coefficients, as well as their sum in each o f the F functions, is o f considerable interest, as w i l l be discussed below. In a hierarchy o f speeds, the state variables, x, are the fastest moving entities in this model. Following the so-called slaving principle (Haken 1982), their behavior is thought o f as being slaved by the parameters [a], which are slow-moving entities, while parameters A vary even at a slower pace.

246

Chaos Theory in the Social Sciences

Scaling

the One-Stock,

Three-Location

Model

Taking location one as the reference (or numeraire) location, one obtains the following repetitive or reproductive process based on comparative advantages set in reference to the numeraire location, found by dividing the comparative advantage functions, F , , by the comparative advantage function o f location one, F > 0: x

* , ( / + 1) =

+ F (t)

+ F (t)]

2

= 1 - x (t

3

2

+ 1) - x (t 3

+ 1)

(8)

x (t

+ 1) = F (t)l[\

+ F (t)

+ F (t)]

(9)

x (t

+ 1) = F ( / ) / [ l + F (t)

+ F (t)]

(10)

2

2

3

2

3

3

2

3

In F (t)

= In s& + a

2 l

In x (t)

+ a

In F (t)

= In s& + a

3 1

In x (t)

+ a

2

2

3

s& , sâ 2

3

> 0;

3

-oo

x

x

22

In x (t)

+ a

In x (t)

+ a

2

32

2

< a , a , a , a , a , a 2 1

2 2

2 3

3 1

3 2

3 3

23

In x (t)

(11)

33

In x (t)

(12)

<

3

3

+oo,

(13)

where a

« y = o ~ a ; i = 2,3 XJ

^

2

= A /A,, A 2

3

j = 1,2,3

(14)

= A /A,.

(15)

3

In addition, and without any loss o f generality, the above can be further algebraically scaled by assuming that A , = 1 and that all a = 0, that is, by assuming without imposing any restrictions on the model's dynamics that F = 1. XJ

x

Configuring the discrete dynamics system o f a single stock and three interdependent (perfectly competitive in the case o f relative dynamics) loca­ tions i n such a manner supplies all the dynamical features recorded in the nonlinear dynamics (mathematical) literature: a wide battery o f bifurcations (see Thompson and Stewart 1986; Devaney 1989) are encountered in transi­ tions from any state to another by crossing critical thresholds o f distinctly different dynamical behaviors. A l l possible events that can occur in t w o - or three-dimensional maps (based on the horseshoe map, the hyperbolic toral automorphisms, and the various forms o f attractors; see Devaney 1989, 159) do occur in this map as w e l l . However, the topological properties o f the map w i l l not be addressed here. A complete menu o f dynamics is offered, inclusive o f fixed-point (asymptotic or not, stable or unstable, cyclical, nodal, or saddle) attrac-

Cities as Spatial Chaotic Attractors

247

tors/repellers, periodic movement and cascades o f period-doubling cycles, quasi-periodic (toroidal) motion, and finally aperiodic (chaotic) behavior. A n extraordinary variety o f quasi-periodic motions, and numerous types o f chaotic movement (involving fractal and nonfractal, strange and nonstrange attractors or containers) is present i n this map (Dendrinos 1991a, 1991b, 1992a). The discrete iterative process configured as a single-stock, three-location case is universal because, first, the map's menu o f dynamical behaviors and their list o f bifurcations is comprehensive; and second, because the variety o f spatial patterns it can generate and thus describe is as extensive as any k n o w n map. W i t h the one-stock, three-location configuration o f the multiple-stock, multiple-location map, one has universality within universality in the discrete iterative process. The Multiple-Stock, Multiple-Location Case The configuration in this case, where one wishes to describe the interdepen­ dent evolution o f many stocks allocated into many interdependent and inter­ acting locations, goes as follows: {t

Xij

+ 1) = FipyY.fifi);

i = 1,2, . . . J;j

= 1,2, . . .

J (16)

l (t) iXij

= l;j

= 1,2, - · · J.

Under this configuration, index i stands for locations and index j for stock type; the functions F could be specified equivalently to those specifications given for the case o f a single stock and multiple locations. H o w this multiple-stock and multiple-location case could be used w i l l be briefly outlined later in the chapter. However, since this configuration w i l l not be used extensively i n this paper (which deals w i t h only a simple case), not much detail w i l l be supplied here. The interested reader is referred to Dendrinos and Sonis 1990 for more. The Universal M a p a n d Physical Space In what follows, the subject o f evolution in the form o f human settlements is addressed. A l l movements and bifurcations in phase space o f the universal map o f discrete relative dynamics have been interpreted so far as associated with changes in the relative distribution o f one (or more) stock(s) as found in two or more locations. For example, human population could have been the entity modeled (described) in the case o f a single stock. Population, wealth, and capital stock could have been the physical entities abstractly and compre­ hensively modeled in the case o f multiple stocks, among two or more loca­ tions. Phase space in these iterative discrete dynamics has been abstract

248

Chaos Theory in the Social Sciences

relative stock size space, and not physical space. Indeed, locations have been merely represented by an index, i, i = 1,2, . . . ,/. Next, a different angle to the universal map is taken, whereby its phase space is thought to be physical space. A n attempt is made to describe the formation and evolution o f human communities or settlements comprehen­ sively in the form o f spatio-temporal chains, or a series o f points (sites) inhabited by human population w i t h i n a specific area. Sites have different temporal vintage, some being inhabited before others. These chains are pro­ duced as either fixed stable and unique points; periodic, attracting, stable foci; quasi-periodic, cyclically formed rings; or strange, aperiodically formed attractors in physical space. The settlement formation (chaining) process to be presented next de­ scribes the form o f communities w i t h i n urban areas and produces intra-urban allocations in the shape o f urban patches. I t also describes the formation o f urban settlements w i t h i n nations or regions, producing interurban allocations in the form o f urban strings. Hence, it can depict formation o f microsettlements o f different vintages w i t h i n metropolitan areas, as well as the rise o f a series o f interconnected urban macrosettlements appearing as linked clusters o f urban agglomerations in the context o f nations or even continents. Put in slightly different terms, what is shown next is the formation o f human settle­ ments (cities) w i t h i n spatio-temporal niches. Patterns in the evolution o f these niches are also shown. A question not often raised by spatial analysts is whether the formation o f settlements at particular sites in space is due to forces operating exactly and solely at these sites or whether the appearance, g r o w t h , and extinction o f urban settlements at these sites is due to forces operating at a distance from them. Obviously, there are endogenous as well as exogenous forces at work at the sites in question, and the framework shown next is helpful i n putting this issue in a spatial (physical) context. Spatial allocations o f human population are influenced by the presence o f topographical features (such as rivers, shores, mountains, valleys, lakes, etc.), climatic conditions, or man-made landmarks, such as highways. Alter­ natively, transportation arteries and corridors (in the form o f rivers, shores, or highways), can be looked at as spatial attractors or containers along which human settlement activity periodically or aperiodically occurs. One o f the principles embedded in the spatial shaping o f form emerging from the universal map o f discrete iterative dynamics is the following: spatial form, which contains components o f different vintage, is neither shaped nor does it evolve in a locationally contiguous manner. Rather, it rises, transforms itself, and disappears in a spatially discontinuous way. A broadly based prin­ ciple may underscore this finding: that spatial diffusion processes do not occur through locational proximity, but rather through spatial ecological propin­ quity, which might involve spatial discontinuities (Dendrinos 1992b).

Cities as Spatial Chaotic Attractors

249

Stock Allocation and Conventional Analysis: Statics Allocation o f a stock (for example, residential consumption or industrial production activity) in a two-dimensional physical space (usually confined to a triangle defined by points identified either as sources from which inputs flow, or sinks toward which outputs flow) and based on economic agents incurring some form o f a spatial transaction ( i . e . , transportation) cost and a set o f spatially defined prices has been a long-standing problem i n spatial sciences. I n spatial economics and geography, static long-run market (unique) equilibrium (or welfare optimum) spatial allocations o f economic (production and/or consumption) activities have preoccupied analysts for more than a century. Most o f this work is based on the (interurban) spatial economics o f Alfred Weber (see Friedrich 1929 and Isard 1956), to the extent that produc­ tion is concerned. Intra-urban, as well as interurban, allocation o f a residential or farming activity around a single point in homogeneous physical space under isotropic transportation conditions is another closely related problem, due to Von Thunen (see Alonso 1964). These classical problems in spatial economics have preoccupied spatial analysts for decades. The Alternative: Dynamics A dynamic, much broader, comprehensive, rich, and abstract albeit descrip­ tive view o f spatial allocations in physical space is supplied next. The context in which the model can be applied is both the inter- as well as the intra-urban scene. The method goes far beyond a mere reliance on (slow-changing) trans­ port costs and/or (fast-altered) spatial prices. There is no requirement that unique, long-term static spatial configurations be generated. There is also no need to explicitly model the vastly complex interdependencies and ensuing interactions among a variety o f spatial agents involved in shaping spatial form through vastly complex consumption and production as well as market, ex­ change (trade), and governmental intervention mechanisms. Instead, a small number o f variables and parameters are chosen to effec­ tively model or mimic reality, producing a state o f virtual reality, but not reality itself. Interdependencies among these variables are captured in the most general manner, attempting to describe a comprehensive menu o f physi­ cal forms o f human settlements. Deductive explanation, in the form o f ex­ plicit behavior by the many agents responsible for these motions, is largely absent from the analysis. Many factors are embedded in the bundlelike parameters included in the configurations o f the universal map. They form the basis for a reductionist explanation, which in turn is based on the notion o f (fast-moving) locational comparative advantages. A few comments on this central notion o f compara-

250

Chaos Theory in the Social Sciences

tive advantages seem warranted, in view o f its rather confined definition and use w i t h i n the theory o f international trade in economics. Locational comparative advantages is a Darwinian-like notion in the evolution o f human settlements. A m o n g other things, it calls for the survival o f the most advantageous o f all spatial settlements, much like Darwin's notion o f survival o f the fittest among various animal and plant species. Embedded within this assertion is Darwin's principle o f evolution o f species through a natural selection process. The process o f selection acquires a broader frame­ work under the mathematical properties o f chaos, as also does the notion o f Malthusian-type dynamics, a notion that considerably influenced D a r w i n originally. Selection involves processes o f optimization and search for optima. Questions regarding the optimization processes employed by social agents individually and/or collectively (objective functions and constraints), and numerous search mechanisms applicable thereon resulting in a variety o f locally (as opposed to globally) optimum outcomes in socio-spatial evolution are not the concern here. Mathematical chaos may be capable o f shedding additional light on such socio-spatial processes, including processes equiva­ lent to those associated w i t h neural nets, cellular automata, and processes o f evolution in genetics. But the central concern o f this paper is much more narrow, and such considerably more elaborate discussions are found elsewhere (Dendrinos ( w i t h M u l l a l l y ) 1985; Dendrinos and Sonis 1990; Dendrinos 1992b). Spatial

Triangles

and Their

Antitriangle

Assume the presence o f three interdependent points (nodes) o f interest, defin­ ing an arbitrary triangular area. This triangle could be inside an urban area; it could be formed by nodes found i n a spatially extensive national landscape among widely separated urban areas; or it could represent the area o f an island. Reconstructing a map o f archeological remains on the island o f Sicily could provide a good forum to test the model, for example. The problem is to derive a dynamic process that w i l l generate a map o f human settlement activity, as it serially appears over a time horizon. Forma­ tion o f spatio-temporal chains o f human settlement activity inside the area o f the triangle is defined partly by the location o f the three nodes and the forces operating at them. Because the spatio-temporal chains are closely dependent upon the exact location o f the three nodes delineating the area chosen to reproduce the locational chains, choice o f nodes is critical. Where a settlement appears at a point in time is subject to two sets o f forces. Certain forces locating settlements at various points in the space defined by the three nodes are exogenous to the specific location o f the cities (or city) inside the triangle and attributed to its nodes. Other forces that affect settlement formation inside the triangle are those

Cities as Spatial Chaotic Attractors

251

associated w i t h the previous settlements. W i t h i n its borders an original state o f some human activity must be asserted. This initial condition affects the location inside the triangle o f the subsequent settlements; it constitutes the endogenous force. As it turns out, the original state (initial location) o f the human stock i n the triangle is critical for the dynamics o f human settlement formation at certain instances, whereas in others it matters little. As a result o f spatial attributes, heterogeneity o f space, and the presence o f the initial stock at some point within the triangle, spatial interdependencies appear. Interdependencies among the three points (nodes) involve forces lead­ ing to a net pull or push force attracting toward or repelling from these nodes human activity during subsequent time periods. This net pull/push force is a result o f a tug-of-war between two opposing component subforces: attraction, acceptance, and affinity demonstrated toward the (homogeneous) stock on the one hand at a node or repulsion, expulsion, and hostility toward the stock at another. The three points (nodes) that define the spatial extent o f the physical space under consideration are located in a two-dimensional Euclidean physi­ cal space that is characterized by some form o f spatial heterogeneity (or anisotropy). These three points might be intersections o f a network involving transportation corridors o f various forms (rivers, highways, ravines, etc.), or peaks or valleys i n contour maps o f any kind (topographical, economic, or any other type). A n y social stock (say, human settlement activity) moving into the triangle is subjected to normalized net repulsion from or attraction to forces the three nodes o f the triangle exercise on i t . Before taking a look at the details o f these forces, the geometry o f the problem w i l l be addressed. In a three-dimensional space the three normalized forces form a vector lying w i t h i n an equilateral triangle in R (fig. 11.1). This equilateral triangle can be transformed into an orthogonal triangle with unitary orthogonal sides by a mere projection into R ( f i g . 11.2). The original three points can repre­ sent, as already stated, any arbitrary Euclidean triangle in R space (see fig. 1 1 - A l a in app. A ) . Such an arbitrary triangle can be transformed into a Euclidean right-angle isosceles triangle (see fig. 1 1 - A l b in app. A ) ; under the transformation, proportionality in distances o f any point inside the triangle to its nodes is preserved. This problem and its associated proposition are pre­ cisely stated, demonstrated and proved in appendix A . 3

2

2

2

Hence, any triangle in R Euclidean space can be represented by a rightangle triangle w i t h unitary orthogonal sides ( f i g . 11.2) and by a one-to-one correspondence, shown in appendix A , the orthogonal triangle can be trans­ formed into an equilateral triangle o f the type shown in figure 11.1. A vector in R identifies the location o f a settlement in the transformed space, where the point associated w i t h that location is given by the coordinates (normalized distances) from the point o f interest associated with the origin. 2

Let us now revisit and broaden the interpretation o f the universal discrete

252

Chaos Theory in the Social Sciences

X

X2W

2

P

Fig. 11.1. T h e phase p o r t r a i t o f t h e m a p for the one-stock, threel o c a t i o n case

2

Fig. 11.2. T h e phase space i n a n orthogonal triangle

map o f relative dynamics. Since on the horizontal and vertical axis one measures the transformed and normalized distance between the origin (an arbitrary point among the three, say point P in fig. 11.2) and one o f the other points o f interest (P, or P ) , then [1 - *,(<*)] and [1 - x (t)] represent the normalized proximity o f the settlement to the other two points o f interest, correspondingly. Whereas the distance x , ( f ) is proportional to a net repulsion force from the origin jettisoning the stock toward point P,, [ 1 - x , ( 0 ] is a distance proportional to a net attraction force pulling the stock toward point p.3

2

2

3

In R , one can form an equilateral triangle based on forces of net repulsion, the triangle at the northeastern section o f figure 11. A 2 i n appendix B . Further, an antitriangle exists that corresponds to the original net repulsionbased triangle. This antitriangle can be constructed from measures o f prox­ imity based on the equivalent forces of net attraction, and it is the shaded triangle found at the southwestern part o f figure 11.A2. I n appendix B certain analytical properties o f these two triangles are given. Location as a Response and Repelling Forces

to

Attractive

A unit o f human settlement is assumed to have already been located at some point w i t h i n the space outlined by the nodes o f the equilateral triangle in R (fig. 11.1). Thus, it identifies some original state (or initial perturbation) o f the system, and it is responsible for triggering a sequence o f subsequent spatial allocations, that is, a spatio-temporal chain o f settlement activity. 3

Subsequent allocations are the result o f either endogenous growth or net in-migration, although this is a subject outside the scope o f this chapter. A temporal sequence o f spatial allocations can be thought o f as a sequence o f

Cities as Spatial Chaotic Attractors

253

points where new activity by the same stock locates. This is a process o f spatial colonization. The model could also be interpreted as simply represent­ ing a nomadic process, that is, a temporal sequence in the location o f the same stock unit. Interpreting the universal map in physical space, one obtains a process that generates spatial form as a single frame m, m = 1,2, . . . ,M. Each frame corresponds to a particular set o f parameter values, initial conditions, and number o f iterations (time periods), all o f them constituent components o f the phase space. B y consecutive changes in any o f these elements, one obtains a process that includes a temporal sequence o f spatial frames (changes in the spatial form, or phase portrait). I n particular, a change in parameter values or initial conditions can result in a new frame, m, potentially characterized by a qualitatively different dynamic; that is, a different phase portrait may emerge as a result o f a bifurcation. This change constitutes the evolution in spatial form principle. In carrying out such changes in parameter values, a condition can arise according to w h i c h the initial state o f the new frame is the end state o f the last frame. This represents a last-point-out-first-point-in condition. A series o f frames, M, that does not obey this last-point-out-first-point-in condition is a series that contains frames where the slate has been cleaned at the end o f each one and the initial state has been reset. When each frame, m, corresponds to a particular set o f parameter values, frames change when parameter values change. H o w frequently these parame­ ters change must be related to the time horizon, T , o f each frame, m. One could suppose that the more frequently on the average a parameter changes, the shorter the average length o f the r ' s . m

m

3

2

By projecting the study's areal triangle from R into R , quantities x (0) and x (0) become the physical location coordinates o f the first settlement (initial condition) (fig. 11.2). In cases o f quasi-periodic or nonperiodic spatial formations the initial location o f the first settlement in the phase portrait is significant. It loses its significance in case o f periodic or fixed-point spatial forms i f the average length o f the time horizon, T, is long enough. t

2

The next human settlement in the temporal sequence o f spatial alloca­ tions (chains) is assumed to occupy a particular point in the two-dimensional physical space as a one-time-period delayed response to the conditions laid out by the previous allocation, and in proportion to a force o f net attraction toward points P, and P from point P (the origin), given the original perturba­ tion. Stated differently, the next settlement locates at a point at some distance away from the origin proportional to the current net repulsion force along the t w o axes. 2

3

The above is the locational rule underlying the physical configuration o f the universal map o f relative discrete dynamics. In the example that follows,

254

Chaos Theory in the Social Sciences

only units o f human settlement (say, an individual, a family, or any other unit) are thought to be allocated on the basis o f this rule. In case one wishes to allocate at each time period a different quantity o f a single human a c t i v i t y — for example, a varying quantity o f human population—then stock size must be configured in the formulation; then the formulation w i l l include two state variables, one depicting location and the other stock size, as in the case o f two stocks and three locations mentioned above. I f one wishes to consider additional stocks—for example, capital and/or wealth allocation i n these spatial dynamics—then this is easily accomplished by adopting a configuration capable o f accommodating additional stocks. Such configurations w i l l not be elaborated, however, in this chapter. Only the note w i l l be added that such configurations supply the comprehensive treat­ ment o f all land-use models found in the standard literature o f spatial model­ ing. Land-use models involve dynamics, either directly or indirectly, by way of long-run static (equilibrium or disequilibrium) solutions obtainable through some form o f an iterative process. The magnitude o f the net repulsion force from the origin along the horizontal axis (and equivalently the size o f the net attraction force toward P,) is given by F,(r)

= AJ {t)f {t)fp); x

2

t = 0 , 1 , . . . ,T,

(17)

where A , is the level o f relative environmental and access (due to the topogra­ phy and relative location w i t h i n the landscape) favoritism toward location P, and is constant over the time horizon, T, under consideration. Equivalently, F (t) is also specified. 2

Functions f depict location-specific and relative temporal comparative advantages due to current interdependencies as a result o f the current locational choice o f a unit o f human settlement; they are given by the specifica­ tions Infp)

= a

XJ

\x\xp);

j = 1,2,3;

(18)

where coefficients depict net repulsion force elasticities o f response to normalized distance o f the human settlement from three points o f interest (nodes) in reference to point P,, and also i n reference to a numeraire location. Exponents (coefficients), [a], act as weights o f (sensitivity to) nor­ malized distance o f the settlement from the origin along the t w o axes. They identify positive ( i f a > 0) or negative ( i f a < 0) cross- or interspatial external effects (externalities) in reference to the numeraire location. Coeffi­ cients [a] differ partly because o f the heterogeneity o f physical space. These weights (elasticities) do not change w i t h i n the time horizon, T, although they may change faster than those in A . Parameters a, could be construed as (j

y

Cities as Spatial Chaotic Attractors

255

ecological (comprehensive) prices directly responsible for interaction among sites i and j . The magnitude o f these coefficients, individually and in their sum, is o f considerable interest. I f an exponent is in absolute value greater than one, then this implies considerable sensitivity in the connectance among two nodes appropriately weighing the distance in question, and in reference to the numeraire node location. In contrast, in the case o f a magnitude less than one, relative insensitivity must be present. In each F , i f the sum o f the exponents exceeds one in absolute value then one might consider the force F to be under increasing returns; whereas i f the sum is less than one then the force F is under decreasing returns to scale. This horizontal sum depicts the effect o f all nodes upon a specific node. Moreover, i f the exponents are summed vertically, then the effect a node has upon all nodes is captured. Whether the sum is negative or less than one in absolute value is also o f interest. I f the sum is negative, then a competitive association must be inferred between the node in question and its rivals (the other t w o nodes); on the other hand, i f positive, then a cooperative association must be present. I f the sum is less than one in absolute value, then one is confronted w i t h a relatively m i l d interaction among them. It is stressed that these characterizations apply to the nodes and not to the actual location o f the human settlement. Further, it is underscored that spatial agglomeration or déconcentration is not a force under these specifications (that is, a cause for or against concen­ tration o f human settlements in space). Rather, it is the end result or the impact from these spatial forces o f net repulsion or attraction present w i t h i n the triangle at any time period, /. Both o f these statements are to an extent at variance w i t h currently prevailing theories o f settlement formation. The problem thus is to derive a temporal sequence o f unit human settle­ ment allocations w i t h i n the triangle, due to either migration or endogenous growth, and subsequently analyze their form. A number o f cases w i l l be discussed next, indicative o f the variety in form o f spatial agglomerations one can obtain using this generic algorithm. Periodic, Quasi-Periodic, a n d Chaotic Spatial S e t t l e m e n t s In what follows one must keep in mind that the phase portrait represents transformed actual space. First, the case o f a very efficient specification o f the map w i l l be shown. It has only one or two parameters that are not either zero or one (in absolute value) and is capable o f generating periodic and quasiperiodic (in the case o f a single parameter), as well as chaotic (in the case o f two parameters) form. Then, a more complicated and less efficient specifica­ tion o f the map w i l l be shown. Its phase portraits w i l l be derived in a very

256

Chaos Theory in the Social Sciences

small neighborhood o f the parameter space, where various types o f chaos are present. The case demonstrates the encompassing behavior o f the universal map in its capacity to generate and replicate human settlement variety ( i n the shape o f individual frames, or phase portraits) and their evolution.

Efficient Versions of the Map Next, t w o sets o f examples are presented. I n the first set, the case o f periodic movement is shown in a specification o f the map involving only one parame­ ter being neither zero or one in absolute value. H o w this periodic map is transformed into a quasi-periodic one is demonstrated by a very short trip along the dimension o f a single parameter in parameter space. The second case presents the shaping o f spatial form as either m u l t i r i n g , quasi-periodic, or chaotic motions and it involves a two-parameter specification o f the map.

Periodically Derived Forms First, consider the specification o f the universal map o f discrete relative 2

dynamics configured as a three-location, one-stock model in R and specified in the most efficient manner. A n example o f such a model is given by the parameters 0

0

[a] =

-1 .

-.1111 -1

1

0 1 0

A = .

" 1" 1 . 1 _

where the initial state is neutral (JC,-(0) = .333 . . . , i = 1,2,3). The iterative process under this specification is as follows: In

xi(t +

In x2{t

+

1) = - I n

F(t)

1) = - I n jc,(r) -

.1111 In x2(t)

+ In x3(t)

-

In

F(t) (20)

In

x3(t

+ 1) = In JC,(0 -

F(t) = 1 + x2(t)' where xx(t)

and x2(t)

In x2(t)

ilux3(t)/xt(t)

+

In

F(t)

Xl(t)/x2(t),

are plotted in the phase portrait. I n this specification there

is only one parameter, a22,

that appears to affect the system's dynamics.

Location-node "one" is the reference or

numeraire

site. The returns to scale i n

all elements o f F ( i . e . , all three corners o f the triangle) are

not exactly

approximately but

equal to zero, and the environment is neutral to all (as all elements

o f A are equal to one). The dynamics are shown in figure 11.3, panel (a) and

la)

(b) Fig. 11.3. T w o e x a m p l e s d e m o n s t r a t i n g f o r m a t i o n of spatial h u m a n s e t t l e m e n t s (see t e x t f o r p a r a m e t e r specification): {a) o n e - p a r a m e t e r case: 4 - p e r i o d cycle; ib) t w o - p a r a m e t e r case: quasi periodicity.

258

Chaos Theory in the Social Sciences

represent a four-period cycle spiraling from the center o f the triangular space (the initial state) toward four particular points very close to its sides. This example may be modeling the formation o f very stable, highly dense, small in area, fortified frontier cities o f the past. Then, consider the very close neighborhood case o f specifications, i n ­ volving t w o parameters: 0 a] =

0

0

-.77..

-.33.

1

1

-1

0

A

" 1" 1

=

.

. 1 .

where the initial state is x,(0) = .333 . . . , / =

1,2,3. This example can be

thought o f as an evolution o f the first specification through a change in only two

parameters, namely a

and a .

2l

In this case the returns to scale are

22

exactly equal to zero at all locations (nodes), while the environment remains neutral. The phase portrait is shown in figure 11.3, panel b , and it represents the formation o f a quasi-periodic motion (a rotating nine-period cycle) ap­ proaching (but never falling on) the three sides o f the triangle including its three nodes. Such a spatial form may be seen i n , among other cases, trafficways and traffic patterns o f modern day metropolises. It is again noted that the shape shown in figure 11.3, panel b is in transformed physical space. Quasi-Periodicity

and Chaos in an Efficient

Specification

Consider the f o l l o w i n g efficient specification o f the universal map o f discrete relative dynamics, involving t w o parameters and configured in its threelocation, one-stock case and interpreted as a physical space allocation o f a single stock in R 0 [a]

-1

= .

1

2

0

1

0

-1

0

-1

a

A

=

10"

.

33

.

3

1

-

where the neighborhood o f varying the parameter a

is 1.15 < a

33

where the environment is no longer neutral (A specification, the vector in R

3

+ 1) = - I n F(t)

In x (t

+ 1) = - 6 . 9 0 7 8 -

In x (t

+ 1) = In

2

33

< 1.2 and

< 1). According to the above

( f i g . 11.1) is

In (t Xl

2

In x (i) x

-

In x (t)

-

2

In F(t) (23)

3

F(t)

= [(1 +

JC,(0

-

In x (t) 2

10- )/(x,(0* (0)] 3

2

+ a

33

In x (t) 3

+ [(x,(/)A: (0) 33 a

3

-

In F(t)

(x (f))]. 2

Cities as Spatial Chaotic Attractors

259

2

The physical space, in R and in figure 11.2, is given by plotting x {i) and x (t)- Under the above specifications, the environment is handicapping pointnode P , and the comprehensive access (interaction elasticity) depicting pa­ rameter a reveals the presence o f internal increasing returns at the o r i g i n , point P , as a > 1. The returns to scale are negative at P ( F ' s sum o f elements o f [a] is —2), whereas they are positive (but less than one) at P . x

2

2

33

3

33

2

2

3

Two phase portraits from this example are shown in figure 11.4. In figure 11.4, panel a, the case o f a multiring spatial form o f settlements is shown away from node P . Such form can be detected in the location and areal size o f a number o f cities concentrated within a subregion o f the triangle under consideration, thus producing some form o f spatial dualism. 2

Figure 11.4, panel b , demonstrates a regime o f spatial form generated out o f a chaotic motion. This can be seen i n , among many other cases, the suburbanization phenomenon, whereby a number o f initially isolated settle­ ments are gradually j o i n e d by one (or more) spatially expanding core(s). The t w o cases shown in figure 11.4 can be also construed as a set o f t w o successive frames involving a transformation o f spatial form through a trip in parameter space at a very small neighborhood o f a , as the intralocational returns to scale there decrease very slightly. Evolution o f spatial form is depicted by this transformation. It is noted that the chaotic form in figure 11.4, panel b , has fractal properties. 33

In both o f the cases shown in figure 11.4, the initial conditions are not important w i t h regard to the formation o f the final form when vintage is suppressed; however, they are extremely important to the sequence involved in generating the final form. Nonefficient Specifications The f o l l o w i n g cases are presented as indications o f the variety o f quasi- and nonperiodically derived spatial forms embedded in the universal map i n R , as the map loses in efficiency. They are all found in a small neighborhood o f the parameter space o f a five-parameter specification o f the one-stock, threelocation configuration o f the map at the point 2

la] = .

0 .28 -1.11835

0 -1 -2.28

0 «23

-1

A =

1 1 .10" . 5

In figures 11.5 and 11.6 some spatial forms are shown, as phase portraits o f the dynamics corresponding to a set o f parameter values for a in the range 0.6 s a s 1. Figure 11.5 contains four cases o f quasi-periodic motion, whereas i n figure 11.6 four cases o f nonperiodic (chaotic) dynamics are supplied. 2 3

2 3

Fig. 11.4. T w o examples of spatial h u m a n s e t t l e m e n t f o r m a t i o n (see t e x t f o r parameter specification): (a) m u l t i p l e ring structure ( a = 1.188 . .. ); ib) chaos ( a = 1.177 . . . ). 3 3

3 3

Cities as Spatial Chaotic Attractors

261

Transformation o f a smooth ring into an edged but continuous line i n R , which in turn is changed into a set o f ringlets and back into a sharply edged quasi-periodic structure is shown in figure 11.5, as the interactionenhancing coefficient a linking points (nodes) P[ and P increases. Spatial forms i n the shape o f rings are not frequently observed in human settlement formation processes. These must represent fragile spatial forms, not lasting long time periods. 2

3 1

2

Bifurcations involving various types o f chaos are shown in figure 11.6, as the elasticity, a , increases further. The chaotic spatial allocations form a Cantor-type dust cloud, w i t h varying densities i n space. A l l these chaotic attractors have a fractal Hausdorff dimension (see Schuster 1988, 55) greater than one. Their mathematical properties are topological rather than simply algebraic, and they are: (1) sensitivity to initial conditions; (2) topological transitivity; and (3) dense periodic points (Devaney 1989, 169). 3 1

In the transformations shown in figures 11.6, panels (a) through (d), a number o f chaotic spatial allocations and their changes are depicted as one travels through parameter space. Specifically, figure 11.6, panel (c) picks up a short-lived spatial form, whereby spatial consolidation and an increase i n densities is an intermittent event. Intermittent events are novel aspects o f dynamical behavior revealed through mathematical chaos; they point to the possible presence o f many but not long-lasting conditions o f approximate periodicity found in a sea o f chaotic motions on trips through parameter space. Chaotic dust clouds look very similar to spatially distributed human settlements, particularly those found i n early historical and prehistorical sites. They also seem to mimic land-use patterns o f preindustrial or early industrial cities. Such forms are the outcome o f laissez-faire-type market processes, where the influence o f governmental action and urban design is minimal i n individual social motions and actions. The final, and maybe the most important, point to be made from these examples is that recognizable spatial patterns (of human settlements in this case) are equivalent to detectable order in chaos, as emerging from interde­ pendent nonlinear spatial dynamics. Spatial forms o f later vintage may thus simply be remnants o f order in chaotically shaped earlier settlements. For instance, waves and differential settlement densities can be depicted in the examples found in figures 11.6, panels (a), (b), and (d). Conclusions Mathematical chaos has made considerable inroads into social sciences, rang­ ing from marginal effects in economics to possibly basic contributions in geography. A brief commentary has been presented on the possibilities and obstacles currently present in further applying dynamical analysis in social

f

\ \ (c)

id) Fig. 11.5. Four examples i n v o l v i n g m o r e c o m p l e x f o r m s of spatial settlem e n t s g e n e r a t e d b y q u a s i - p e r i o d i c m o t i o n (see t e x t f o r p a r a m e t e r speci f i c a t i o n ) : (a) a = .61 (regular r i n g ) ; [b) a = .671 (edged r i n g ) ; (c) a = .68 (13 r i n g l e t s ) ; (d) a = .70833 (ring). 3 1

3 1

3 1

3 1

266

Chaos Theory in the Social Sciences

fields of inquiry. W i t h i n this context, a universal map o f discrete relative spatial dynamics, developed in the socio-spatial field, has been briefly dis­ cussed. A n extension i n the capacity to interpret the universal map o f discrete spatial relative dynamics has then been presented. Appropriately transformed physical space is seen as the phase portrait o f the iterative process. A number o f quasi-periodic, nonperiodic, and periodic dynamics are considered as the underlying mechanism giving rise to a variety o f physical forms as spatio-temporal chains o f a time series o f population allocations. A n example is given, involving human settlements o f different vintages, as allocated w i t h i n a specified area o f physical space confined by three points o f interest. Evolution i n the physical form o f settlements can then be studied as a case in bifurcating behavior o f nonlinear interdependent dynamics. Statistical testing and verification o f such model forms against actual human settlement formations at various regions and subregions o f the globe and at various epochs seems to be a necessary next step. Tests o f significance involving both locational and temporal proximity o f such chains to actual sequences o f human habitats (or other events occurring in a confined space, ranging from epicenters o f earthquakes along fault lines to points o f riots in urban areas and the allocation o f buildings w i t h different vintage along central city roads) could considerably enhance this effort. It is pointed out, however, that one ought not have high expectations i n such statistical attempts for verification when chaotic regimes prevail, beyond looking for rough qualita­ tive indications. A number o f reasons are given for this. Single-frame dynamics, corresponding to a unique set o f parameter values and initial conditions, are discussed as g i v i n g rise to physical form corresponding to a particular chain o f spatially distributed human settlements and its evolution. However, superimposition o f frames (simultaneously incor­ porating phase spaces corresponding to a range o f parameter values and/or initial conditions) could also be considered. Superimposition o f spatiotemporal chains is an interesting component i n the definition, testing, and interpretation o f socio-spatial dynamics (see Dendrinos 1991a). I t underlies issues o f structural stability o f spatio-temporal forms. Future research along these lines seems quite promising, indeed.

APPENDIX The

A: T H EG E O M E T R Y

OF LOCATION

IN R

2

problem in this appendix is to define a transformation among two

triangles,

triangle A B C in figure l l . A l ( a ) and triangle A ' B ' C in figure l l . A l ( b ) , such that there is a particular one-to-one correspondence between any two points (points D and D')

within them. The particular transformation under consideration is that specific

Cities as Spatial Chaotic Attractors

267

A

Fig. 11.A1. A n a r b i t r a r y t r i a n g l e (a), a n d its t r a n s f o r m a t i o n (6)

ratios of measures are preserved. Given the audience of this book, simple geometry will be used to demonstrate the statements, instead of matrix algebra and topology. For any arbitrary point in triangle A B C , say point D , one can identify the following two ratios (parentheses stand for measure or length): ( B E ) / ( E C ) = a , and ( A D ) / ( D E ) = b. With these two ratios alone, one can uniquely identify the location of point D in the triangle A B C . One can now show that: (1) measures ( B D ) and ( D F ) obey a specific ratio that only depends on the quantities a and b, as do the measures ( C D ) and ( D H ) ; and (2) these ratios are independent of the form of the triangle A B C , say the sine or cosine of each of its three angles or the length of its sides. In other words, given any two arbitrary triangles, A B C and A ' B ' C , to any arbitrary point D in A B C one can assign a unique point D ' in A ' B ' C so that certain ratios are preserved.

268

Chaos Theory in the Social Sciences

T H E O R E M . The ratios (BD)I(DF) PROOF.

Draw line F G parallel to B C . Then ( G D ) / ( D E )

(GF)/a(EC) (AD)[(£> +

and (CD)I(DH) are functions of only a and b.

=

(AG)/a(AE).

l ) / b ] , and b =

Further,

(AE) =

(AD)/(DE)

=

(AD) +

=

(GF)/(BE)

(DE) =

[(AG) + (GD)]/(DE)

=

=

(AD) + (AG)/(DE)

+

( A G ) / a ( A E ) . Thus, b = 6 ( A G ) / ( A D ) + ( A G ) / a ( A E ) and ( A G ) =

+

l / a ( A E ) ] . From the above one obtains: ( A G ) = (AD){[a(6 + 1)]/[1 + a(b + 1)]}.

b/[b/(AD)

Now, one also has: ( B D ) / ( D F ) = ( D E ) / ( G D ) = ( A D ) / & ( G D ) ; since ( G D ) = ( A D ) (AG)

= (AD){1 -

[a(b + 1)]/[1 + a(b + 1)]} = ( A D ) / [ 1 + a(b + 1)], then the

previous expression becomes ( B D ) / ( D F ) = [1 + a(b + l)]/b.

QED.

The proof that the other ratio, ( C D ) / ( D H ) , is also a function of a and b alone is identical to the above and thus omitted. Hence, for a unique point in A B C specified by the ratios a and b, one can locate in any other triangle, including the isosceles orthogonal triangle of figure 1 1 . A 2 , a unique point preserving certain ratios that are functions of only a and b. Instead of working with any triangle A B C , one can, as a 3

result, work with A ' B ' C , that is, the phase portrait of the universal map in R as 2

projected in R .

APPENDIX AND

B: T H E REPULSION

ITS ATTRACTION

TRIANGLE ANTITRIANGLE

In the one-stock and three-location configuration of the universal map of discrete 3

relative dynamics, the phase portrait is contained in the positive domain of R and confined by the shaded triangle at the upper right hand side of figure 11.B1, since 2,.jr,(f) = 1, i = 1,2,3; t = 0 , 1 , . . . , 7 . Such is also the case for the physical space interpretation of the one-stock, threelocation configuration of the universal map. In this case, the set of state variables, x, depict normalized distances away from the respective nodes, the end result of local net repulsion forces operating at the three nodes. Thus, this is a repulsion triangle. By forming the quantities ~x,{t) = [1 - *,(/)], /' = 1,2,3, one obtains normalized (but not relative) proximities to the respective nodes, so that their sum is equal to two. As a result, one can find the iterative dynamics of the normalized proximities on the shaded triangle located at the lower left-hand side of figure 11.A2. The triangle is defined by the condition 2,.-jt,.(f) = 2 , i =

1,2,3; t = 0,1, . . . ,T,

3

in R . This is an attraction of the antitriangle to the repulsion triangle shown above. Its sides are equal to twice the square root of two. In it, the phase portrait of the corresponding dynamics of attraction is contained; on this phase portrait the original

Cities as Spatial Chaotic Attractors

269

1-X,(t) Fig. 11.A2. A n equilateral triangle of repulsion forces in phase space and its attraction forces antitriangle

dynamics from the repulsion triangle are blown up (exactly twice in size) and pro­ jected. As distance from repulsion is given by the general (universal relative dynamic) expression x (t + 1) = F^lzZjFp), proximity is given by ~ (t + 1) = 1 - x£t + 1) = [ 2 , ^ F / r ) ] / [ S , F / / ) ] . It follows that t

[-xff

Xj

+ l)V[ (t Xi

+ 1)] = 2 ^ , . F / 0 ; i =

1,2,3

2 * , ( ' + 1) = [2,x,-(r + 1 ) ] [ 2 ^ , F / 0 I = 2. r

+

By specifying x,(t + 1) = 2x,(r + 1) = 2[1 - ~x,(r + 1 ) ] , then one has: +

2,- x,-(/ + 1) = 2,-*,-(f + 1) = 2. +

Quantity JC,-(0 may be viewed as a measure of local push equal to twice the magnitude of the local repulsive force operating at each node, i.

Part 4 Implications for Social Systems Management and Social Science

CHAPTER

12

Field-Theoretic Framework for t h e Interpretation of t h e E v o l u t i o n , Instability, S t r u c t u r a l Change, and Management of Complex

Systems

Kenyon B. De Greene

Policy making and decision making and other aspects o f the management o f complex systems are becoming increasingly difficult. Management philoso­ phies, approaches, and techniques were developed during simpler times. However, complex systems are dynamic rather than static, evolve or are driven into domains o f instability, and emerge into new structures. There is now a growing gap or loss o f fit between our systems-management capa­ bilities and the real w o r l d . Policymakers and decision makers must deal especially w i t h severely reduced time frames, consequences-of-action uncer­ tainty, and actions that produce diminishing returns. Fortunately, advances in systems theory provide a means for narrowing the gap and providing a better fit between systems management and reality. The theories have multifarious origins and different emphases. This question therefore arises: how can these theories be interfaced and interrelated? M y approach is to use the integrating framework o f field theory. As has histori­ cally been the case with the development o f field theory in the several sci­ ences, field theory is not so much a theory in itself as a framework for theoretical orientation and study. I have coined the term world system-field ( o f forces), which consists o f the dynamically interacting world societal field and the world ecosystem. A field describes the structure and behavior o f space-time and the placement o f objects in space-time. I n fields, myriad state variables depend continuously on the space coordinates. Einstein's field equations relate the properties o f spacetime to those o f matter, but thinking has proceeded quite far beyond Einstein's original rather static formulation that denied irreversibility and evolution (Nicolis and Prigogine 1989). A field can also describe the interactions be­ tween a system at a given hierarchical level o f the organization o f matter, energy, and information (e.g., a human institution) and its external environ­ m e n t s ) . A (turbulent) field can induce structural changes in the various parts 273

274

Chaos Theory in the Social Sciences

o f the environment itself (e.g., De Greene 1990a). A n d in the vertical dimen­ sion, a field can describe micro-macro interrelationships. A field can be viewed as a macrolevel, semiautonomous, collective structure (or order pa­ rameter) that exists over a relatively limited period o f time. Interactions at a more microlevel generate the field via positive-feedback mechanisms, and the field, reciprocally, through negative feedback, delimits the realm o f the pos­ sible at the level o f individual interactions (e.g., De Greene 1989; Weidlich and Haag 1983). Fields show differential sensitivity over time and space, and a fluctuation or perturbation that is usually damped may surprisingly trigger a violent and explosive reconfiguration o f the field. I t is difficult, perhaps almost always impossible, for human actors to know the present stability state o f the systemfield. Hence, policies and decisions must be developed that preclude to the extent possible the precipitation o f the system into an unsuspected realm o f instability and reconfiguration. O n the other hand, these policies and deci­ sions must not result i n rigidity and the stultification o f desirable evolution and structural change. Field theory provides a framework for dealing with these complex issues. This chapter is a presentation, w i t h considerable further thinking and updating, o f my research in this area over the past several years. In the sense of fields within fields within fields, a term sometimes used by systems theo­ rists, the top-level organizing approach is the evolution, stability/instability features, and structural change o f / w i t h i n the w o r l d system-field and the ef­ fects o f these dynamics on constituent systems. W i t h i n this context the next level o f organizational emphasis is the order parameter as a sociotechnical, techno-economic, and macropsychological (collective-cognitive) structure. The third level o f contextual approach is provided by the theory o f the eco­ nomic long wave or Kondratiev cycle/structure. W i t h i n this overall frame­ w o r k , constructs from a number o f systems theories are utilized. Examples include the critical threshold or bifurcation point, far from equilibrium, order through fluctuation, and irreversibility. The constructs are general purpose in physical, biological, and social science, contributing to laws o f nature, but they receive specific meaning only as properties o f the given field o f forces. Deterministic chaos is treated w i t h i n these contexts, but the calculational specifics w i l l not receive attention here because they are discussed more fully in other chapters o f this book. A n additional integrating theme o f this chapter is the wearing out and exhaustion o f given order parameters or paradigms and their replacement by paradigms that are better fitted to the new evolutionary situation. Under such conditions we must reexamine our very methods o f inquiry about nature and society, as w e l l as status quo policies and practices. Paradigm change pro-

Interpretation of Complex Systems

275

vides new, often surprising challenges, opportunities, and choices. The chap­ ter discusses important systems management consequences o f structural change and paradigm shift. The chapter concludes with some thoughts about the meaning o f the new theories to the social sciences. The approach is transdisciplinary. That is, basic constructs are sought from the physical and biological, as well as the behavioral and social sciences. Indeed, many o f the constructs had their origins in the physical sciences. Transfer o f constructs provides an exciting challenge. The bibliography at the end o f this book provides a rather comprehen­ sive literature on systems theories and models, definitions o f terms, and relevant examples. The reader should turn to these references for a fuller understanding o f the basic ideas and o f previous work. Space limitations preclude our starting from scratch in this chapter. E v o l u t i o n , Instability, a n d S t r u c t u r e of C o m p l e x S y s t e m s A l l systems evolve, although the rates o f evolution may vary over time both between and w i t h i n systems. The rate o f evolution is a function o f both the inherent stability o f the system and changing environmental circumstances. But no system can be stabilized forever. For the universe as a whole, an isolated system, time's arrow points toward greater and greater breakdown, leading to complete molecular chaos, m a x i m u m entropy, and heat death. For open systems, including the l i v i n g systems that are o f major interest to us and that interchange matter and energy w i t h their external environments, time's arrow points to evolution toward greater and greater complexity. Thus, the universe consists o f islands o f increasing order in a sea o f decreasing order. Open systems evolve and maintain structure by exporting entropy to their external environments. The concept o f evolution is finding increasing application in physics, chemistry, astronomy, and astrophysics, as well as, o f course, in biology, geology, and paleontology. But it appears that all too few behavioral and social scientists use the evolutionary framework, and that all too many theo­ ries, hypotheses, and empirical research efforts are directed toward the static, the cross-sectional, the linear, the equilibrium seeking, the stable, the revers­ ible, and the structurally constant. These efforts, as interpreted here, operate w i t h i n a prevailing but exhausted paradigm. This chapter and this book repre­ sent attempts to move beyond this paradigm. In organic evolution, as summarized by Neo-Darwinism, two principal processes are involved. First, random mutations lead to genetic differences among organisms. Second, natural selection tends to enhance the likelihood

276

Chaos Theory in the Social Sciences

of genes that produce adaptive characteristics. Some parts o f the genome may be more susceptible to mutation than are others, and environmental factors may also affect mutation rates. But any given mutation is considered to be independent o f any adaptive or survival value imparted to the organism. Whether Neo-Darwinism provides a complete answer to organic evolution remains to be seen, and the issue continues to be debated. Alternative expla­ nations are often dismissed as harking back to Lamarckism and teleology. But it seems intuitively unlikely that the organic evolutionary system would func­ tion open loop; that is, w i t h no feedback between effect and cause. D i s s i p a t i v e - S t r u c t u r e Theoretical I n t e r p r e t a t i o n s Following dissipative-structure theory (Nicolis and Prigogine 1977, 1989; Prigogine and Stengers 1984), the evolution o f open systems can be inter­ preted as follows. Movement o f the system away from equilibrium, associ­ ated w i t h some internal irreversible processes, increases the rate o f dissipation as measured by the entropy production. Instability, triggered by nonequil i b r i u m environmental conditions, leads to further dissipation and entropy production; this in turn leads to the appearance o f further instabilities. Farther from equilibrium, the probability increases that the system, with its internal processes, is unstable w i t h respect to given fluctuations. This concept o f evolutionary feedback regards energy dissipation as the driving force o f evolution. I t is characterized by nonequilibrium conditions leading to the system's crossing a critical threshold. Beyond this threshold the system becomes structurally unstable with regard to the fluctuations, which leads to increased dissipation and, in a positive-feedback loop, change i n the threshold. Thus, there is an acceleration o f irreversible evolution over time. Each new stage o f organization has the potential for further evolution. A n important aspect o f dissipative-structure theory and o f its and other interpretations o f evolution is the emphasis on the complementarity o f stochasticity and determinism (or, put another way, chance and necessity, nov­ elty and confirmation). Fluctuations (e.g., genetic mutations) arise randomly. Far from a critical threshold, called a bifurcation point, the larger system tends to express average behavior; that is, to follow the law o f large numbers, with the damping o f fluctuations. But near a bifurcation point, the fluctuations may self-amplify or cross-amplify (autocatalysis and cross-catalysis) via posi­ tive feedback to produce a nucleation. The nucleation may then enter into conflict w i t h the larger system or external environment. The latter may still try to squelch the g r o w i n g nucleation, but sometimes the nucleation prevails and becomes a new self-organized system. Dissipative-structure theory is also called order through fluctuations. Self-amplification is an example o f non­ linear behavior, a topic widely discussed throughout this book.

Interpretation of Complex Systems

277

Logistic E v o l u t i o n and t h e Constraints of Occupied Space A further kind o f evolution, logistic evolution, has been studied by dissipative-structure theorists (e.g., Prigogine and Stengers 1984), as well as others. The evolution o f species can be interpreted as the filling or creating o f ecological niches by successive species, each o f which is better fitted than was its predecessor. The well-known mathematical model, the logistic differential equation, which defines exponential growth limited by a carrying capacity or saturation level, expresses population growth w i t h i n the niche. But the carry­ ing capacity need not be fixed, so it is the sequence or family o f logistic functions (or better, logistic functions followed by domains o f diminishing returns) that is o f the greater interest in this chapter. The question must be raised, however, as to how much more morpho­ logical diversification is possible, once most biospace has been occupied. Put another way, as a system-field achieves structure, does not this structure now constrain further differentiation? W i t h regard to the evolution o f arthropods, most divergence occurred i n the Cambrian period, early in the evolution o f multicellular animal organisms. A rigid evolutionary constraint has precluded further diversification o f body plans. A l l possible body plans had become exhausted by the end o f the Cambrian, and further evolution has involved only convergences and reversals. B y the end o f the Cambrian, the limits to d i ­ vergence had been reached. The reasons for this constraint are unknown. (See the Technical Comments, "Cambrian and Recent Morphological Disparity," Science, 258, 1816-18, 11 December 1992.) Societal evolution, as interpreted in this chapter, utilizes the ideas just summarized. The fluctuations can be scientific discoveries, inventions, tech­ nological and social innovations, and great persons who appear at the right time and place in history. Logistic evolution with diminishing returns is ex­ pressed as the evolution o f successive Kondratiev cycles/structures. The satu­ ration and constraining impacts o f collective cognitive (macropsychological) space and the exhaustion o f ideas and practices, at least occasionally and temporarily, are emphasized. Order Parameter as Macropsychological S t r u c t u r e M u c h o f science is concerned w i t h the interrelationship between the micro­ scopic and the macroscopic. Evolving systems show feedback between macro­ scopic structures or collective fields and the events o f individual interactions at the microlevel. Macrostructures or fields emergent from the microlevel in turn modify the individual interactions at each stage o f irreversible evolution. This is a vertical perspective, like that o f the familiar hierarchy/emergence

278

Chaos Theory in the Social Sciences

theory. For systems management purposes, it should be noted here that things often look normal at the macrolevel o f the established order parameter, but that at the microlevel things are seething w i t h incipient change. Preceding a bifurcation o f the macrostructure, the system has been reorganized via longrange correlations (see below). A field i n the horizontal sense at any given time reflects the competition between the stability provided by communication (and information) and the instability that can follow fluctuation and nucleation. The result dictates the threshold o f stability. The threshold o f stability reflects the interplay among forces. I f the fluctuating area lies, say, below some critical point, the area w i l l regress; i f it lies above, the fluctuation and nucleation can spread across the entire field or take over the entire system. The field need not be limited to two dimensions (physicists, for example, provide both t w o - and three-dimensional models for many systems that show phase transitions.) A n order parameter (De Greene 1989, 1993; Haken 1983; Nicolis and Prigogine 1989; Weidlich and Haag 1983; Wilson 1979) is such an outgrowth of micro-macro interactions. It is a macrostructure or macrovariable that emerges along with a great reduction in the number o f degrees o f freedom o f a system. The emergence o f an order parameter can also be viewed from the horizontal perspective, but the vertical sense receives more emphasis in this chapter. The S t u d y o f Phase Transitions The term order parameter arose i n the study o f the physics o f the equilibrium phase transitions mentioned above. It was used to describe a number o f phenomena that are the properties o f collectivities o f subatomic particles, atoms, or molecules, rather than o f the sometimes randomly ordered particles themselves. Order typically emerges at a critical threshold. Examples o f physical order parameters include spontaneous magnetization, correlation length, the properties o f metal alloys, density differences between the liquid and gaseous phases o f a fluid, and concentration differences. The strong fluctuations that characterize these phase transitions take place in the orderparameter field. It is not necessary to know the precise form o f the strongly nonlinear interactions. Collective behavior arises at a statistical level, and only probabilistic predictions can be made as to the behavior o f any given element. The system may display, as a function o f fluctuations o f state near the critical point, pockets o f structure or order embedded in regions o f disorder embedded in still greater domains o f order, and so o n . Well above a critical point (say the Curie point o f temperature that is associated w i t h magnetiza­ tion), the system's elements are arranged essentially randomly, and the system shows only short-range order. A s the critical point is reached from above,

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larger-scale order begins to emerge. A t the critical threshold itself, discon­ nected patches o f order expand to infinite size, but fluctuations o f lesser scales still remain. A critical point is a discontinuity, above and below which the system is qualitatively different (e.g., paramagnetized and magnetized). Associated w i t h the description o f the instability o f fields and the emer­ gence o f new structure w i t h i n fields is the concept o f correlation. I f a single element in the system is perturbed, the disturbance may propagate, neighborto-neighbor, across the entire field. Distant elements, now structured in the same way, have become correlated. Correlation length indicates the maxi­ mum distance over which correlation can be determined. Near the critical threshold, the correlation length grows rapidly. Long-range order has emerged out o f the same short-range forces. A t the critical point itself, the correlation length becomes infinite. Phase transition has not yet taken place, but the system is now hypersensitive to very small perturbations, the effects o f which can now nearly instantaneously explode across and engulf the entire system. B i f u r c a t i o n a n d Order Parameter Under nonequilibrium conditions, the amplitudes o f the long-range correla­ tions increase with distance from equilibrium. Even before bifurcation at the macrolevel, the system has achieved a prestructure via the long-range correla­ tions. The phase as well as the amplitude o f a bifurcating branch can be an order parameter. Indeed, the very phenomenon o f bifurcation can be de­ scribed in terms o f the order parameter rather than in terms o f the state variables originally present. Larger numbers o f equations can describe the interactions among several order parameters. A s suggested above, order pa­ rameters can characterize the transition o f a system from an uncorrelated state to a correlated state far from equilibrium. Near to a bifurcation point, a multivariable system may be describable in terms o f a limited number o f collective variables or order parameters (Nicolis and Prigogine 1989). According to synergetics also, the nonlinear, coupled order-parameter equations may themselves permit bifurcations. The order parameter, the am­ plitude o f a bifurcating macro variable, slaves the individual constituents as some control parameter is changed. Equations can also show the competition among order parameters, and that elements obey that order parameter that wins the competition (Haken 1983). Societal a n d M a c r o p s y c h o l o g i c a l Order Parameters Closer to the emphasis o f this chapter, Haken (1983) offers this example o f an order parameter. Languages are the order parameters that slave the subsystems that consist o f individual human beings. The language changes very little over

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a given person's lifetime. After his birth, a person learns a language ( i . e . , he is slaved by i t ) , and over his lifetime he contributes to the survival o f that language. Order parameters are considered to be unstable modes that slave stable modes. This makes sense in the longer term because Homo sapiens change less rapidly than do the myriad languages the species speaks and has spoken. L i k e languages, I have proposed, the great religions, the scientific and technological paradigms, and the Kondratiev structures are order parame­ ters. M i n d is also an order parameter, emergent out o f a field o f forces in the brain as the brain interacts w i t h its internal and external environments. Be­ yond the scope o f this chapter is my contention that studying the brain as a w i r i n g diagram and the mind as a set o f logical rules and procedures is fundamentally incorrect; beware o f the claims for artificial intelligence and expert systems (De Greene 1991a, 1991b)! A societal field emerges out o f the myriad person-person and personmachine interactions o f a sociotechnical system; the field reciprocally then constrains the realm o f the possible at the microlevel. There may be a general principle o f short-range randomness and activation at the microlevel and longrange order and inhibition at the macrolevel. The randomness provides the innovation necessary to explore the field or space, and the inhibition main­ tains a collective stability over a considerable amount o f space-time. A n order parameter is a collective phenomenon. It is emergent beyond the obvious other structures o f organizations and societies. I n my writings the order pa­ rameter is macropsychological, and it shows such characteristics as collective m i n d , collective intelligence, collective perception, collective belief struc­ tures, and collective anxiety. Moods o f the time and overall social climates are expressions o f macropsychological order parameters. Group and organiza­ tional cultures, climates, and cognitive styles are order parameters that may encourage or impede problem solving, learning, competition, and adaptation. Like other evolutionary features, order parameters rise and fall as the stability dynamics o f the situation change. Experience shows that these cultures and climates cannot be directly managed, although they may be damaged or destroyed by management. We shall return to this topic at the end o f this chapter. The Kondratiev C y c l e / S t r u c t u r e in Recent Societal E v o l u t i o n The Kondratiev cycle has been classified as one o f the business cycles. Cycles of about three-to-four or four-to-seven, and ten, twenty, and fifty-five years mean duration have been identified. The first is the business cycle per se and the last is the Kondratiev cycle or economic long wave. Cycles o f 150 or more years duration have also been identified. A large literature has now been built up in this area (e.g., Berry 1991; De Greene 1988a, 1988b, 1989, 1990d,

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1992, 1993a, 1993b, 1993c, 1993d, 1994; Kondratiev 1984; Schumpeter 1939; Van Duijn 1983; Vasko 1987; Vasko, Ayres, and Fontvieille 1990). The term cycle, however, is insufficient. I view each cycle as an order parameter that fits into even longer order parameters, such as the hegemonic cycle, which describes the rise and fall o f nations ( v i z . , the great powers) and the life histories o f the Newtonian paradigm, our phase o f w o r l d civilization, and w o r l d civilization as a whole (for the last, see Tainter 1988). Hence, my use o f the term Kondratiev cycle I structure (and hegemonic cycle/structure, etc.). The Kondratiev cycle/structure can be viewed as a link between the study o f science and the study o f history. It is an important organizing prin­ ciple for the study o f the evolution o f sociotechnical or techno-economic macrosystems. The Three-Tiered S t r u c t u r e o f a M o d e r n Economy In all the sciences it is difficult to explain a system simply on the basis o f function or behavior. A n understanding o f structure is necessary. A modern economy can be viewed as three-tiered, w i t h the most basic and constraining order-parameter field placed for convenience on the bottom. Here we are not dealing w i t h individuals versus collectivities but rather with collectivities o f different levels o f profundity. The top tier is the world o f mainstream eco­ nomics, o f prices, wages, money supply, inflation, simple supply and de­ mand, hiring and firing, and so on, and o f behaviors like the business cycle, per se. This is the w o r l d o f the recessions and recoveries that presently draw the attention o f policymakers, decision makers, and the media. The middle tier deals w i t h the means o f production and consumption—factories, equip­ ment, tools, kinds o f skilled workers, and model changes w i t h i n the prevail­ ing paradigm. The intermediary Juglar and Kuznets cycles arise here. This is no longer the w o r l d o f mostly the economist. Industrial engineers, behavioral and social scientists, and ergonomists play important roles here. The third and lowest tier involves the basic capital, energy, and technology structures o f society. I t is the world o f the eventual exhaustion o f capital and o f past investments and commitments, the wearing out o f one technology and its replacement by another (Schumpeter's creative destruction), and the substitu­ tion o f one energy source (and now informational structure) by another. The way o f thinking in society and the prevailing paradigm may become exhausted (De Greene 1993a, 1993c). This is the w o r l d o f the Kondratiev cycle/ structure. The Phases of t h e K o n d r a t i e v s Four Kondratiev cycles/structures have been identified, w i t h the first begin­ ning with the Industrial Production Revolution about 1785. We are now, an

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increasing number o f authors believe, in the depression phase o f Kondratiev number four, not in the state o f recovery from a recent business-cycle reces­ sion. Kondratievs are conveniently divided into phases o f recovery, prosper­ ity, recession, and depression. The fourth Kondratiev started around 1940 at the end o f the Great Depression o f the 1930s. The rising leg o f recovery and prosperity lasted until about 1970, when the system entered the Kondratiev phase o f recession. The Great Depression o f the 1990s probably began about 1989-90. Assumptions obviously help structure policies and decisions as well as the associated intermediary models. Most economists, policymakers, decision makers, and the media assume the continued truth o f the present para­ d i g m . But i f the w o r l d economy is in reality in a great depression that w i l l accompany the end o f meaningfulness o f this paradigm, then it is likely that the continuation o f present policies and the formulation o f new policies based on the o l d paradigm w i l l lead to counterintuitive results. Problems o f knowability and controllability arise. The Kondratiev phenomenon, with its proper­ ties o f nonequilibrium, nonlinearity, instability, and structural change, is far from transparent and far from conducive to personal psychological comfort. A business cycle is easy to understand and seemingly receptive to control ac­ tions. We shall return to this challenge to systems management later in this chapter. Features of a Kondratiev Phase of Depression A Kondratiev depression phase has a number o f identifiable features. The overall picture is one o f overcapacity, overextension, maturation o f industries, saturation o f markets, misalignment o f subsystems, and wearing out. Indus­ tries, from hotels and office space to aerospace, automobiles, and o i l , now show overcapacity. Services, from schools to health care, are overextended and cannot meet the needs o f constituents. Stock markets are bloated with speculator shares. Trade and internal deficits have mounted. The markets for the products o f a wide range o f industries, from ships to commercial aircraft to most computers, are saturated or nearing saturation. Products are cloned and typically distinguished by packaging rather than by inherent worth. M y r ­ iad factories and most infrastructure are worn out. Maintenance is deferred. Labor has been replaced by capital, yielding an increased capital intensity. Unemployment has increased and remains recalcitrant to reduction. M a n y kinds o f jobs have been eliminated, never to return. Relatively good jobs in manufacturing have often been replaced by relatively poor, minimum-wage jobs ("McJobs") in the services; and the proportion o f involuntary part-time workers has increased. Wealth has generally flowed in the direction o f the already well-to-do. Just as before the great depressions o f the 1890s and the

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1930s, the number o f millionaires has greatly increased. Productivity shows mixed patterns, but white-collar and professional productivity, including that in computerized operations, is largely stagnant. Organizational bloat, conse­ quent to increased administrative overhead, is held to account for much o f this stagnation. Whereas there are local, regional, and national differences among the individual symptoms, the overall picture is worldwide. Indeed, differences are a source o f further instabilities, as is attested to by continued deterioration in Eastern Europe and the countries o f the old Soviet Union and by large-scale migration from poorer regions to regions that are better off. See De Greene (1990c) for a discussion o f factors common to capitalistic as well as to communistic collapse. Explanation of the Kondratiev Cycle/Structure The Kondratiev cycle/structure can, in the cyclic sense, be explained in terms o f the expansion and contraction o f the basic capital sector (see the threetiered economy discussed above). In the phase o f recovery, new investment opportunities are perceived and investment decisions made, perhaps to a considerable extent on the basis o f innovations that have "bunched" late in the phase o f depression. New technologies and energy sources are developed. Factories are built, new kinds o f jobs are created, and people are hired. The overall economy, that is, the sociotechnical or techno-economic macrosystem, grows exponentially as these elements and subsystems reinforce one another via positive-feedback loops. But eventually environmental and carrying-capacity constraints (social, economic, natural, etc.) and saturation effects operate as negative-feedback mechanisms to slow the rate o f growth past some inflection point. Growth then slows to an asymptotic value, and may become negative in a domain o f diminishing returns. This is a simple verbal description o f logistic/diminishing-returns growth. A family o f these curves can depict the four Kondratievs. See the discussion o f logistic evolu­ tion earlier in this chapter. Considerable evidence has suggested the bunching or clustering o f inno­ vations in the phase o f depression (e.g., Schumpeter 1939). I interpret the phase o f recession as a time o f very rigid societal and individual thinking, when great efforts are expended to reconfirm the status quo. But in the phase o f depression, it appears that society has given up and is ready to try just about anything to remedy the misery. A technological innovation is defined as the first commercial application o f a scientific discovery or invention. During the phase o f depression, entrepreneurs take the latter and develop the former, helping to stimulate recovery and the beginning o f a new cycle as noted above. Just as important to true progress, however, are the social and institu-

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tional innovations and societal leaders that often appear to lag behind the new technological capabilities. As mentioned earlier, scientific discoveries, inventions, innovations, and great persons are mutations in societal evolution. The dynamics o f evolution, as interpreted in terms o f dissipative-structure theory and order-parameter theory, are held to apply here. Each Kondratiev structure is bounded by i n ­ stabilities. Each Kondratiev structure is an order parameter or macrovariable, initially emergent out o f interactions at a more microlevel and sustained as a field that offers both opportunities and constraints. The order-parameter field defines the realm o f the possible. During much o f its fifty-five or so years duration, the field functions as a mean field and average behavior dominates, w i t h the damping o f fluctuations. Finally, in the search for a basic causality, the order parameter is viewed as primarily macropsychological, and collective perceptions, intelligence, anxiety, and so on shape the investment decisions and other features discussed above. D e t e r m i n i s t i c Chaos a n d Other Nonlinear Behaviors In deterministic chaos (not to be confused w i t h molecular chaos), the output wave forms or the phase portraits o f a model look stochastic or unexpectedly complex even though the causal mechanisms o f the iterated model are com­ pletely deterministic. Deterministic chaos is associated with very strong feed­ back and therefore w i t h strong nonlinearities. I n deterministic chaos the system is extremely sensitive to initial conditions, and trajectories w i t h m i ­ nuscule initial differences diverge exponentially fast. Small random fluctua­ tions are amplified over several iterations and eventually take over the system. Deterministic chaos can be observed in discrete-system simulations involving only one difference equation (e.g., the logistic difference equation) and in autonomous (not externally forced) continuous-system simulations w i t h three or more coupled differential equations (e.g., the Lorenz weather model). Deterministic chaos has been identified in real-world simple systems like pendulums and lasers. However, its existence in real-world complex social and societal systems is still questionable and a matter o f debate. To the extent that models are used for prediction, chaos does place a definite limit on their usefulness. The identification o f chaos may be useful in distinguishing be­ tween normal and pathological states. Also important from my perspective is the extent to which deterministic chaos represents a kind o f instability o f fields that plays a major role i n systems evolution. Chaos arises from selfamplification and cross amplification, perhaps o f the hypercycle k i n d , o f subsystems. These subsystems could, for example, be social and techno­ logical or demographic and natural environmental. Indeed, the related dy-

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namic behavior o f turbulence has been interpreted as arising at the boundaries of m o v i n g subsystems. Consideration o f the routes to chaos such as the Feigenbaum scenario, use o f Poincare maps, and calculation o f Lyapunov exponents is beyond the scope o f this chapter. These matters are dealt with in other chapters o f this book. However, a word o f caution is in order. Critics point out, for example, that calculation o f the Lyapunov exponents itself is insufficient, in the absence o f some explanatory theory, for the identification o f chaos. Beware the i m ­ pulses to calculate prematurely and to overcalculate. In system-dynamic computer simulation models, instability and deter­ ministic chaos are often associated with negative-feedback loops possessed o f relatively high loop gain together with appreciable delays, which permit the amplification o f the small, random fluctuations. Alternatively, in simple models o f the Kondratiev cycle, instability and chaos arise out o f positivefeedback loops (Mosekilde, A r a c i l , and A l l e n 1988). A n example is the selfamplification o f the capital sector mentioned above. See also Forrester 1977. Berry (1991) reports deterministic chaos in Kondratiev series o f annual growth rates in wholesale prices. He generated a phase space by plotting each relevant year, t, against the previous year, t — 1. Rosser (1991) and Zeeman (1977) also discuss nonlinearities and discontinuities in economic behavior, particularly o f the catastrophe theory k i n d . Gilmore (1981) provides a mathe­ matical treatment o f catastrophe theory, an important topic that space limita­ tions place beyond the scope o f this chapter. Later in this chapter, we shall discuss the evolution o f information struc­ tures (macropsychological order parameters) in relationship to both the Kondratiev cycle/structure and the several kinds o f attractors, including cha­ otic attractors, and their behavior in phase space. On Paradigm Change A paradigm can be defined as a set o f interrelated Weltanschauungen: theo­ ries, models, practices, findings, explanations, values, beliefs, and feelings that characterize a given culture at a given time. This definition is much broader than most. A simpler definition might equate a paradigm to a theory or a model. Some authors (e.g., Kuhn 1970) describe different paradigms w i t h i n a scientific field like physics and present seminal interpretations o f the nature o f scientific revolutions. Because science and society or culture interact over long periods o f time, and because scientists have all too human limita­ tions, it is necessary to provide the broad definition given above. The main paradigms dealt with in this chapter are the Newtonian para­ d i g m , which has come under increasing scrutiny and criticism (e.g., Checkland 1981; De Greene 1993b and the many references therein; and Prigogine and

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Stengers 1984), and the "new" paradigm (which might be called the paradigm o f evolution and reconfiguration). The Newtonian paradigm may be viewed as part o f the macrostructure o f information and knowledge that emphasizes and usually tolerates only the following: rationalism; reductionism; parts isolated from wholes; detached objectivity o f observation and measurement and separation o f the observer from the observed system; simple causality; logical, steplike but iterative analysis; deduction o f rules, procedures, and algorithms; m a x i m u m use o f numbers; emphasis on average behavior; equilibrium; fixed, inviolable laws; reversibility; denial o f variety and ambiguity; denial o f subjectivity; and con­ vergent focus on the correct answer or solution. The Newtonian paradigm is best fitted to a static or slowly changing w o r l d o f stability and structural continuity, not one o f evolution, instability, and structural change. In contrast, the new paradigm encompasses nonrationality, nonlinearity, mutual causality, nonequilibrium, irreversibility, stochasticity/determinism, uncertainty, opportunity and choice seen in fluctuations and apparent noise. Moreover, the dominance o f exceptions near critical thresholds, the genera­ tion and maintenance o f variety, structural change, divergent thinking, and the recognition that there can never be eternal truth and reality but only different perceptions o f such, also represent crucial elements o f the new paradigm. F o l l o w i n g the broad definition above, paradigms structure the very way that science is conducted. Karl Popper emphasizes testability over truth because truth may never be reached. Further, scientific theories can never be proven or validated through experimental tests; they can only be disproven or falsified. Science thus cannot be reduced to the formal, logical method o f the positivists. Popper considers conceptual structures like Marxism and psycho­ analysis to be outside the realm o f science because they cannot be tested and falsified. I t appears to me that this orientation rules out o f scientific inquiry all the really interesting systems—the large-scale, complex, unstable, evolving, self-organizing, structurally changing systems—of concern in this chapter. Because o f instability, chance, fluctuation, bifurcation, and chaos, each such complex system is unique. Both organic and societal evolution are summa­ tions o f chance occurrences over time. Even i f testing o f large-scale l i v i n g systems were possible, the social objections and social and environmental dangers and costs w o u l d most likely preclude such activity (see, for example, many o f the criticisms o f the Strategic Defense Initiative). In some cases small, experimental studies (e.g., the comparison o f small ecosystems w i t h and without a given species) can be suggestive, but even here the generaliza­ tion o f the results requires caution. A great deal o f research in the behavioral and social sciences, both in the laboratory and in the field, has followed the Newtonian paradigm. One o f the

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major purposes o f this book can be the rethinking o f the adequacy o f this kind of scientific method. A t the present time many scientists are taking a new look at the very purpose, meaning, and practice o f science. This discussion is pursued further elsewhere (De Greene 1993b). A t the level o f societal policy making, decision making, and design, different paradigms or subparadigms are needed, depending on the apparent linearity, equilibrium state, stability, and predictability o f the system (De Greene 1987, 1990d, 1991b). These features can also be related to the phases o f the Kondratiev cycle/structure. The Challenge t o t h e M a n a g e m e n t of C o m p l e x S y s t e m s A sociotechnical or techno-economic macrosystem is a dissipative structure in the sense that high-quality inputs (energy and matter) are converted to lowquality outputs like heat and waste, w i t h an increase in disorder and entropy. W i t h i n this overall process, o f course, low-quality raw materials are con­ verted into high-quality finished products, but these themselves eventually break d o w n , yielding further entropy. This is one way to look at the product life cycle. The G r o w t h o f Exhaustion o f I n f o r m a t i o n / K n o w l e d g e at t h e End of t h e Kondratiev C y c l e / S t r u c t u r e To the extent that information is the negative o f entropy (see the Shannon and Boltzmann formulas), we can also speak o f an information life cycle and a growth o f exhaustion o f information as the end o f the cycle, identified by this author (De Greene 1993a) as the recession-early depression phase o f the Kondratiev cycle. A t the end o f a Kondratiev, entropy is high and information level is low. The progressive uniformization o f the system is associated w i t h the loss o f information. In the stages o f recovery and prosperity, information is increasing as the new cognitive space or field is explored, and disorder and entropy are decreasing as new designs are implemented. In the phase o f recession, when the field has been explored to its boundary, energy is dissi­ pated and information decreases as confirmation o f the status quo, w i t h the aid o f proliferating bureaucracies, becomes the established modus operandi. But eventually, reinforcement no longer works, real underlying instability arises, and the system starts oscillating, with the emergence o f fads and hysterias as society attempts to preserve and regain order. We note here that entropy has a material meaning as well as informational and strictly energy-based mean­ ings. The untoward consequences o f the degradation o f matter and energy by

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the economic process have been beautifully explicated by Georgescu-Roegen (1971). Nicolis and Prigogine (1989) conclude that chaotic motions necessarily imply that phase-space volume is expanding in certain directions and contract­ ing i n others. Variety and choice are continuously generated along the expand­ ing directions o f the motion, and predictability increases along the contracting directions. Thus, chaotic attractors, which possess asymptotic stability, can be generators and processors o f information. Lyapunov exponents can be used to describe the mean rates o f expansion or contraction. It is tempting, therefore, to divide the Kondratiev cycle/structure into four epochs that are offset some­ what from the four phases, w i t h chaotic attractors associated w i t h innovation in the late depression and recovery phases, limit-cycle attractors and oscilla­ tory behavior associated w i t h both the early prosperity and late recession phases, and point attractors and equilibrium-seeking behaviors characterizing the late prosperity and early recession phases around the inflection point o f the logistic curve. M o d i s and Debecker (1992) have identified chaotic wave­ forms at both the beginning and the end o f the logistic function. The overall Kondratiev system w o u l d evolve in the following manner: from chaotic attractor to limit-cycle attractor to point attractor to limit-cycle attractor to chaotic attractor, w i t h successive growth o f exhaustion o f informa­ tion and innovation (De Greene 1993a, 1993c), and so on. Evolution w o u l d show movement from nonequilibrium to equilibrium to nonequilibrium, and so o n . Because o f the irreversibility o f structural change, o f course, the specific structures w o u l d not be the same. Moreover, as noted earlier, the Kondratiev system overlaps or operates within still larger cycle/structures. In addition, the Kondratievs are bounded by a hyperbolic (or exponential) enve­ lope curve depicting overall sociotechnical evolution (De Greene 1982). Fur­ ther, similar curves depict changes in natural environmental variables, so that the overall picture is likely one o f humanity moving farther and farther from equilibrium and closer and closer to the absolute limits o f sustainability and stability, as they can realistically be perceived today. A n y thinking about a fifth Kondratiev has to consider the major structural changes in the larger sociotechnical and natural environments o f the present Kondratiev (De Greene 1993d). Potential exploratory space, in both the physical and abstract senses, may be much less than humanity might hope. Let us look more closely at the growth o f exhaustion o f human cognitive models and artifacts, along w i t h effective information and knowledge, that reflects the increasing loss o f divergent thinking and the increasing dominance o f convergent thinking at the macropsychological level o f a nation or an interlinked group o f dominant nations (De Greene 1993a, 1993c). Instead o f nation(s), the words the economy, the sociotechnical macrosystem, and the

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world system-field, and so on, could o f course be substituted. Information and knowledge (information integrated and fitted into a context) are more than just flows in an information-processing system; it is better to think in terms o f evolving (and devolving) structures. I n the evolution o f a Kondratiev, users over time tend to select the more stereotyped forms o f information and k n o w l ­ edge. People seek the expected, the riskless, the tried and true. There is a great deal o f cloning o f designs, processes, products, services, and organiza­ tional practices, and a great deal o f bandwagon activity. Older organizations may lose their original purposes or, perhaps worse, may continue rigid, stereo­ typed behaviors derived from original purposes when dynamic, evolution­ ary changes in the external environments have long since passed them by. Information and knowledge embodied in the original designs gradually dissi­ pate as the designs wear out. Nevertheless, in modern societies equilibrium structures like machines and technology i n general can perpetuate the exis­ tence o f an o l d system long after it w o u l d have expired naturally. A n d the collective mind or consciousness possesses tremendous inertia, introducing time lags into attempts at innovative systems management. Such powerful and well-known organizations as the United Nations, the World Bank, the Inter­ national Monetary Fund, N A T O , and N A S A may fall into this category. Experience shows that it is quite difficult to j u m p start a renaissance in ingrained, postmature organizations. In addition, there is the vast morass o f collapse that is the relic o f the old Communist w o r l d . This suggests a w o r l d o f worn-out ideas, designs, and practices; a w o r l d awaiting transformation (De Greene 1993b). Following Shannon (1948), total certainty is equivalent to no informa­ tion. New information can be interpreted as reconfirming and strengthening or stabilizing existing structures. Following Thorn (1975), reduction o f informa­ tion to its scalar measure (given in bits) means loss o f almost all the meaning that might be imparted to a message. Also, no information exchange is pos­ sible between static forms. A message that conveys what is already known provides no change in knowledge. A known answer provides no uncertainty and no information. Further, information is lost when a system has diffused into its environment, and it is no longer possible to distinguish the system from its environment. For example, an innovation may be viewed in a figure/ ground relationship; but after the innovation has diffused throughout the soci­ ety, figure merges into ground. Information and knowledge are more than flows in informationprocessing systems. Information and knowledge are themselves structures that show collective or coherent behaviors at a more macrolevel emergent out o f interactions at a more microlevel. The resulting cognitive field then helps regulate behavior at the more microlevel. The field can spontaneously and

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irreversibly reconfigure into different populations or complexions that display different information and energy qualities. See again our discussion o f order parameter. S y s t e m s M a n a g e m e n t at a T i m e of T r a n s f o r m a t i o n a l Change Organizations and societies thus seek continuously to reconfirm themselves, which leads to a selective use o f information and a dangerous narrowing o f perception (De Greene 1991a). The evolution o f new knowledge systems can be stifled in the search for continued equilibrium. The economy, for example, aims toward an equilibrium state dominated by confirmation. Predictability is thought by increasingly conservative mainstream policymakers and decision makers to be enhanced by attempts to restructure and therefore to control the overall environment and to maintain a static security. Control or apparent control reduces anxiety. Yet when the information/knowledge structure evolves into a stage o f saturation or even diminishing returns following diffu­ sion into the surroundings, most people do not perceive the incipient struc­ tural changes. The present epoch o f diffusion saturation o f innovations re­ sponsible for previous economic growth began about 1970, around the inflection point o f Kondratiev number four. Policymakers, planners, and deci­ sion makers, and their analyst advisors, continue to apply the same tired, old logic and management methods to the increasingly evident symptoms o f the reconfiguring field. As discussed above, an attractor may over time degrade from chaotic to l i m i t cycle to fixed equilibrium point. This process can yield a cer­ tainty, confirmation, and predictability that no longer possess information/ knowledge. O l d , decrepit systems may be more easily captured by an equi­ l i b r i u m attractor. Contrariwise, the emergence o f a new self-organizing collective m i n d , which can provide a driving force for the next (now the fifth) Kondratiev, can follow the smallest o f stimuli or fluctuations. Nucleations can then spread explosively across the field. Newly formed collectivities can stand out in a figure/ground sense from established structures and can thereby encourage a growth in knowledge. The insight learning interpreted by the Gestalt psychologists as a sudden reconfiguration o f the cognitive field may represent capture by a chaotic attractor and its associated basin o f attraction. Referring back to the popularly understood meaning o f information, massive investment in this technology has yielded no increase in the produc­ tivity o f information workers. From 1980 through 1989, office equipment increased from 3 percent to 18 percent o f U . S . capital equipment. But the average productivity o f the workers was about the same as in the 1960s in spite o f the huge increase i n the amount o f computer power. Lack o f improve-

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ment appears to be due to a considerable extent to increases in organizational bloat. See the discussion and references in De Greene 1990b. Further, computerized information systems can constrain and channel thinking, so that innovation is lost. The latest technologies can rigidify manage­ ment functions like communications, coordination, and control (De Greene 1991a, 1991b). Such reinforcements can be part o f a larger evolution toward greater standardization and reduced variety during the Kondratiev phases o f recession and depression. The Office o f Technology Assessment (cited i n De Greene 1990b) comments that organization and management styles appear to be becoming more and more identical across such sectors and institutions as banking, insurance, large farms, supermarkets, construction, textiles, hospi­ tals, and automobiles. Kinds o f jobs and skills become increasingly identical, and responses to forces in the external environment like financial markets and w o r l d trade are increasingly standardized and stereotyped, as rationalized or optimized best fits diffuse and are implemented. Moreover, over the next decade or t w o , most applications o f new technology w i l l involve doing some­ thing familiar in a better way. These trends imply reconfirmation o f the status quo and growth without innovation. I n evolution, however, processes, mechanisms, elements, and inter­ actions are added or subtracted, and evolution selects for variety. Optimiza­ tion and improved efficiency make the organization more vulnerable as orga­ nization, environment, and their interrelationships change over time. The efficiency o f a single perspective or rationality may improve the organization only for a specific purpose and w i t h i n a limited time. These matters are persuasively argued further by A l l e n (1993). Each Kondratiev is characterized by a primary energy source, primary technologies, and primary institutional forms. However, any o f these features can spill over to the next Kondratiev. The railroads and the telegraph were primary technologies o f Kondratiev number t w o , and improved means o f organizational communication arose out o f the need to coordinate the far-flung operations o f the railroads. Internal communications, coordination, and con­ trol also changed greatly, and these changes spread to manufacturing around 1880. Yates (1989) discusses the rise o f the American system o f management (my term, see her subtitle) over the years 1850-1920 based on the philosophy o f system and efficiency. She states that managerial control is essentially management as we now know i t . The important emphasis here is that the transformation o f American firms was essentially complete by the end o f World War I , the end o f the phase o f prosperity o f the third Kondratiev. Subsequent advances in computer-communications technologies have rein­ forced the dominant managerial philosophy and organizational design and have enhanced the efficiency (but not necessarily the effectiveness) o f organi­ zational processes.

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The American system has been a dominant institutional form over part o f the second and essentially all o f the third and fourth Kondratievs. It has, o f course, spread around the globe. The myriad criticisms o f Tayloristic/Fordian work-system designs and their deleterious impacts on worker motivation, morale, j o b satisfaction, and productivity (e.g., De Greene 1982) can be noted here but are beyond the scope o f this chapter. The Japanese system appears presently to be at least a partial alternative to the American system, but this topic, which has also received a tremendous amount o f attention, is also beyond the scope o f this theoretically based chapter. I n keeping w i t h other interpretations made herein, it is my belief that the American system o f management is part o f the exhausted machine model o f humanity, part o f the overall Newtonian paradigm. New theories, methods, observations, and find­ ings should not be distorted and made to fit the old paradigm and outmoded designs. Finally, a dynamic field o f forces cannot be controlled or managed; but it can be perturbed, leading to surprising nonlinear, unstable, and reconfigurational results. These results could come about because critical thresholds were passed, the delicate balance among feedback loops disrupted, and the bound­ aries o f stability passed. On the other hand, reconfiguration is natural in evolution and can provide new choice and new opportunity. The distinction is management w i t h i n the field, not o f the field (De Greene 1990b). The chal­ lenge for Americans, and for humanity in general, is to try to induce climates of creativity, small at first, that might generate desirable innovative fluctua­ tions that could trigger the emergence o f new order parameters better fitted to evolutionary reality. Note the absence o f deterministic language and the ab­ sence o f any implication o f ultimate human understanding and control. Concluding Remarks and R e c o m m e n d a t i o n s How valuable are chaos and nonlinear systems theory to the social sciences? The answer depends on the nature o f the driving force. I f these theories act as perturbations and fluctuations, driving a restructure o f social science, and i f they help generate new paradigm thinking, then the future can indeed be promising. I f , however, the new theories function just as new tools (like a new form o f regression analysis), then the social sciences may find that the excit­ ing and challenging aspects o f social reality have been usurped by the more dynamic, the more imaginative, and the more adventuresome, and that tradi­ tional economics, sociology, political science, and so on have become i n ­ creasingly irrelevant. The same k i n d o f situation pertains for the high-level systems practices of management (in this chapter policymaking and decision making in particu­ lar) and design. A time o f major transformation o f the existing force field

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always provides greater opportunity as well as greater challenge. A time o f major transformation in societal as well as organic evolution leads to emer­ gence o f new forms as well as extinction or submersion o f older or less adapted forms. The next few years may provide a window o f opportunity that may not return for decades, i f ever. This is because each force field is spatially and temporally unique, and because the magnitude and acceleration o f the intersecting environmental and societal forces is probably now the greatest in the experience o f humanity. I f , say, policymakers can see the opportunity o f chance is today's instability and transformation, then the chaos o f messes and problems may be, as a whole, amenable to a new and enlightened manage­ ment. I f , on the other hand, energy and resources are directed to reconfirming the status quo, then it is likely that these efforts w i l l compound present difficulties via acts o f omission and commission linked to bad timing and bad placement. The theory o f dynamic, reconfiguring fields having only temporary sta­ bility is antithetical to concepts o f absolute and everlasting truth and perma­ nency o f methods. The policy community must understand the cognitive/ emotional ties to such recently and presently powerful belief systems as humanity's place in nature, organized religion, communism, capitalism, de­ mocracy, the market economy, free trade, and so forth. It is, o f course, easy to argue for understanding. But (selective) understanding and not understanding can have apparent survival value to individual and society. Most o f us feel comfortable w i t h the familiar, the k n o w n , the tried and true. Most o f us build up elaborate systems o f rationalizations and systems o f denial to defend ourselves against perceived vulnerability and psychic insult. For example, most mainstream economists deny the limits to growth, deny that free-market practices can lead to environmental destruction and societal impoverishment, and indeed deny the existence o f uncontrollable situations. Because o f time lags associated w i t h growing up, education, and experience, it w i l l probably be the rare individual who understands and grasps the opportunities now emerging through chaos in the broadest sense o f the term. History shows that great leaders arise at times o f crisis. From a systems theoretic standpoint, the damping and stabilizing effect o f average behavior (the law o f large numbers) weakens as control parameters (e.g., human popu­ lation g r o w t h , atmospheric change, biodiversity loss, and inequitable distri­ bution o f wealth) approach bifurcation points. This time around, because o f the intensity o f planetary crisis, the leaders may emerge too late, may not fit the field, or may be o f the kinds that present thinking would find objection­ able. Therefore, using perhaps overworked terms, we must be anticipatory and proactive. T h i n k i n g must be radical and revolutionary. But again this is easier said than done. There is no simpleminded, linear advice that can be offered like that o f the evolved-away-from, left-behind, equilibrium-centered

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w o r l d o f the 1950s and 1960s: use risk analysis, perform a decision analysis, optimize, reduce the scope o f concern to the immediately manageable. There is no meaningful list o f steps that can be taken. There are no real-worldsensitive panaceas that can be offered. A field o f forces is imprecise. As system dynamics, dissipative-structure theory, and catastrophe theory all show, and as experience indicates, a field o f forces can surprisingly swallow policy and decisional perturbations, or it can be destabilized by such perturba­ tions. What happens is fortuitous and depends on the t i m i n g and placing. The task confronting us now is a highly individualistic and lonely one, but poten­ tially a most creative one. Leaders and changers w i l l be those persons who can overlap their cognitive/emotional models, themselves internal fields o f forces, w i t h the w o r l d system-field or a constituent part o f that field. In the latter case, an energized local field could excite, or even inflame the entire field. Yet in spite o f the formidable nature o f evolution and reconfigurational change in the w o r l d system-field and in spite o f the great difficulty in mapping external reality onto the m i n d , certain general advice can be offered to those w i l l i n g and able to accept it: recognize human limitations, do not compart­ mentalize your thinking, avoid linear thinking about cause and effect, respect and learn from history, expect the system to behave counterintuitively, expect surprises and anticipate the opportunities that surprises can provide, expect policy and decision actions to assume new meanings when imbedded in a field context, think instability, instability, instability. As emphasized here, rare persons w i l l intuitively understand these matters, but this advice should help others. Finally, we can note that there are implications for education. Once again the questions o f appropriateness in a field and o f direction o f causality arise. A t times o f instability and chaos, rigidities such as programs, plans, curricula, learning objectives, fixed assignments, tests, and administrative control are likely to be indifferent at best and harmful at worst. Educational needs, expressed as fluid learning experiences, w i l l evolve and emerge from the pioneering behavior o f those who can sense and move w i t h the dynamics o f the field. I f and when things quiet d o w n , some further formalization o f methods and designs w i l l probably be desirable. But formalization should never again be allowed to reach the scope o f the Newtonian paradigm o f science, culture, and society.

CHAPTER

13

S o c i a l S c i e n c e as t h e S t u d y o f C o m p l e x S y s t e m s David L . Harvey and Michael Reed

Despite three decades o f development in the physical sciences, the social sciences are only now coming to grips with deterministic chaos and its worldview. Several explanations have been given for this hesitance. Positivists, appealing to Comtean doctrine, conceptualize the sciences as constituting a developmental hierarchy. Mathematics, physics, and astronomy, envisioned as the most mature o f the sciences, are at the top o f this hierarchy, while the immature social sciences are situated on the bottom rungs. Given their relative youth, it is only natural, from the positivists' perspective, for the social sciences to master scientific advances at a slower rate than their elders. After all, they do not yet possess the mathematical or technical sophistication o f the older sciences and are not prepared, therefore, to take advantage o f their most recent innovations. A second and more practical account o f this scientific lag appeals to the formidable mathematics o f deterministic chaos theory. It claims that only a handful o f applied mathematicians working within the social sciences are capable o f fully grasping the fundaments o f the new science. Since these scholars stand on the margins o f the human sciences, it is no wonder that the chaos paradigm has made little headway in these disciplines. A third and final interpretation o f the slow reception o f chaos theory into the social sciences underscores the fact that in the last thirty years science itself has become déclassé in many circles. The cultural revolution o f the sixties identified science with the repressive and dehumanizing tendencies o f modernism. Influenced by this radical humanism, many social scientists re­ jected quantification w i t h a vengeance and have opted for a hermeneutic method in their research. In the process, they have steadily retreated into the deconstruction o f texts, or into the study o f postmodern and postindustrial mentalités. As is often the case, none o f these accounts can be fully refuted. Positiv­ ists defend their position by citing a long history o f the social sciences borrowing from the physical sciences. They have trouble explaining, how295

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ever, w h y chaos scholars reject the classical paradigm o f Newtonian science, and actually seek the company o f humanists when exploring the full implica­ tions o f the new science o f chaos. Similarly, those who cite the intimidation o f mathematics ignore the significant headway chaos theory has already made in econometrics and in urban ecology. A n d , finally, those underscoring the re­ cent antimodernist drift in the social sciences fail to realize that chaos theory has its cultural roots in the revolutionary sixties (Gleick 1987; Loye and Eisler 1987) and that the theory—at least in its more popular renderings—has even been used to justify the subjectivism and antirationalism o f the postmodernists (Young 1991). W h i l e all three positions give credible accounts o f chaos theory's slow penetration into the social sciences, all ignore a fundamental ideological problem that often emerges when a new methodenstreit erupts. Most parties to such debates i m p l i c i t l y assume that there is only one social science and only one scientific method. This paper challenges this monological assumption and argues that no single method can fully appropriate the manifold complexity o f social life. I t suggests, instead, that social scientists have a plurality o f methods from which to choose when they research a subject, and it is their responsibility to select the one method that best fits the ontological contours o f the problem they are studying. Hence, asking why the social sciences are slower than the physical sciences in adopting the deterministic chaos para­ d i g m , or w h y sociology, anthropology, and the cultural sciences are lagging behind econometrics and social ecology misses the mark. What should be asked, first, is whether the social phenomenon a scientist chooses to investi­ gate imposes ontological limits on the methods he or she can employ. In this essay we w i l l take up this latter question. We w i l l explore the circumstances under which research strategies employing the deterministic chaos paradigm can and cannot be deployed in the human sciences. I n particu­ lar, we w i l l look at how the ontological structure o f institutional life restricts when and where a given method can be applied. In exploring these issues, we w i l l steer a course midway between those positivists who would use chaos theory to revivify an exhausted scientism and those postmodernists who reject quantification "on principle." Our middle course is grounded in the philosoph­ ical work o f the British philosopher Roy Bhaskar (1978, 1989a, 1989b). Bhaskar's philosophy advocates a realist ontology and a modified scientific naturalism in the area o f epistemology. As we w i l l show, Bhaskar's "critical naturalism" is especially sensitive to the differences between the w o r l d o f nature and the social realm. It is this sensitivity to the structural differences that exist in the w o r l d that allows us to wed Bhaskar's realist ontology and epistemology to a dissipative systems approach. Such a wedding, we believe, can give rise to a new social scientific w o r l d v i e w — o n e grounded in the axioms o f deterministic chaos, and which is cognizant o f the profound differ-

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ences separating the social world from that o f nature. Such a dynamic realism is capable o f sustaining the particularity and plurality o f the social w o r l d while preserving rational canons o f scientific understanding. A n y hope we have o f realizing this new social scientific worldview is tempered, however, by our realization that the ontological structure o f social phenomena places objective limits upon the types o f methods that can be legitimately deployed when studying the social realm. Hence, the naturalism we advocate demands social scientists employ a methodological pluralism when studying institutional life from a dissipative systems perspective. I n line w i t h these reservations, we make two assumptions. First, w i t h the notable exceptions o f its judicious application in such areas as demography, social ecology, and urban place theory, the simple incorporation o f the deterministic chaos paradigm into social scientific research w i l l prove nugatory. Second, i f the actual mathematical models o f deterministic chaos and the concrete find­ ings o f the physical sciences have limited value in their direct application to the social sciences, they can still provide a rich heuristic base from w h i c h social scientists can w o r k . Our vehicle for exploring these theses is a matrix that matches modeling strategies used in the social sciences with the different ontological levels that characterize social systems. The matrix is constructed in three steps. We begin by identifying six modeling strategies currently used in the social sciences: (1) predictive models patterned after the methods o f classical mechanics; (2) statistical models commonly employed in the social sciences; (3) iconological models recently developed by mathematical chaos theoreticians; (4) structural models popular i n contemporary anthropology and sociology; (5) ideal type models utilized in comparative economics and sociology; and, finally, (6) historiographical models that concentrate on the ideographic description o f temporally ordered concrete events. The second step in con­ structing the matrix focuses upon the development o f a framework that reflects the ontological stratification o f the social world. Borrowing from Kenneth Boulding's (1968, 3 - 1 0 ) work on hierarchies o f complexity and Neil J. Smelser's (1963, 1-46) reading o f Parsonsian systems theory (Parsons 1961), we develop a model o f nested social system hierarchies that emphasizes their ontologically layered nature. This conception o f social systems sees them as consisting o f emergent layers that are ontologically distinct and irreducible. The third and final step in this process is the construction o f the matrix itself. After completing the matrix, we illustrate its use by assessing the relative efficacy o f each o f the six strategies for studying chaos. We make no exaggerated claims for this matrix or for the resulting typology. Ours is a first effort, and like all initial formulations it invites comment and revision. Rather than presenting a finished product, we offer the matrix as a tentative first statement to be tested and revised in practice.

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Moreover, our perspective is grounded in the assumption that a simultaneous convergence in several scientific fields is occurring around the new chaos paradigm (Reed and Harvey 1992). Before presenting the matrix proper, however, we must briefly describe the philosophical foundation o f our project, beginning w i t h a discussion o f Bhaskar's critical realist perspective. We then examine the dissipative systems perspective (Prigogine 1980; Prigogine and A l l e n 1982; Prigogine and Sten¬ gers 1984) that is at the heart o f our modeling/ontology matrix. Finally, we take up the problem o f social ontology, and in doing so move to the presenta­ tion o f the matrix itself. Philosophical A s s u m p t i o n s Bhaskar's critical realism begins with an immanent critique o f scientific method and scientific findings. That is, in order to philosophically explore the ontological foundations o f the w o r l d , he assumes that the methods and find­ ings o f science are essentially correct. Working from this presupposition he then asks what the ontological structure o f the real w o r l d must be in order for science to sustain its truth claims. I n this way he moves quite logically from questions o f scientific method, per se—how scientific knowledge is produced—to philosophical inquiry into the structures that underlie and orga­ nize the visible w o r l d . What are the results o f Bhaskar's immanent critique? First, science and its findings, despite the claims o f some o f its apologists, is a human product and bears the same indelible markings as all else that is produced by human hands. L i k e other human endeavors, science is a historically open and contin­ ually evolving enterprise. It has an ideological dimension, and its practices and findings reflect the cultural commitments o f the society that houses and nourishes i t . To this extent, science is a historically conditioned, transitive activity. But i f that were all there was to science, it would be indistinguishable from other ideological systems. Critical realism assumes that there is more to science than this, and it distances science from simple ideologies by insisting upon the obdurate nature o f the real w o r l d . For critical realism, science is anchored in an intransitive domain, in a w o r l d whose autonomous constitution stands independent o f the knowing subject and his or her parochial interests. It is the intransitive nature o f science's object that prevents its full incorporation into the partial ideologies that constitute a society's belief system. Because o f this anchoring in the external w o r l d , and the constraints that follow from that anchoring, scientific findings cannot be dismissed out o f hand as mere selfinterested assertions. The affirmation o f science's intransitive object serves as a grounding for the second result o f Bhaskar's immanent critique. He then begins a dissection

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of the interior o f that object by examining the commonplace observation that knowledge, especially in the physical sciences, possesses a cumulative, often progressive, nature. That is, the findings o f the physical sciences seem to build upon one another and to form a more or less irreversible compilation o f factual knowledge about the structure and functioning o f the w o r l d . Working from these mundane observations, Bhaskar deduces a second set o f ontological propositions about the w o r l d . Science's cumulative laying up o f k n o w l ­ edge from one generation to the next proceeds by progressively stripping away layer after nested layer o f reality. W i t h each new advance, science probes deeper into nature and uncovers ever more fundamental levels o f physical reality. I f scientific knowledge has this cumulative structure, then reality itself, Bhaskar argues, must be organized in a similar fashion. Moreover, this ontological layering forms a hierarchy o f open, nested structures so that, as science penetrates each new layer, it discovers the ontological base o f those entities and structures it has previously uncovered. For example, as science has penetrated deeper into the structure o f matter, the perspectives of, first, classical physics, and then molecular chemistry, gave way to quantum mechanics and the study o f the subatomic world. Upon deeper probing, quantum mechanics yielded to particle physics and the subse­ quent explosion o f a myriad o f even more fundamental structures o f matter. In the last t w o decades, the study o f these fundamental particles exposed a new stratum o f the w o r l d — t h e curiously constrained universe o f colored and flavored quarks. A t each step new levels o f material reality emerged, only to be displaced by even more fundamental entities and processes. Hence, the his­ tory o f physics and chemistry in this century and the last attests to the hier­ archical and nested structure o f physical reality. The world is composed o f layers o f matter emergently related and nested one within another. A n d this ontological deepening o f reality parallels the evolution o f modern science. The metaphor o f peeling away successive layers o f an onion is appropriate for describing this process. Moreover, whether we are talking about the w o r l d o f matter or the w o r l d o f social systems, this ontological nesting o f irreducible layers seems to apply w i t h equal force. This hierarchical ordering o f disjunctive layers is held together by loosely bounded processes o f determination and overdetermination, thus forming a multilevel system. I f this system did not exist, even in a loosely formed state, there could be no coherent body o f scientific knowledge, and certainly no common ground for communication between the various sci­ ences. This point has been made by P. A . Anderson (1972). He argues the existing scientific division o f labor points to an ontologically layered uni­ verse, not unlike the one Bhaskar assumes. Running the gamut from elemen­ tary particles to social structures, Anderson suggests that each layer o f reality is structurally unique, operates according to its o w n laws, and demands its

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o w n investigative protocols. I n explaining the origin o f these various levels, Anderson underscores the role played by broken symmetries. Symmetrybreaking mechanisms provide the ontological foundations for the emergence o f new levels o f reality from established levels. Symmetry breaking allows each new ontological level to organize itself around its o w n set o f irreducible principles. Since different principles regulate the activities o f each level, no level is reducible to those from which they have emerged. The resultant pyramiding complexities produced by the geometry o f broken symmetries is the source, therefore, o f our ever-deepening modern division o f scientific labor. Thus, Anderson explains and then schematizes this irreducible relation be­ tween the various sciences and their ontologically unique objects as follows: The behavior o f large and complex aggregates o f elementary particles, it turns out, is not to be understood in terms o f a simple extrapolation o f the properties o f a few particles. Instead, at each level o f complexity entirely new properties appear, and the understanding o f the new behaviors re­ quires research which I think is as fundamental in its nature as any other. That is, it seems to me that one may array the sciences roughly linearly in a hierarchy according to the idea: The elementary entities o f science X obey the laws o f science Y. X solid state or many-body physics chemistry molecular biology cell biology

psychology social sciences

elementary particle physics many-body physics chemistry molecular biology

physiology psychology

But this hierarchy does not imply that science X is "just applied Y . " A t each stage entirely new laws, concepts, and generalizations are neces­ sary, requiring inspiration and creativity to just as great a degree as i n the previous one. Psychology is not applied biology, nor is biology applied chemistry. (Anderson 1972, 393) Combining Bhaskar's a priori deductions and Anderson's supporting remarks, we can sum up the philosophical ontology employed in this paper by listing five fundamental assumptions concerning the ontological structure o f the everyday w o r l d : (1) the w o r l d is real in the sense that it exists and operates independently o f our knowledge o f i t ; (2) the w o r l d is ontologically layered,

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that is, it is made up o f structurally irreducible levels that form a loosely organized hierarchical system; (3) while not specified by Bhaskar, there is strong scientific evidence that symmetry breaking is central to producing this stratified complexity; (4) the progressive accumulation o f scientific k n o w l ­ edge points to an irreversible time frame through which the world develops; and (5) the emergence o f one layer from another suggests that the w o r l d is an open, evolving entity. I m p l i c a t i o n s o f Critical Realism f o r Social Scientific M e t h o d The above ontology entails certain methodological principles as w e l l . From critical realism's perspective, there is the real world and what we say about that w o r l d . Contrary to idealist epistemologies, our statements about the structure o f the w o r l d are not the world as such, and the two must never be confused. B y the same token, the empirical appearance o f reality should never be mistaken for those generative structures and mechanisms whose powers and limitations actually produce the world o f appearance. While experience can often be the point o f departure for scientific inquiry, empirical presence alone cannot give us access to the lawful structures that produce those appear­ ances. To penetrate to that more fundamental substratum o f causal forces, more than controlled observation is required. Just as Kant rejected Hume's empiricism, so critical realism rejects the claim that the experimental method, in and o f itself, can provide logically feasible laws. Causal explanations derived from empirical evidence are necessary, but never sufficient, for the identification o f lawlike principles. The same assertion holds equally for social scientific surveys, demographic analyses, or participant observer studies. The empirical w o r l d can reflect, but never give us direct access to the causal mechanisms o f reality. Bhaskar is by no means among the first to argue against the idea that classical prediction is the final arbiter for establishing scientific truth claims. He has, however, contributed significantly to our philosophical understanding of this problem by providing an ontological framework that establishes the limits o f predictive validity in open evolving systems. The universal claims o f prediction in such systems had already been seriously challenged by quantum theory early in this century (Peitgen, Jiirgens, and Saupe 1992, 11-14). Heisenberg's uncertainty principle denied that scientists could ascertain a dynamic system's initial conditions at the subatomic level. It was impossible to measure simultaneously and w i t h equal precision a particle's position and its momentum. As a result, classical mechanics' claim that given any system's position, momentum, and the dynamic laws, it could predict the system's future state was denied. By the midsixties, chaos theory had further limited 1

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the range o f classical causation's explanatory scope. By then, the science o f nonlinear dynamics had demonstrated that even i f a scientist could pinpoint a nonlinear system's initial conditions, there were many instances in which precise forecasting o f future behavior was all but impossible. O n l y under the constraints o f certain Lyapunov-specific conditions and w i t h the application o f various renormalizing techniques could chaotic systems be analyzed—and even then only under the most restrictive o f circumstances (Wolf 1986; Peitgen, J ü r g e n s , and Saupe 1992, 5 0 9 - 2 0 , 7 0 9 - 2 0 ) . Dissipative Social S y s t e m s Following Bhaskar's logic, what is now needed is a scientific worldview capable o f filling in the philosophical gaps. Such a worldview is already at hand, we believe, in the science o f nonequilibrium thermodynamics. This new science studies dissipative systems. Dissipative systems embrace a wide variety o f chaotically driven systems: cross-catalytic chemical processes that often seem to imitate life itself; physically and biologically constituted evolu­ tionary systems; and, as we w i l l see in the next section, social systems. Dissipative systems, first and foremost, are natural systems. They are materially and thermodynamically constituted entities. This means their inter­ nal structuration and development, as well as the processes by which they are born, evolve, and die, are regulated by transfers o f energy from their immedi­ ate environment. The most important feature o f the thermodynamic com­ position o f dissipative systems is their negentropic potentiality. Negentropy, or negative entropy, and its opposite, positive entropy, are irreversible thermodynamic processes. Positive entropy refers to the universal tendency o f thermodynamic structures to evolve irreversibly toward a stage o f maximum disorder called thermodynamic equilibrium. The most effective measure o f this increasing disorder is the progressive degradation o f the system's internal structure. Thus, by the time thermal equilibrium is reached, the energy mak­ ing up the system has been smoothly and equiprobably distributed throughout its boundaries. Thermal equilibrium was long thought to be the fate o f all living things and, indeed, o f the universe itself. In fact, the idea o f thermal equilibrium became a controversial staple o f popular culture i n the late nineteenth century as speculation arose concerning the eventual "heat death" o f the universe. In the last few decades, however, research into systems ordered by negentropic processes has changed our ideas about irreversible thermodynamic processes and the role they play in a wide range o f evolutionary phenomena. Ne­ gentropy, unlike positive entropy, occurs in a limited set o f circumstances, but when it does occur its ramifications are profound. As the name implies, negentropy represents a tendency that runs counter to that o f positive entropy,

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in that systems i n which it is produced are capable o f forestalling their descent toward thermal equilibrium. Negentropic processes are, therefore, the mate­ rial foundations for the growth and evolution o f thermodynamic systems. They enable dissipative systems, via internal metabolic mechanisms, to trans­ form free environmental energy into increasingly more complex structuration. But for dissipative systems to sustain their g r o w t h , they must not only increase their negentropic potential, they must also eliminate the positive entropy that naturally accumulates over time and that degrades the system's internal structuring. That is, as dissipative systems grow and become more complex internally, the price o f their increasing structural complexity is mea­ sured in terms o f their rising levels o f positive entropy. The latter is a natural waste by-product o f the process by which dissipative systems develop. I f allowed to accumulate over time, positive entropy w i l l eventually nullify the system's tendency toward increasing differentiation and w i l l move the system toward thermal equilibrium. Negentropic systems must, therefore, develop a means o f eliminating their accumulated positive entropy. This is usually achieved by the system transporting its waste energy to its immediate environ­ ment. For this reason, any long-lived dissipative system must fulfill two pre­ requisites i f it is to persist: (1) it must be able to convert free environmental energy into ever more elaborate forms o f internal structuration; and (2) it must transport thermal disorder into the environment. Dissipative systems are thus characterized by a dynamic tension between their ability to accumulate negentropy and their need to transfer their positive entropy into the environ­ ment. I f they can sustain this tension, then under proper circumstances they can achieve a state o f net negative entropy and persist. I f they cannot, they w i l l evolve to a state o f thermodynamic equilibrium. On the basis o f their ability to achieve or not achieve a state o f net negative entropy, then, we can analytically partition thermodynamic systems into two sets: those dissipative systems that are capable o f sustaining their internal growth and those evolving toward a state o f thermal equilibrium. Because dissipative systems can increase their negative entropy, they are capable o f evolution. Two o f their properties, in particular, allow this to occur. First, dissipative systems are peripatetic in the sense that they constantly seek new organizational states. Second, dissipative systems are capable o f achiev­ ing a far-from-equilibrium state that allows them to move to reference states so distant from thermal equilibrium they can transform themselves into more complex entities. Both properties, internal peripatetic boundary testing and far-from-equilibrium dynamics, are sources o f instability and, hence, o f po­ tential change. When these t w o instabilities combine, they give dissipative systems an evolutionary potential that differentiates them from other forms o f organization. Let us see how this is done.

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The internal source o f a dissipative system's instability resides in its boundary-testing propensities. That is, dissipative systems are constantly try­ ing to transform themselves—moving away from their present point o f equi­ poise to some alternative state. M o r e often than not, however, this boundary testing goes for naught since the environment fails to provide a sufficient energetic base to sustain an evolutionary transformation. Periodically, how­ ever, this internal boundary testing coincides with a sudden, sustained i n ­ crease i n environmental energy. Readied to change by internal boundary testing and suddenly perturbed by an environmental flux, the far-fromequilibrium potential o f the system can destabilize i t , force it to abandon its previous reference state, and begin to evolve to a new configuration. This evolutionary process begins with the appearance o f bifurcational behavior, that is, a fluctuating behavior that sends the system into an oscillat­ ing movement between t w o or more new points o f possible equilibrium. This cyclic oscillation indicates the dissipative system in question has destabilized and entered a chaotic phase. A t this point one o f two things can happen: the system can remain chaotic, oscillate ever more rapidly, and eventually destroy itself, or the fluctuation w i l l dampen as the system settles into a new configu­ ration. When this happens, we speak o f the dissipative system as having evolved: it has reorganized itself around a new reference point from which it w i l l once again begin its peripatetic boundary-testing behavior. This testing w i l l continue until a new perturbation o f sufficient force occurs, and a new conjunction o f internal and external conditions once again pushes the system into a new evolutionary trajectory. Dissipative systems are, hence, chaotically structured, far-from-equilibrium entities. Their evolution proceeds from two crucial preconditions. First, dis­ sipative systems exhibit what is called a sensitive dependence on initial condi­ tions, a property that is a cardinal indicator o f chaotic structures (Ruelle 1990; Devaney 1989, 4 4 - 5 3 ) . As we have seen above, the evolution o f dissipative systems requires a coincidence o f internal boundary testing and external per­ turbations to push it into a new evolutionary path. I f this flux were to impinge on the far-from-equilibrium system either a moment sooner or a moment later, this lag or acceleration could either suppress a possible evolutionary tendency or accelerate i t , sending the system into a path it might otherwise not have followed. This sensitive dependence on initial conditions is a key indicator o f deterministic chaos. Its dynamic presence is a major reason why evolving systems rooted in the cyclical and self-feeding dynamics o f deterministic chaos are unpredictable. It is noteworthy, therefore, that with the concept o f sensitive dependence on initial conditions we find a major point o f conjunc­ ture between our philosophical ontology and the scientific ontology o f dissipa­ tive systems.

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A similar convergence holds for the second crucial precondition—the process o f symmetry breaking. To say that far-from-equilibrium systems evolve, and that they do so through symmetry-breaking processes, is to state the same fact at t w o different levels o f abstraction. A t the first level, symmetry breaking indicates the crucial fact that far-from-equilibrium systems are not conservative entities. To understand what this means, let us imagine that the normal growth and evolution o f a system takes place in an n-dimensional space. We w i l l call this space, for simplicity's sake, a phase space. This phase space can be visualized as being structured by such dimensions as time and space coordinates, the degree o f its internal complexity, the nature o f the cycles governing the internal dynamics o f the system, and the total environ­ mental energy available to the system. A system is conservative when any evolution it undergoes takes place within the existing parameters o f its phase space. The growth o f conservative systems preserves the dimensionality o f its phase space. In that far-from-equilibrium systems are not conservative constellations, they do not preserve their phase space during their evolution. Instead, dissipative systems i n a far-from-equilibrium state are predicated upon bifurcational processes i n which new dimensionalities emerge to foster change. Put another way, when a system's evolution requires the breaking o f old symmetries as a conditio sine qua non for pursuing its o w n far-from-equilibrium path, then it is not enough to speak just o f the evolution o f an individual system; we must also recognize a second, global process—the evolution o f evolution itself. This second evolution is the concrete condition that separates the evolution o f far-from-equilibrium systems from all other forms o f change. The evolution o f evolutionary space allows us not only to differentiate between the evolution of the physical universe, molecular evolution, the evolutionary emergence o f new species, and, finally, the evolution o f dissipative social systems; it also reminds us that each different form o f evolution represents discrete moments of the same self-organizing, evolutionary process that categorically unites all dissipative systems into a meaningful category. Social O n t o l o g y : The S t r u c t u r e o f Dissipative Social S y s t e m s Dissipative social systems are a subset o f dissipative systems. This means much o f what is said o f natural dissipative systems is also true o f dissipative social systems. Hence, dissipative social systems have many o f the charac­ teristics o f natural entities, and, like them, they are grounded in the material w o r l d . Consequently, any paradigm that studies dissipative social systems must begin with an accounting o f the economic and ecological components that mediate between societies and their natural milieu. This does not mean,

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however, that an ecological approach to the study o f dissipative social systems is a warrant for ignoring those symbolically constituted elements that com­ pose a society's culture, nor does it entail the reduction o f the cultural domain to the material processes and interests that gave rise to i t . It can be demon­ strated, in fact, that cultural systems, as linguistically mediated, symbolic constellations, have many o f the same dissipative traits as the human commu­ nities that produced them. There is nothing in a materialist, dissipative sys­ tems paradigm, then, to prevent us from dealing forthrightly w i t h cultural phenomena. In addition to their material constitution, dissipative social systems dis­ play far-from-equilibrium behavior. That is, their evolution is chaotically driven and is sensitive to initial conditions. Further, the structure o f dissipa­ tive social systems is generated by symmetry-breaking mechanisms, and is, consequently, ontologically layered. These evolutionary properties establish the foundations for the historicity o f dissipative social systems. Hence, any ecological or cultural analysis o f institutional life must take into account both the origins and the developmental history o f the entities and events under investigation. When these attributes are gathered together and applied to the study o f society, a portrait o f dissipative social systems emerges that presents us w i t h the image o f an inherently historical entity whose evolution is driven as much by internal instability as by external perturbation. Moreover, the grounding o f dissipative social systems in nature and i n the dynamics o f deterministic chaos demands a materialist interpretation o f dissipative social systems not unlike that developed by critical M a r x i s m . Despite their commonalities, however, there are important differences separating dissipative social systems from their physically constituted counter­ parts. Most o f these differences hinge on the fact that societies and their institutional activities are constructed by the collective action o f human be­ ings, and, thus, are profoundly influenced by the way in which humans subjectively define themselves and their actions. This fundamental difference has already been expressed in Bhaskar's critical naturalist paradigm, for when he describes society and its functions he underscores the " w i l d card" nature o f human beings and their innovative abilities. This same exceptionality has long been recognized in dissipative systems theory, and can be neatly inserted into the paradigm advocated by Prigogine and the Brussels School (Prigogine and Stengers 1984; Nicolis and Prigogine 1989). Thus, as long as the existence and impact o f this w i l d card on institutional life is kept i n the forefront o f one's analysis, it is possible to assume that dissipative social systems can be studied in a manner similar to the study o f natural dissipative systems. Failure to recognize the indeterminate aspect o f human nature, however, w i l l result in that reification that is so common in positivist and behaviorist interpretations o f social life (see Bhaskar 1989a, 38).

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Strategies o f M o d e l i n g a n d Their O n t o l o g i c a l L i m i t s Having specified our metatheoretical premises, we can now describe how we constructed the modeling/ontology matrix. The matrix is displayed i n figure 13.1. Its six columns, labeled "Levels o f Modeling Abstraction," cover a wide range o f techniques currently in use i n the social sciences. Starting w i t h predictive models, these strategies are ordered so that as one moves from left to right there is a decreasing tendency for the strategy in question to produce

Hierarchy of Ontological Complexity in Social S y s t e m

Modeling Strategies for Studying Chaotic Social Systems: Arrayed by Decreasing Determinist Presuppositions (horzontal axis) a n d by Levels of S y s t e m Specificity (vertical axis)

Process I I : Societal evolution via historical m o d e s of production Process 1: C l a s s struggle. Conflict over cultural h e g e m o n y 12

Values I I : H e g e m o n i c culture & subcultural b a s e s of resistance

11

V a l u e s 1: Struoale of hegemonic vs. s u b t e r r a n e a n world views

10. N o r m s II: Allocation of relative p o w e r a m o n g social institutions 9.

N o r m s 1: Personal c o n f o r m i t y to g e n e r a l h e g e m o n i c s t a n d a r d s

8.

Roles I I : Intraoraanization allocation of roles & resources

7.

Roles 1: Distribution of material r e w a r d s & e s t e e m

6

Facilities I I : T e c h n i c a l division of labor in productive sphere

5.

Facilities 1: S o c i o t e c h n i c a l infrastructure of organization

4. Ecological organization of institutional t i m e & space 3.

Ecological organization of local biotic c o m m u n i t y

2. Biological evolution a s a series of assisted bifurcations 1.

Determinant regularities of the physical universe Predictive Modeling

Statistical Modeling

I c o no logical Modeling

Structural Modeling

Ideal T y p e Modeling

Historical Narratives

Levels of M o d e l i n g A b s t r a c t i o n

Fig. 13.1. P a r a d i g m s h o w i n g general range o f f i t b e t w e e n m o d e l i n g s t r a t e g i e s a n d d i f f e r e n t levels o f o n t o l o g i c a l c o m p l e x i t y i n t h e social system

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lawlike, scientific propositions. Conversely, moving in the opposite direction, one moves from modeling strategies designed to understand particular people and events to those that formulate abstract, general propositions. Hence, nomothetic methods occupy the left half o f the matrix, while ideographic methods populate the right. I t should be emphasized from the start, however, that our ability to locate these various modeling strategies on a continuum, should not obscure the fact that there are radical differences between these strategies. As we w i l l point out later, we encounter several epistemological breaks as we move along this abstraction dimension. The fourteen rows o f the matrix are organized from bottom to top i n terms o f increasing ontological complexity. As noted i n the introduction to this chapter, this ontological array has been constructed using the works o f Kenneth Boulding (1968, 3 - 1 0 ) and Neil J. Smelser (1963, 1-46). Rows one through four o f the matrix are adaptations o f Boulding's original w o r k , while rows 5 through 12 are adapted from Smelser and used to elaborate upon the "undeveloped upper end," so to speak, o f Boulding's grand ontological scheme. Similarly, Boulding's typology is used to specify more precisely those ontological levels to which structural-functional analysis has tradi­ tionally given short shift, that is, the sociotechnical and ecological founda­ tions o f the social system. Hence, the natural and biological milieu in which dissipative social systems operate occupies the lower rungs o f the hierarchy. The ecological and sociotechnical spheres o f social life mediating between society and its natural environment come next, and, moving up the hierarchy, we encounter the institutionalized systems o f resource distribution, social organization, and moral evaluation. A t the top o f the hierarchy are the general regulatory processes that structure modern societies and their evolution. In this context, class conflict and social evolution are not ontological levels, but dynamic processes that drive the evolution o f the social system as a whole. As noted earlier, the ontological levels represented in the row structure o f the matrix are systemically integrated. Lower levels form the loose founda­ tions and conditions by which higher levels emerge and operate. Concomi­ tantly, the higher levels, once established, feed back upon and delimit the operation o f those levels that undergird their very existence. We w i l l not, however, dwell for long on the dynamic integration o f this hierarchy, since we are more interested in them as static configurations and in the way the struc­ ture o f these configurations brackets the scientist's choice o f method. Finally, the last feature o f the matrix is the shaded cells running diag­ onally from the lower left-hand corner to the upper right-hand corner. These cells are hypothesized areas o f optimum fit between modeling strategies and the different ontological levels o f dissipative social systems, while those projects falling on either side o f the diagonal are seen as analytically problem­ atic. We w i l l take up the interpretation o f these areas in more detail later on.

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M o d e l i n g Strategies and Levels of A b s t r a c t i o n We have been careful in this chapter to distinguish between deterministic chaos theory and the modeling techniques actually used to study chaotic phenomena. Models, as opposed to theories, are well-formed metaphors and analogies. They do not claim to express the truth o f the w o r l d , but merely to provide heuristic insights. While theories claim to actually explain reality, models are only partial, fictitious constructions. They speak a language o f "as i f , " not "what i s . " But i f models can make few explanatory claims, they are rich in the conceptual materials upon which they can draw and are freer to organize those materials i n a manifold o f different directions ( L l o y d 1986, 124-37; Bhaskar 1978, 143-63). Unconstrained by a formally operationist logic or the presuppositions o f a well-articulated paradigm the way theories are, modeling freely participates in acts o f imagination to produce a wide range o f alternative insights to old problems. These are fairly standard observations concerning models and their gen­ eral scientific functioning. Kaplan (1964) has taken these observations a step further. He claims models have as their basic focus the mapping o f formal similarities and the establishment o f strict isomorphisms between systems. According to Kaplan, once these isomorphic similarities are established, the goal o f modeling is to map specific structural correspondences between t w o or more systems (Kaplan 1964). There are several ways to establish these correspondences in the social sciences. The column structure o f figure 13.1 lists what we believe to be the major families o f modeling strategies. Predictive models are the most famil­ iar, for they have been procedural paragons in the social sciences for almost two hundred years. As noted above, this strategy is being eclipsed due to discoveries that have been made both by quantum theorists and by chaos researchers. Its insistence that the test o f a theory is its ability to make longterm predictions is not compatible with the dynamics o f dissipative systems, and should be handled gingerly by all those engaged in chaos research. The second family o f models depicted in our matrix is statistical modeling. I t focuses on fitting data to a set o f mathematical curves. Here the criterion o f adequacy is not prediction o f a system's future state, but the post hoc close­ ness o f fit a researcher can forge between longitudinal data and a given mathematical function. W h i l e predictive and statistical models are well established in the social sciences, a third modeling strategy, iconological modeling, is only now com­ ing into its o w n . It represents a radical epistemological break in the type o f knowledge it provides to researchers. Iconological modeling is rooted in a pictorial method, in visual correspondences rather than in deductive reason­ ing. Iconological modeling is a recent innovation, originating in the iterative

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mapping o f complex systems o f equations—such as the nonlinear differential equations that generate the quadratic iterator, or the so-called strange attractors. Iconological mapping arose in large part as a necessary response to the complexity o f the patterns generated by the iterative analysis o f these equa­ tions. Their evolution becomes so complex, so rapidly, that they overwhelm human comprehension. Thus, graphical techniques assist the researcher in visually tracing the chaotic trajectories o f these iterative systems. Indeed, Gregerson and Sailer (1993) suggest this form o f modeling would have been all but impossible three decades ago, for much o f the computer know-how had not yet been perfected to achieve such graphics. Thus they note: M u c h o f the research performed on chaotic systems i n the physical and biological sciences has benefitted from visual representations o f chaotic behavior using high-powered computers w i t h advanced graphic capa­ bilities. . . . I n order to better understand chaos, it is useful to have a way to produce a visual representation (derived from mathematical models) o f chaotic behavior. (Gregerson and Sailer 1993, 7 8 3 - 8 4 ) Iconological modeling thus represents a break with earlier modeling strategies and has been made possible by recent technical progress. Its novelty, how­ ever, resides in the priority it gives to the visual over the analytic. In philo­ sophical terms, we could say that in iconological modeling the gaze is more important than deductive logic in grasping the evolution o f a chaotic structure. There is something refreshing yet discomfiting about this almost ad hoc exper­ imental approach. W h i l e the iconoclastic style o f those engaged in this form o f modeling may be a remnant from the sixties, there is something new and significant here that was not possible i n the past. The term we have chosen to describe this modeling strategy has been borrowed from the Renaissance art historian E r w i n Panofsky (1972, 3 - 3 1 ) . Iconological analysis, according to Panofsky, is preeminently visual and phenomenological in its exercise. It employs various levels o f interpretive an­ alyses that range from the recognition o f simple visual forms to attempts to capture the Weltanschauung o f an epoch. More precisely, iconology is the interpretive study o f aesthetic objects—their visual style and presence, their formal structure, the cultural symbolism employed in their construction, and the social forces and personalities that produced them. The uniqueness o f the iconological method lies in its ability to recognize how an aesthetic object's phenomenal appearance can communicate all aspects o f itself, and the ways in w h i c h they express those different aspects. In a real sense, Panofsky's aes­ thetic epistemology parallels the phenomenological and analytic process the scientist uses when trying to interpret the meanings o f the graphic images whose unfolding maps the chaotic evolution o f a system.

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As we move to our fourth family o f research strategies, structural model­ ing, we encounter a second radical break. W i t h structural modeling we meet a strategy that in articulating and testing its results largely discounts the evi­ dence o f the senses in favor o f purely formal analysis. Structuralism holds that the truth o f the w o r l d , including that which is presented by deterministic chaos, can be captured by axiomatic analysis alone. That is, Platonic-like intuitions are sufficient to grasp the truth o f models constructed using axiom­ atic methods. Such refined intuitions guide the researcher past empirically based deceptions and pitfalls, and are able to immediately grasp the structural essence o f the object itself. This ultra-idealism is given its classic statement by Thorn (1983) when he argues against those statistical analyses o f chaos that begin and end w i t h empirical reality. Thorn believes that to understand chaos, one must not reify the empirical patterns themselves, but unearth the "simple mathematical forms" o f which they are composed. Employing an ontology that echoes Plato's vision o f the w o r l d as being structured by a concatenation o f ideal geometric forms, Thorn writes: Here I am a Platonist: we forever see only the shadows o f things, and, it is necessary for us to go beyond the cavern's w a l l . . . , hence to imagine a reality o f that space beyond, projecting itself on observed phenomenol­ ogy. . . . Replace the complicated visible by the simple invisible: . . . it is also necessary to think o f simple mathematical forms which generate complicated empirical datum. (Thorn 1983, 80) This faith in the ability o f mathematics to penetrate to the very core o f our world is not new. I t has been with us almost as long as the species has been engaged in speculative reasoning. In modern times, since the Renais­ sance and the age o f Newton, and into the present, there has been a cult whose professions o f faith in the power o f mathematics have been profound. One o f the most touching o f these bears a mystical awe that is hard to find in an age o f opportunism and high-profile cynicism such as ours. Thus, Nick Herbert quotes the physicist Eugene Wigner in discussing mathematics and its relation to the real w o r l d . In the seventeenth century Galileo, Newton, and the other natural philosophers discovered that an enormous body o f physical facts could be encompassed in a few mathematical formulas. For instance w i t h only three mathematical laws Newton could explain all motion in heaven and on earth. W h y should mathematics, developed primarily to keep track o f human business transactions, have anything at all to do with the way the non-human w o r l d operates? Nobel laureate Eugene Wigner refers to the magical match between human mathematics and nonhuman facts as "the

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unreasonable effectiveness o f mathematics in the natural sciences." "This unreasonable effectiveness," writes Wigner, "is a wonderful gift which we neither understand nor deserve." (Herbert 1985, 2) When Wigner speaks o f that "wonderful gift which we neither under­ stand nor deserve," he expresses the deep emotions we have all felt when we first glimpsed the elegance o f self-similar constructions (Peitgen, Jiirgens, and Saupe 1992, 6 3 - 1 8 2 ) or the manifold possibilities that Feigenbaum's perioddoubling law, w i t h its suggestive cascading o f bifurcation points, might hold. The same astonishment accompanies our discovery o f the wholly unexpected behavior a quadratic iterator displays when we vary its control variable through the smallest range o f numerical values (Baumol and Benhabib 1989; Peitgen, Jiirgens, and Saupe 1992, 5 0 7 - 7 6 8 ) . A n d certainly one cannot sit at a monitor and watch the unfolding orbit o f Lorenz's strange attractor without being deeply moved. The social sciences, to be sure, are not immune to such enthusiasms. Positivists have for years reined mathematics, applauding it as being the pinnacle o f knowledge, and urging the social sciences to emulate its example. To match the mysticism and awe o f Wigner in the social sciences, however, we need go no further than the work o f the French structuralist Claude Lévi-Strauss. Lévi-Strauss's (1955; see also Badcock 1975, 3 3 - 6 7 ) structural analysis o f m y t h , for example, resonates w i t h Thorn's Platonic manifesto. Lévi-Strauss begins his analysis by collecting different versions o f a single myth drawn from a number o f different groups or societies. He then abstracts from this manifold o f empirical detail the key elements o f the myth he is studying. I n that none o f the original myths contain all o f its essential elements, it is the j o b o f the social scientist to assemble the fully developed true myth. This recon­ structed myth is an idealized, composite configuration containing all o f the essential components. Working w i t h these ideal elements, the scientist builds a relational model o f the myth and shows how its various elements are oppositionally j o i n e d to one another. Since the original versions o f the myth are seen as incomplete or distorted by the accidents o f local history, they are treated by the scientist as inferior to his or her rationally reconstructed model. This reconstituted myth is the equivalent o f Thorn's "simple invisible." It is the intellectually fulfilled object from which the chaos o f the different empirical myths has emerged. It is this logically precise and coherent myth that is the proper object o f a theoretical science. Ideal typical modeling constitutes the mirror image o f structuralist anal­ ysis. Scientists w h o employ ideal typical modeling in their research are not interested in structural commonalities. Quite the contrary, they are interested in the singularities that create deviations from homogeneous patterns. The

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structuralists, as we have just seen, garner as many different versions o f a class o f phenomena as possible. From these they construct a consistent core o f attributes. They then discard or ignore the variations and turn to the logical essence or generative form o f the materials being studied. Their goal is to capture the logically pure form, for therein lies reliable, theoretically articu­ lated knowledge. Those engaged in ideal typical modeling also gather as many different empirical examples as possible. They also seek commonalities among the class o f objects surveyed and, like the structuralists, construct from those commonalities a logically pure type that captures the essence o f the class o f entities under study. But here the similarities between the two methods end, for after the ideal type model is constructed, it is "discarded," so to speak, as the scientist turns his or her attention to the deviant elements that cannot be subsumed under the logically pure type. Using the ideal type to identify these singularities, the scientist focuses upon the nonlogical elements that set each instance off from the logical pattern. In these nonlogical exceptions, the scientist discovers what is historically and developmentally unique i n each instance. The ideal typical modeling strategy, in sum, produces a variety o f ideographic knowledge that can serve as the basis for a comparative history and social science. Historical modeling is close in its form and content to ideal type model­ ing. Like the latter, historical narrative seeks to understand the singular truth o f events and the manner in which they unfold. What separates ideal typical modeling and historical narrative from one another is the stubborn insistence o f historical modeling on the singularity o f the events and personages it is reconstructing. The purpose o f historical narratives is to render as close a reading as is possible o f concrete events. From this reading, statements o f empirical causation may well be constructed. What is denied, however, is that the causes creating one chain o f events can be compared with the causes giving rise to other chains o f events. From this perspective, narrative history is possible, but a comparative science o f history is not. W i t h the consideration o f narrative history, we conclude our survey o f the construction o f the methods/ontology matrix. This matrix compares mod­ eling strategies w i t h the different ontological levels o f dissipative social sys­ tems. In identifying different ontological levels w i t h various modeling types, we suggest that level and type can be related to one another in this fashion because each is ultimately grounded in its o w n dimensionality. I n the case o f ontological layering, this integrating dimension is lodged in the structure o f those broken symmetries that internally differentiate the world o f experience. In the case o f modeling strategies, the common dimensions stratifying the models spring from the varying abstractness o f the knowledge they produce.

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Having identified these dimensions, we are now in a position to assess the relative fit between a given modeling strategy and the ontological composition o f the societal phenomenon being studied. Promise a n d Paradox: The Future of t h e Paradigm in t h e Social Sciences Returning to figure 13.1 and its shaded diagonal, we can now imagine its matrix as being partitioned into four quadrants. The lower left quadrant and the upper right are areas o f relative fit between the dissipative system's ontol­ ogy and the modeling methods o f those quadrants. The upper left and lower right quadrants, by contrast, are areas o f modeling misapplication—that is, areas in which basic methodological fallacies are most likely to occur. Work­ ing from this interpretation o f the matrix and its partitioning, the following rules apply: 1. Predictive, statistical, and iconological modes o f chaos modeling should be restricted to those ontological levels in which collective social phenomena can be legitimately treated as a statistically aggre­ gated phenomenon, that is, as being composed o f additive, numer­ able, and interchangeable individual units. 2. Structural, ideal typical, or narrative history models o f chaotic pro­ cesses can be deployed w i t h maximum effectiveness in those very areas where predictive, statistical, and iconological patterns are least suited. That is, structural, ideal typical, and historical models are best suited to those upper ontological levels in which cultural products, historically specific events, and collective symbolic processes are the rule. 3. I f predictive, statistical, or iconological models o f chaos are used to research norms, values, and ideographic or historical phenomena, then the researcher runs the risk o f reifying the social and cultural phenomena under investigation. The term reification in this context refers to the fallacy o f treating humanly produced conventions, insti­ tutions, and historically complex events as though they were natural objects governed by recurrent processes and universal laws. Such reifications always fall short o f the mark, in that they ignore the role human intention and agency play in constructing these complex do­ mains. The various fallacies o f reification would fall in the upper left quadrant o f the matrix. 4. To the extent structural, ideal typical, or narrative history modeling strategies are employed exclusively by chaos researchers to explain natural, biological, and ecological phenomena, they run the risk o f

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committing what we call a mythopoetic fallacy. In this context, mythopoesis refers to that tendency in humanist research to mistakenly treat obdurate facts o f nature as though they were purely discretionary human constructions, that is, objects whose form and content were w h o l l y dependent on the intentionality and interpretive activities o f persons or their communities. This fallacy would fall in the lower right-hand corner o f the matrix. These four possibilities form a sufficient, i f not exhaustive, framework for evaluating the various forms o f chaos research currently being carried out in the social sciences. Beginning w i t h predictive modeling, let us see what this framework allows us to say about the present practices and future prospects o f the six modeling strategies delineated i n our matrix. Predictive Modeling Consistent w i t h the above four rules, predictive modeling is optimally applied when researching the lower ontological levels o f the dissipative social systems hierarchy. The further one moves up the ontological hierarchy, the more one runs the risk o f falling v i c t i m to reified constructions and misunderstandings. This generalization reflects the fact that predictive methods are most effec­ tively deployed in research areas in which the units o f analysis, be they natural or human objects, are: (1) homogeneous and, hence, interchangeable; (2) capable o f being linearly aggregated into a representative type; and (3) not internally integrated by higher levels o f interaction into complexly structured collectivities. Such ontological assumptions characterize the physical sci­ ences, neoclassical economics, and certain recent approaches to demography and social ecology. As noted above, quantum mechanics and chaos theory have both dealt severe blows to the aspirations o f the predictive sciences. To the extent these sciences base their reputations on their ability to predict future events, they are now in the midst o f a crisis. O f course, social science forecasting technologies have been relatively underdeveloped. Recently, however, efforts have been under way to redefine the mission o f forecasting by taking into account the insights o f chaos theory (Loye and Eisler 1987; Baumol and Benhabib 1989; Huckfeldt 1990; Gordon and Greenspan 1988). Hence, the revised program for improved forecasting now lies in the partitioning o f unexplained predictive variance into its various components. Huckfeldt (1990), to take but one ex­ ample, has identified four sources o f predictive error: measurement error, error caused by the omission o f crucial explanatory variables, error caused by the presence o f stochastic variation, and error caused by chaotic indeter­ minacy. It is his hope that by classifying concrete sources o f predictive error

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into one o f these four categories, the forecaster can improve his or her overall predictive performance. W h i l e forecasting may not be quantitatively i m ­ proved in the sense o f increased precision by following Huckfeldt's path, the art o f forecasting can achieve a more realistic sense o f its o w n possibilities and options. B y identifying the presence o f stochastic or chaotic variation in forecasting data, a more realistic set o f expectations can be made possible by which forecasters and their publics can judge the efficiency o f their results. In this way the art o f forecasting can be evaluated against a pragmatic back­ ground o f the limitations imposed upon it by the stochastic and chaotic struc­ ture o f its data. Even this paring o f expectations may not suffice, however, for there are momentous technical problems associated with being able to differentiate between stochastic errors and chaotic indeterminacy. Data-based deficits haunt all such efforts at chaos modeling—not just forecasting. There is, for example, a g r o w i n g awareness among researchers that chaos modeling is a data-hungry process that requires inordinately long time series. In most cases such series are virtually impossible to obtain in common research settings. For example, the use o f Lyapunov exponents is a crucial test in determining the presence o f deterministic chaos. Without i t , and without the possibility o f periodic renormalization, the possibilities o f the mathematical modeling o f social chaos are severely constrained. Yet, as W o l f (1986) and B r o w n (see chap. 6 i n this volume) have noted, adequate databases are usually not avail­ able for such testing. The problem o f insufficient databases, o f course, can be technically resolved w i t h computer modeling. A n d while this allows re­ searchers to explore chaotic models in the abstract, it does little to solve the problems o f building a tradition o f empirical chaos research. Finally, going beyond the current predicaments o f improved forecasting and computer simulation in econometric modeling, there are additional prob­ lems. These problems have been raised for more than a decade by M i r o w s k i (1988, 1989, 1990, 1993). He maintains that most o f the use o f partial differential equations in modern econometrics cannot be justified. Such prac­ tices, M i r o w s k i claims, are based on an unsubstantiated premise: the principle o f the conservation o f energy. This principle was at the heart o f nineteenthcentury classical mechanics and was borrowed by neoclassical economists in an attempt to assume the trappings o f genuine science. The validity o f the conservative premises adopted, however, has yet to be proven. Unless they can be formally justified, the entire enterprise o f mathematical economics and its use o f integrable functions to model markets and entire economies, chaotic or otherwise, w i l l collapse. To our knowledge, M i r o w s k i ' s institutionalist critiques o f neoclassical theory and method have not been adequately ad­ dressed and, until they are, the work o f those economists and social scientists

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w h o are attempting to construct mathematical models o f chaotic social sys­ tems lacks a foundation. Statistical Modeling The predominant use o f statistical modeling in chaos research these days attempts to use curve-fitting techniques to identify latent recurrent patterns in longitudinal data. The classic case o f this type o f modeling is found i n at­ tempts to explain the generative structures that produce the now famous Kondratiev economic cycles. According to Kondratiev, capitalist economies moved through fifty-year cycles o f high productivity and wealth and then decline and relative poverty. He attempted to explain these cycles by appeal­ ing to technological factors, that is, to long-term waves o f rising and declining technological capacities. Subsequent research, while provisionally accepting his findings, has suggested alternative, sometimes chaos-based, explanations. Thus, in this volume, Berry and K i m (see chap. 10) take longitudinal data on economic growth and decline in the United States over the last two centuries and argue that American economic life conforms to a series o f Kondratiev waves that often exhibit chaotic properties. I n contrast to Kondratiev's expla­ nation, however, they locate the origins o f these cycles not in the mode o f production, but in historical contingencies and fiscal policies regulating the ebb and flow o f the money supply. Working in a broader interpretive frame­ w o r k , Weber (1981) and Namenwirth (1973) have applied Kondratiev's ideas to the analysis o f cyclic shifts that have occurred in the political and cultural spheres o f Great Britain and the United States. Both have found patterns o f variation in political thought that correlate with Kondratiev-like economic cycles. Using computer-structured content analysis techniques, Weber re­ searched British speeches from the throne over a period o f some 180 years, and found a fluctuating cycle o f political concerns. He found this cycle corre­ sponded closely to Kondratiev patterns that were discernable in Britain's economic history. Namenwirth, on the other hand, analyzed the thematic content o f Democratic and Republican Party platforms in this country and found thematic cycles roughly correlating w i t h economic patterns established by Kondratiev scholars. After fitting his data to a complex trigonometric function, and after experimenting with several theoretical interpretations that could explain the overlapping o f the economic and thematic cycles, Namenwirth finally settled on a culturological interpretation drawn from Parsonsian systems theory. Following Parsons, he argues that the cycles o f thematic shifts reflect the natural phase transitions through which social sys­ tems pass as they adapt to new environmental challenges. Statistical modeling is, o f course, an attractive way to examine historical

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time series. Indeed, it can give the illusion that patterns in history come and go w i t h an almost sidereal certainty. I f not handled with caution, however, such quantitative modeling can become a Rorschach Test in which the re­ searcher, once having found what appears to be a pattern, can interpret it almost any way he or she chooses. I f we are correct, such modeling is most appropriate when working in the lower levels o f dissipative social system specificity. Otherwise, the risk o f reification increases as one moves to the higher levels o f the system. The shaded cells in figure 13.1 for statistical models (column two) show how we would ontologically delimit such efforts in order to avoid the reification o f institutional life. Iconological Modeling The process o f pictorial recognition and the researcher's ability to interpretively link graphically produced images o f chaos to actual social processes or structures is at the heart o f deterministic chaos research. Ironically, as sophisticated as the mathematics are that generate the pictorial patterns, only a fraction o f that sophistication is needed to properly interpret the graphics. Hence, an intuitive feel for quadratic iterators, for Feigenbaum's (1978) anal­ ysis o f period doubling, for the self-similar structure o f bifurcational pro­ cesses, or for the trajectories o f strange attractors is obtainable through their visual representations. The articles in this volume abound w i t h pictorial repre­ sentations that attempt to communicate the reality o f chaos. N o mere decora­ t i o n , these graphics appeal to the power o f visual representations to communi­ cate facts and to convince the reader. This reliance upon iconic imagery to communicate facts about chaos and its constituent dynamics is ubiquitous i n the new science. There is a growing reliance upon the visual to both grasp and communicate findings where words and mathematical accounts fail. H o w better to describe the self-similarity o f the bifurcation process that quadratic iterators undergo than through pictures? Economists make extensive use o f the quadratic iterator and its visual repre­ sentations when they explore the chaotic dimensions o f markets and national economies. Indeed, Rosser's (1991b) recent volume on catastrophic and cha­ otic modeling in economic systems is a veritable cornucopia o f graphic repre­ sentations depicting what might otherwise be obscured by purely analytic description. Similarly, M a y ( M a y 1977; M a y and Oster 1976) has used graphics w i t h telling effect to explain the deterministic chaos inherent in the mathematics o f growth and decay among animal populations. A n d , in a triumph o f computer simulation, A l l e n and Sanglier (1978, 1979) almost two decades ago defined urban areas as dissipative entities and used computer modeling to explore urban place theory. Through the power o f computer

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graphics, they were able to reproduce pictorially many o f the patterns o f urban structure first discovered in the social sciences by the Chicago School. Finally, we would be remiss not to mention what is, perhaps, the most powerful icon o f a l l , the folded catastrophe surface that was popular a decadeand-a-half ago. Although it has fallen into disfavor in recent years, we can think o f fewer images that visually communicate the ideas o f nonlinear change, sensitive dependence on initial conditions, and the processes o f dy­ namic bifurcation. Whatever the cause o f its fall from favor, two points are obvious: (1) the mathematics o f catastrophe remain valid; and (2) the visual power o f the catastrophe surface is still a potent stimulus for promoting metaphorical and analogical thinking—especially in the early phases o f con­ ceptualizing or outlining a research problem. Structural Modeling As we saw in the last section, structural modeling is grounded in a rationalist worldview. R e n é Thorn's work is the epitome o f this approach. Indeed, his modeling o f the morphogenesis o f living forms is a classic expression o f the position that sees the w o r l d as conforming to structures and processes that are embedded in reality itself. Devaney's (1989) work in chaos theory using what he calls "symbolic dynamics" can also be subsumed under the structuralist rubric. Professor Richards's chapter in this volume (see chap. 5 herein) does yeoman w o r k in explaining Devaney's approach to chaos and in showing how it can be applied to research problems in political science. When this axiomatic paradigm is applied to social science research, its basic assumption o f a deductive, deterministic order underlying phenomenal reality has its parallel in structuralism. We earlier explored the association between Thorn's worldview and that o f Lévi-Strauss. As the reader might guess, this modeling strategy also has an affinity with Parsonsian systems theory and its homeostatic assumptions concerning social system stability and equilibrium. Axiomatic theory's assumption that there is a logically tight structure underlying everyday reality, when translated into sociological termi­ nology, finds resonance w i t h the idea that social systems are naturally selfequilibrating entities. The affinity between Parsonsianism and approaches using axiomatic methods to generate a conception o f chaos is most evident in the literature on bureaucratic organizations. There is now a substantial literature i n which scholars have applied chaos theory to the study o f those processes governing the rise, efflorescence, and decay o f imperatively coordinated organizations. There is an implicit assumption in such studies that bureaucracies are inher­ ently orderly and self-equilibrating systems located in turbulent environ-

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ments. A t times the degree o f environmental turbulence may be seen as a function o f the organization's ability to control or exploit those environments, but i n this case, the organization is still seen as a conservative self-organizing entity whose activities are largely conservative and reactive, ratherthan evolvng, boundary-testing constellations. The works o f Leifer (1989) and De Greene ( 1 9 8 1 , 1988b, 1990a, 1991a, chap. 12 herein) represent this approach. W h i l e they have been able to avoid the usual Parsonsian pitfalls, they still assume in one way or another bureaucratic organizations are fundamentally reactive, stability-seeking entities that under duress w i l l opt for minimal change and no more. The application o f this modeling strategy is hence limited by its assump­ tion that organizations and social systems are inherently adaptive entities. The inherent disorderliness o f modern organizational life, however, especially as it evolves under the pressures o f flexible accumulation (Harvey 1989), is given short shrift in their analyses. Because o f its metatheoretical presuppositions concerning the naturalness o f system integration and the reactive, adaptive goals organizations pursue vis-a-vis their social and technical environments, structuralist research is usually restricted to the middle ontological levels depicted in our matrix. Ultimately, this family o f models is limited in its ability to grasp technical and ecological processes for the same reason that Parsonsian analysis falters in these areas. Concomitantly, this strategy's fail­ ure to recognize and deal with organizational entropy or class conflict as natural processes places an upper bound on the efficacy with which it can grasp the macrosociological dynamics o f dissipative social systems. Ideal Typical M o d e l i n g I f the structuralist modeling strategy is extreme in its formalist commitments, then the ideal typical strategy is extreme in the opposite d i r e c t i o n — i n its preoccupation with the ideographic and existentialist provinces o f social life. As noted earlier, ideal typical strategies follow the structuralist path to the extent that they, too, construct ideal, logically integrated models. They differ, however, in wanting to grasp the historically specific and often deviant singu­ larities that evade common, everyday practices. They are interested in conti­ nuity i f it gives them insight into the social sources o f chaos. But their ideographic propensities incline them toward the study o f situations in which the cognitive maps a culture provides its people suddenly collapse, or other­ wise lead them into a terra incognita where there are no clear guideposts or commonsense precedents by which to measure reasonableness, morality, or truth. Examples o f this approach abound in the chaos literature. Arthur's (1988, 1990) work on positive feedback points in history emphasizes the

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singularities o f social life. Indeed, one can find few better illustrations o f the operation o f sensitive dependence on initial conditions than the stimulating examples he provides. Gemill and Smith's (1985) work on organizational evolution also emphasizes the disjunctive. Their provocative suggestions con­ cerning the role o f symmetry-breaking processes in the evolution o f organiza­ tions have yet to be fully appreciated by other analysts o f formal organization. Indeed, the implications o f symmetry-breaking processes for deterministic chaos theory have just begun to be explored by the social sciences. The psychiatrist M o n y Elkai'm (1982, 1985,1987), working in the general systems tradition o f Von Bertalanffy, and the family systems approach to therapy established by Bateson and his colleagues, has approached chaos theory from a similar point o f view. Emphasizing existential and nonlinear social inter­ action processes as they arise in the therapeutic situation, Elkai'm uses dissipative systems analysis to map the psychodynamic transition to chaos in the client-therapist relation. Emphasizing the nonlinear aspects o f this relation­ ship, Elkai'm maps the singularities o f interaction underlying the phase devel­ opment o f the "talking cure." The ideal type strategy is grounded in an existentialist celebration o f the particular and the deviant. It epitomizes the iconoclastic spirit that gave birth to chaos theory some thirty years ago. Its ethos stands in strict opposition to the Newtonian nemesis o f uniformity, gradualism, and a universe bound by absolute parameters. It embraces the premises o f deterministic chaos and embodies its commitment to seek out and understand the nonlinear dimen­ sions o f social life lying just below the surface o f seemingly staid, equili­ brated institutions and groups. Historical Narratives The historical study o f chaos is presently undeveloped. Artigiani's (1987, 1991) work on modern revolutionary movements was an early landmark and still exemplifies what deterministic chaos research should look like when taken up by historians. Working in the demanding medium o f historical detail, Artigiani identifies chaotic elements in his narrative treatment o f revolutionary movements. He treats them as compact "modules," so to speak, constellations that illuminate portions o f the larger unfolding narrative. He never succumbs, however, to the temptation o f placing the narrative and its elements on a Procrustean bed o f formally predetermined complexes. Yielding at times to comparative asides, Artigiani nonetheless allows the singular substance o f his subject matter to dictate the form his chaos analysis w i l l take. His method, in short, embodies the methodological pluralism this paper has advocated by letting the ontology o f the subject matter dictate the range o f methods em­ ployed, and not vice versa. He has followed a path consistent with the onto-

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logical turn advocated by Bhaskar's critical realism, and exemplifies the direc­ tion deterministic chaos theory must take i n the cultural sciences i f they are not to succumb to scientism when adapting the perspective o f deterministic chaos to the study o f history. S u m m a r y a n d Conclusions The mounting volume o f writing in deterministic chaos theory seems to belie the title "new science" that is often used to describe the deterministic chaos paradigm. Yet, when we survey that literature, the variety o f subjects encoun­ tered, the tentativeness o f the findings reported, and the personal uneasiness many practitioners express both in their writings and in their conversations with other researchers, we see that the worldview o f deterministic chaos is still undecided as to its final course. Another sign o f its fledgling status lies in the feeling among its practitioners that all is still possible, that nothing is as yet "set in stone," and that nothing, as yet, is fully forbidden. This liberating sense often fills chaos researchers w i t h a certain abandon and recklessness that both rankles and frustrates their elders. The very scope o f this chapter, as well as its efforts to demarcate the ontological and epistemological boundaries o f the new science, is an added sign o f the newness o f the deterministic chaos paradigm. In developing our conception o f a dissipative social system, we have attempted to provide chaos researchers in the social sciences w i t h philosophical and scientific rationales for their w o r k . For these reasons we choose to end this essay as we began— by issuing a caveat. The matrix developed in this essay is a first hypothesis, not a finished product. Its number o f ontological levels and their specific content, for example, are still open to debate. Thus, the levels can be altered to facilitate innovative research or to gain fresh insight into an old problem. What is not negotiable, from our perspective, is the ontological structure o f the dissipative social system itself—its hierarchical layering and the irreduc­ ible, singular nature o f each o f its levels. Also not negotiable is the systemic and emergent nature o f the ontological levels and their loose hierarchical integration. These elements must remain intact, not just for the sake o f empir­ ical verisimilitude, but because they reflect the ontological preconditions for the possibility o f deterministic chaos emerging naturally from the defining dynamics o f dissipative social systems. The same is true o f our portrayal o f the modeling strategies: their number and definitions are largely ad hoc and were developed out o f our o w n experi­ ence. They may be revised and new ones added, but their relative position vis-a-vis the nomothetic-ideographic/abstract-concrete parameters along which they have been arrayed cannot. A n d , finally, there needs to be extended debate over the boundaries o f ontological fit we have portrayed. The shaded

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cells may be added to or subtracted from. I n an attempt to avoid premature closure at any level, we decided to be overly inclusive as to where the diagonal shading should begin and end. What cannot be contested—again— is the idea o f the necessity for an ontological fit between modeling strategies, on the one hand, and the ontological properties o f the social entity being investigated, on the other. We can be as open to revision as we are because there is still much to be done. What must be avoided i n constructing the new social science, however, is methodological reification and premature rejection o f experimental strate­ gies for chaos research. A dogmatic insistence that only mathematically bracketed chaos studies can qualify as valid in a social science grounded in chaos must be avoided. Such a stance is too narrow and risks excluding the valuable perspectives the cultural sciences and the humanities offer to social science research. A minor squabble o f this nature has already erupted between chaos mathematicians and physical scientists working in the area o f dissipative structures ( M o r i n 1983; Thorn 1983; Ruelle 1991). The mathematicians c l a i m , w i t h justification, that physical scientists, as they operationalize their ideas in concrete research settings, are making a mockery o f their precise formulations. In making such charges, the mathematicians hold up an abstract ideal and use it to denigrate the accomplishments and interpretations o f exper­ imental scientists. W h i l e these protests may be technically correct, this un­ abridged idealism can stifle discoveries that might otherwise flow from mar­ ginal application o f these new ideas. We have seen similar imbroglios in the social sciences in the past, and even now are paying the price. Those dedi­ cated to studying society from the perspective o f deterministic chaos would do well to avoid these schismatic encounters and rebukes.

NOTES 1. For an explanation of Heisenberg's uncertainty principle and its contribution not just to physics but to Western culture generally, we have profited from two biographies. The first is Abraham Pias's (1991) biography of Neils Bohr. The second is David C . Cassidy's (1992) biography of Werner Heisenberg. For a more technical discussion of the formal foundation of quantum physics, including the uncertainty principle, Abner Shimony's 1989 article is both helpful and challenging.

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Contributors

Brian J . L . Berry is the Lloyd Viel Berkner Regental Professor and Professor of Political Economy at the University of Texas at Dallas. He is widely published in the area of urban economics and geography. His most recent research examines the role that economic long cycles play in technological and other social transformations. His most recent publications include Long-Wave Rhythms in Economic Development and Political

Behavior

(1991) and America's

Utopian Experiments

(1992).

Professor

Berry is a member of The National Academy of Sciences. Thad A. Brown is Professor of Political Science at the University of M i s s o u r i Columbia. His recent work has focused on the modeling of political phenomena using innovative new methodologies for dealing with complex nonlinear systems. He is the author of Migration and Politics (1988) and has published in British Journal of Political Science and Western Political Quarterly, to name just a few. Kenyon B. De Greene is currently Professor in the Institute of Safety and System Management at the University of Southern California. He has also worked at R A N D , Northrop, and other corporations. He has published extensively; his books include Sociotechnical Systems: Factors in Analysis, Design, and Management (1973) and The Adaptive Organization: Anticipation and Management of Crisis (1982). He is also editor of the recently published A Systems Based Approach to Policymaking (1993). Dimitrios S. Dendrinos is Professor and Head of The Urban and Transportation Dynamics Laboratory at the University of Kansas. His research has primarily addressed the applications of nonlinear dynamics to population dynamics. He is the author of The Dynamics of Cities (1993) and the coauthor of Urban Evolution (1985) and Chaos and Socio-Spatial

Dynamics (1990), as well as over 100 papers on chaos

and socio-spatial sciences. Euel Elliott is Associate Professor of Government, Politics, and Political Economy, and

Director of the Master of Public Affairs program at the University of Texas at

Dallas. His research interests span a number of areas, including electoral behavior, public opinion, and regulatory policy making. A recurring theme is the nature and dynamics of political change. His work has appeared in Social Science Western Political Quarterly, and Journal of Theoretical Politics.

Quarterly,

He is also the author

of Issues and Elections (1989) and coauthor of Controversies in Political

Economy:

Canada, Great Britain and the United States (1992).

347

348

Contributors

David L . Harvey is Professor of Sociology at the University of Nevada, Reno. He has written in the areas of alienation and reification, Marxist political economy, and poverty.

He has recently published an ethnography

of a poor white com-

munity, Potter Addition: Poverty, Family, and Kinship in a Heartland Community, and is presently completing the second volume of that work, a social history of Potter Addition. Current research interests include the exploration of social evolution and Marxist political economy using the theories and modeling tools of deterministic chaos theory. Ted Jaditz is an economist at the United States Bureau of Labor Statistics. He is a past president of the Society for Nonlinear Dynamics and Econometrics, and has published widely on nonlinear time series, seasonality, and forecasting. L . Douglas Kiel is Associate Professor of Public Administration and Political E c o n omy at the University of Texas at Dallas. His research involves applications of chaos theory to the social sciences and to organizational change. His articles have appeared in Journal of Public Administration Research and Theory, Public Administration Review, and Social Science Quarterly. His book Managing Chaos and Complexity in Government: A New Paradigm for Managing Change, Innovation, and Organizational Renewal (1994) received the best book award for 1994-95 from the public and nonprofit sector division of the Academy of Management. Heja Kim received her doctorate in political economy in 1988 from the University of Texas at Dallas. Since then, she has worked as a postdoctoral fellow in the School of Social Sciences. Her research interests lie mainly in the analysis of long economic cycles. She has, with Brian Berry, coauthored articles that have appeared in Technological Forecasting and Social Change. Along with Brian Berry and colleagues at the University of Texas at Dallas, she is completing a book manuscript that takes an indepth look at the relationship between long waves and social and political outcomes. Michael McBurnett holds a P h . D . in Political Science from Washington University and is currently an executive with Illinois Power Company. In 1991 his dissertation was nominated for the prestigious E . E . Schattschneider award. His articles have appeared in the American Political Science, and Public Opinion

Science Review, American Journal of

Political

Quarterly.

Michael Reed is Professor of Economics and Dean of the College of Business Administration at the University of Nevada-Reno. He has published in the areas of economic history, political economy, and chaos theory. Diana Richards received her doctorate from Yale University and is currently Assistant Professor of Political Science at the University of Minnesota. Her work has focused on the application of nonlinear dynamics for understanding a range of political phenomena, including game-theoretic behavior and empirical work examining international behavior. Her work has appeared in a number of journals, including Behavioral

Sci-

ence and American Journal of Political Science, among others. J . Barkley Rosser, Jr. is Professor of Economics and Holder, Kirby L . Kramer, Jr. Chair of Business Administration at James Madison University. He has published

Contributors

349

extensively on applications of chaos, catastrophe, and evolutionary theory to understanding a variety of economic phenomena. His articles have appeared in Journal of Economic Theory and Journal of Economic Behavior and Organization, among others. He has also recently published Catastrophe to Chaos: A General Theory of Economic Discontinuities

(1991).

A l v i n M . Saperstein is Professor of Physics at Wayne State University in Detroit, Michigan, and Fellow of its Center for Peace and Conflict Studies. Saperstein was among the first natural scientists to apply nonlinear dynamics to social and political phenomena, such as international arms races. His work has appeared in Nature, The Journal of Conflict Resolution, and Mathematical and Computer Modeling, in addition to numerous articles in the physics literature.