Chaos Theory
Naval War College

Newport, Rhode Island

Center for Naval Warfare Studies

Newport Paper Number Ten

October 1996

The Newport Papers are extended research projects that the editor, the Dean of

Naval Warfare Studies, and the President of the Naval War College consider of

particular interest to policy makers, scholars, and analysts. Papers are drawn generally from manuscripts not scheduled for publication either as articles in the Naval War College Review or as books from the Naval War College Press but that

nonetheless merit extensive distribution. Candidates are considered by an editorial board under the auspices of the Dean of Naval Warfare Studies.

The views expressed in The Newport Papers are those of the authors and not

necessarily those of the Naval War College or the Department of the Navy.

Correspondence concerning The Newport Papers should be addressed to the

Dean of Naval Warfare Studies. Requests for additional copies or for permanent

distribution should be directed to the President, Code 32A, Naval War College,

686 Cushing Road, Newport, Rhode Island 02841-1207. Telephone (401) 841-2236

or fax (401) 841-3579 or DSN exchange 948. E-mail: press@USNWC.edu

President, Naval War College

Rear Admiral J.R. Stark,

U.S. Navy

Dean of Naval Warfare Studies

Professor Robert S. Wood

Naval War College Press

Editor: Professor Thomas B. Grassey

Managing Editor: Pelham G. Boyer

Associate Editor: Patricia A. Goodrich

Production

Publication: Diane M. Côté

Graphics: Joseph R. Nunes

Cover: Joan Y. Mikosh-Johnson

Patricia A. Goodrich, associate editor of the Naval War College Press, performed

the principal editing and proofreading of Newport Paper Number Ten.
Chaos Theory

The Essentials for Military Applications

Glenn E. James

Major, U.S. Air Force

§

Printed in the United States of America

to my patient family

Patricia Ann

Christine Marie

Phillip Andrew

who continue to follow me

bravely into Chaos

Contents

Illustrations..................................... ix

Foreword ...................................... x

Rear Admiral J.R. Stark, U.S. Navy

Preface ....................................... xi

Acknowledgements ................................. xiii

Executive Summary ................................ xiv

Introduction

Welcome and Wonder ............................ 1

Part One

What IS Chaos? .............................. 7

Chapter I

Demonstrations .............................. 9

Chapter II

Definitions, Tools, and Key Results................... 25

Part Two

Who Needs Chaos Theory? Applications ................. 47

Chapter III

Expect to See Chaos: Specific Military Systems and Technology ..... 49

Chapter IV

How Can We Use the Results? Exploiting Chaos Theory ........ 55
Part Three

What Next? A Road Map to More Chaos ................. 93

Chapter V

Suggestions for Further Reading..................... 95

Chapter VI

Further Questions to Research ..................... 101

Chapter VII

Conclusion ................................ . 105

Appendix .................................... . 107

Notes....................................... . 109

Bibliography .................................. . 115

About the Author................................ . 123

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Illustrations

Figure 1. Hypothetical Missile Test Data: What Went Wrong?...... 2

Figure 2. U.S. Aircraft Losses in Vietnam ................. 2

Figure 3. Simple Pendulum, No Wobbles Allowed! ............ 11

Figure 4. Stable and Unstable Equilibrium Points............. 12

Figure 5. Formation of Water Drops from a Spout............. 16

Figure 6. First-Return Maps for the Dripping Faucet ........... 18

Figure 7. Night-light with Feedback .................... 20

Figure 8. Finding the Feigenbaum Constant in the Nightlight Experiment .......................... 22

Figure 9. Lorenz’ Weather-in-a-Beaker .................. 27

Figure 10. Chaotic Trajectories of the Iterated Logistic Map........ 31

Figure 11. Fractal Dimension: Always More Detail When You Zoom . . 34

Figure 12. Fractal Boundaries Between Basins of Attraction........ 35

Figure 13. Graphical Iteration of the Logistic Map ............. 36

Figure 14. Distribution of Logistic Map Dynamics............. 43

Figure 15. Chaotic Time Series for the Logistic Map.... What

Comes Next? ............................ 64

Figure 16. Solid Lines Indicate Predicted Values. Dotted Lines Trace

Actual Data ............................. 65

Figure 17. The Logistic Map, Stabilized with Chaos Control ....... 70
Foreword

I take great pleasure in presenting a remarkable addition to our Newport Papers series. While Major Glenn E. James, the author, received support and assistance from sources within his own service, the U.S. Air Force, the final research

and the paper itself are the products of his term in the Advanced Research Program at the Naval War College. This paper typifies the quality of work and capabilities of our students from all the services here at the College. It is an excellent

example of the benefits we derive from the close collaboration between our academic and research departments.

Chaos Theory: The Essentials for Military Applications is a highly challenging

work, one which demands—but amply repays—close attention. It asks for imagination to envision clearly the military applications for which the author argues.

Major James hopes that his efforts can help those of us who labor in the field of national security to appreciate that Chaos theory is a valuable discipline. While

many of the applications of this new field remain conjectural and as yet unclear,

Major James has written a pioneering work which invites military officers to understand the principles of Chaos and to look for applications. I commend this

Newport Paper in particular to policy-level readers, who will find it a useful and

understandable overview of the subject, and to the faculty members of all of the

service war colleges for whom we offer this as a useful text.

J.R. Stark

Rear Admiral, U.S. Navy

President, Naval War College
Preface

Before You Begin...

Before you start into this report, it may help to relax and to prepare to be patient.

Be Patient with the Material...

Chaos as a branch of mathematics is still very young. The first concrete results

surfaced only thirty years ago. Enormous opportunities for new research remain

unexplored. As of yet, not all the bodies of interested researchers know one another or exchange (or search for) information across disciplinary lines. This paper

represents my effort to continue the published conversation on Chaos applications. I’m inviting you to eavesdrop, because the issues are crucial to the military

profession.

Be Patient with the Essay...

Several officers learned of my background in mathematics, and as I left for the

Naval War College, they asked me to consider how Chaos theory influences the

military profession. I examined the published resources that were being used and

felt compelled to correct some serious errors. Many publications overlook key results, make fundamental technical mistakes, or scare the reader with the complexity of the issues. While the progress documented in those papers is

noteworthy—many well-intentioned efforts were made under severe time constraints—we are overdue for a mid-course correction to prevent the errors from

propagating further.

My own Chaos research began in 1987 in my Ph.D. studies at Georgia Tech,

where Professor Raj Roy introduced me to Chaos in lasers. Since then, I have

taught mathematics for four years at the Air Force Academy, including three special topics courses on Fractals and Chaos. This past year, I gave formal presentations to the Air Command and Staff College student body and to two small

seminars of Naval War College faculty. This paper grew out of those talks, subsequent questions, and my continuing research.

I have aimed this report at the broad population of students attending the various war colleges. I have made the format conversational so I may talk with them,

not at them, since this essay takes the place of what I might discuss in a more personal, seminar environment. I struggled to strike a useful balance, sometimes of-
fering many examples so that I can reach this broad audience, and at other times

foregoing extended illustrations on behalf of brevity. I have assumed a minimal

technical background, and resort to an appendix only where absolutely necessary.

I also offer a substantial bibliography of what I consider to be the best available

references for the reader who is anxious for more.

Be Patient with Yourself...

Finally, relax. Chaos isn’t hard to learn—it’s only hard to learn quickly. The important results are often abstract generalizations, but we can arrive at those conclusions via examples and demonstrations that are not difficult to visualize. Allow

yourself to wonder.

In his splendid book, Fractals Everywhere, Michael Barnsley warns:

There is a danger in reading further. You risk the loss of your childhood vision of

clouds, forests, galaxies, leaves, feathers,... and much else besides. Never again will

your interpretation of these things be quite the same.1

I will also warn you of the risks of not reading further: you may fail to understand phenomena that are essential to decision makers, particularly in an era when

the speed and volume of feedback can drive the dynamics of our physical and social—hence, our military—systems into Chaos.

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Acknowledgements

My thanks go first to Colonel John Warden, U.S. Air Force, Commandant of

the Air Command and Staff College (ACSC), who first asked me to consider this

research topic. I gratefully acknowledge the Center for Naval Warfare Studies at

the Naval War College, which sponsored my work under the Advanced Research

Program, and Lieutenant Colonel Roy Griggs in the Air Force Strategic Planning

office, who sponsored this work along with Lieutenant Colonel Jeff Larson

through the USAF Institute for National Security Studies at the U.S. Air Force

Academy.

I am greatly indebted to my friend Major Bruce DeBlois, U.S. Air Force, Professor of Air and Space Technology at the School for Advanced Airpower Studies,

Maxwell Air Force Base, Alabama, who fought hard to incorporate this research

into the Air University curriculum, and to Lieutenant Colonel Rita Springer and

Lieutenant Colonel Gus Liby, U.S. Air Force, for arranging my first of several presentations to the ACSC student body.

I especially appreciate the extensive research and the encouragement of my

friend, Commander Bill Millward, U.S. Navy. He contributed a vast amount

of preliminary research and references that made this project possible in the

Advanced Research trimester at the Naval War College.

My thanks go also to many individuals who patiently critiqued my seminars

and drafts: the Naval War College faculty in Strategy and Force Planning; the Rational Decision Making faculty; Dr. John Hanley, Program Director of the Strategic Studies Group, and the students in his Decision Theory Elective; Lieutenant

Colonel Lloyd Rowe, U.S. Air Force; and Major Tom Tomaras, U.S. Air Force.

Finally, my deepest thanks to Dr. Stephen Fought, my advisor, for his brave support and constant encouragement.
Executive Summary

This paper distills those features of Chaos theory essential to military decision

makers. The new science of Chaos examines behavior that is characterized by erratic fluctuations, sensitivity to disturbances, and long-term unpredictability.

This paper presents specific ways we can recognize and cope with this kind of behavior in a wide range of military affairs.

Designed for courses at the various war colleges, the paper makes three new

contributions to the study of Chaos. First, it reviews the fundamentals of chaotic

dynamics; the reader needs no extensive prerequisites in mathematics. Much

more than a definition-based tutorial, the first part of this paper builds the

reader’s intuition for Chaos and presents the essential consequences of the theoretical results. Second, the paper surveys current military technologies that are

prone to chaotic dynamics. Third, the universal properties of chaotic systems

point to practical suggestions for applying Chaos results to strategic thinking and

decision making. The power of Chaos comes from this universality: not just the

vast number of chaotic systems, but the common types of behaviors and transitions that appear in completely unrelated systems. In particular, the results of

Chaos theory provide new information, new courses of action, and new expectations in the behavior of countless military systems. The practical applications of

Chaos in military technology and strategic thought are so extensive that every

military decision maker needs to be familiar with Chaos theory’s key results and

insights.
Introduction

Welcome and Wonder

Physicists, mathematicians, biologists, and astronomers

have created an alternative set of ideas.

Simple systems give rise to complex behavior.

Complex systems give rise to simple behavior.

And most important, the laws of complexity hold

universally, caring not at all for the details of a

system’s constituent atoms.2

James Gleick

Wake Up and Smell the Chaos

The contractor for the operational tests of your new missile system has

just handed you the chart in figure 1. He ran two tests, identical to six

decimal places, but the system performance changed dramatically after a few

time-steps. He thinks there was a glitch in the missile’s telemetry or that somebody made a scaling error when they synthesized the data. Could it be that the

data is correct and your contractor is overlooking something critical to your

system?

Your wargaming staff is trying to understand and model the time dependence

of American aircraft losses in Vietnam. They look at the data in figure 2 and quit.

It’s just a random scatter of information, right? No patterns, no structure, too

many variables, too many interactions between participants, too large a role

played by chance and human choice. No hope, right?
The results of the new science of Chaos theory offer some intriguing answers to

questions like these. Moreover, the theory has profound implications for the dynamics of an enormous variety of military affairs. In fact, the applications of Chaos in military technology and strategic thought are so extensive that every military decision

maker needs to be familiar with Chaos theory’s key results and insights.

Why Chaos with a Capital “C”?

Chaos, as discussed here, is not social disorder, anarchy, or general confusion.

Before you read on, set aside your connotations of the social (small “c”) chaos reported on the evening news. Chaos is a relatively new discipline of mathematics

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3
with boundless applications; to highlight the difference, I will keep this special

use capitalized throughout.

Chaos theory describes a specific range of irregular behaviors in systems that

move or change. What is a system? To define a system, we need only two things: a

collection of elements—components, players, or variables—along with a set of

rules for how those elements change—formulas, equations, recipes, or instructions.

A remarkable feature of chaotic change is its contrast with “random” motion.

We generally label as random many irregular changes whose dynamics we can not

predict. We will find, as this report progresses, that Chaos displays many of the

same irregularities, with one important difference: the apparently random motion of a chaotic system is often described by completely deterministic equations of motion! Several specific examples of chaotic systems in this will illustrate this point.

The term “Chaos” was first applied to such phenomena fewer than thirty

years ago—that’s a hot topic for mathematics! James Yorke characterized as “chaotic” the apparently unpredictable behavior displayed by fluid flow in rivers,

oceans, and clouds.4 Today, artificialsystems move and reactfast enough to generate similar, erratic behavior, dynamics that were seldom possible before the advent of recent technologies. Nowadays, many military systems exhibit Chaos, so

we need to know how to recognize and manage these dynamics. Moreover, the

universality of many features of Chaos gives us opportunities to exploit these

unique behaviors. Learn what to expect. This is not a fleeting fad:real systems really behave this way.

What’s New in This Essay?

Although numerous Chaos tutorials are available in various disciplines, there

are three main deficiencies in the available resources:

• Many tutorials require an extensive background in mathematical analysis.

• Many works do not focus on useful applications of Chaos theory; they simply

offer a smorgasbord of vocabulary and concepts.

• Some contain major technical flaws that dilute their potential application or

mislead the reader.

So, the immediate need is threefold: we require an accessible bridge to connect

us with the basis of Chaos theory; we seek some in-depth demonstrations of its applications; and we must avoid fundamental conceptual errors.

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Chaos Theory
Who Cares?

Even if Chaos can help military analysts, why should everyone be exposed to the

theory? After all, there is a balance here—you don’t need to know quantum physics to operate a laser printer, right? This paper will show that Chaos occurs in virtually every aspect of military affairs. The 1991 Department of Defense (DOD)

Technologies Plan, for instance, set priorities for research spending.

5 It ranked

the following technologies based on their potential to reinforce the superiority of

U.S. military weapon systems:

1. semiconductor materials and microelectronics circuits

2. software engineering

3. high-performance computing

4. machine intelligence and robotics

5. simulation and modeling

6. photonics

7. sensitive radar

8. passive sensors

9. signal and image processing

10. signature control

11. weapon system environment

12. data fusion

13. computational fluid dynamics

14. air-breathing propulsion

15. pulsed power

16. hypervelocity projectiles and propulsion

17. high-energy density materials

18. composite materials

19. superconductivity

20. biotechnology

21. flexible manufacturing

Every one of these technologies overlaps with fundamental results from Chaos

theory! In particular, the chaotic dynamics possible in many of these systems arise

due to the presence of feedback in the system; other sources of Chaos are discussed

elsewhere in this report. In this paper, you will discover that the fundamentals of

Chaos are as important to military systems as Newton’s laws of motion are to classical mechanics.

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Numerous systems tend to behave chaotically, and the military officer who

does not understand Chaos will not understand many of the events and processes

that mark the life of today’s competent military professional. Look again at figure 2.Nottoo long ago,if we had measured outputlike figure 2 in any scenario, our

analysts would have packed up and gone home, dismissing the data as random

noise. However, it is not “noise” at all. Chaos theory helps us to know when erratic

output like that in the figure may actually be generated by deterministic (non-random) processes. In addition, the theory provides an astounding array of tools

which make short-term predictions of the next few terms in a sequence, predict

long-term trends in data,estimate how many variables drive the dynamics of a system,

and control dynamics that are otherwise erratic and unpredictable. Moreover, this

analysis is often possible without any prior knowledge of an underlying model or

set of equations.

Applied Chaos theory already has a growing community of its own, but the

majority of military decision makers are not yet part of this group. For example, the Office of Naval Research (ONR) leads DOD research into Chaos applications in engineering design, but more military leaders need to be involved and

aware of this progress. Beyond the countless technical applications, many of

which readily translate to commercial activities, we must concern ourselves with

strategic questions and technical applications that are unique to the profession of

warfare. Chaos theory brings to the table practical tools that address many of these

issues.

Why Now?

As long as there has been weather, there have been chaotic dynamics—we

are only now beginning to understand their presence. Some scientists, like

Jules-Henri Poincaré in the late 1800s, had inklings of the existence of Chaos,

but the theory and the necessary computational tools have only recently matured sufficiently to study chaotic dynamics. In 1963 Edward Lorenz made his

first observations of Chaos quite by accident when he attempted simulations

that had become possible with the advent of “large” computers. Currently,

high-speed communications, electronics, and transportation bring new conduits for feedback, driving more systems into Chaos. Consider, for instance,

the weeks required to cross the Atlantic to bring news of the American Revolution to Britain. Now, CNN brings updates to decision makers almost instantaneously.

Until recently, observations of the irregular dynamics that often arise in rapidly fluctuating systems have been thrown away. Unless we train decision makers

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Chaos Theory
to look for specific dynamics and the symptoms of imminent behavior transitions,

erratic data sets will continue to be discarded or explained away.

6

Clear Objectives

As the preface suggested, Chaos theory is not difficult to learn—it is only difficult to learn quickly. Am I violating this premise by trying to condense the essentials of Chaos into this single paper? I hope not. I am trying to build a bridge and

sketch a map. The bridge spans the gap that separates physical scientists on one

side—including analysts in mathematics, physics, and electrical engineering—and humanists on the other—experts in psychology, history, sociology, and

military science. The bibliographical map identifies specific references on issues

that interest segments of the broad audience that I hope to reach with this paper.

My intent here is to teach the reader just enough to be dangerous, to highlight

the places where Chaos happens all around us. The results of Chaos theory can improve military decision making and add new perspectives to creative thought. I

also will offer enough examples and applications so that readers can recognize

chaotic dynamics in common situations. Eventually, I hope the reader will grasp

the key results and apply them in various disciplines. My ultimate aim is to offer a

new perspective on motion and change, to heighten your curiosity about Chaos,

and to provide adequate tools and references to continue the deeper study that is

essential to fully understanding the fundamentals of Chaos.

Here’s the plan. In chapter I we start with Chaos that can be demonstrated at

home, so skeptics will believe Chaos is more than a metaphor, and so we all can visualize and discuss important issues from a common set of experiences. I do not

want you mistakenly to believe that you need access to high-technology circuits

and lasers to concern yourself with Chaos—quite the contrary. Then we’ll add

some detail in chapter II, complementing these intuitions with better definitions.

In chapter III, we consider the pervasiveness of Chaos in military systems. Chapter IV offers practical means for applying Chaos theory to military operations and

strategic thinking. Most of the discussions proceed from specific to general in order to lend a broad perspective of how Chaos gives new information, new options for

action, and new expectations of the dynamics possible in military systems.

In the end, I hope you will learn to:

• Recognize chaos when you encounter it;

• Expect chaos in your field, your organization, and your experiments; and,

• Exploit chaos in your decision making and creative thought.

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Part One

What IS Chaos?

Somehow the wondrous promise of the earth

is that there are things beautiful in it,

things wondrous and alluring,

and by virtue of your trade

you want to understand them.7

Mitchell J. Feigenbaum

I

Demonstrations

The disorderly behavior of simple systems...

generated complexity:

richly organized patterns,

sometimes stable and sometimes unstable,

sometimes finite and sometimes infinite,

but always with the fascination of living

things.

That was why scientists played with toys.8

James Gleick

DEFINITELY Try This at Home!!!

The simple demonstrations in this chapter offer visualizations of a wide

range of chaotic dynamics. They also provide a good introduction to the

methods and tools available to observe, measure, and analyze these dynamics. My

main goal is to build the reader’s intuition of what Chaos looks like.

For any skeptical reader, these examples represent the first exhibits of the extensive evidence I will produce to demonstrate how prevalent chaotic dynamics

are. For all readers, this chapter outlines common examples that provide a context

useful for discussing definitions, tools, key results, and applications in subsequent chapters. We begin with demonstrations to set up at home in order to show

that access to high technology is not needed to observe Chaos. Quite the contrary:
Chaos arises in some of the simplest physical systems. This brief exposure to chaotic dynamics may also spark imagination about the systems where Chaos may

operate in particular areas of interest. A little later, after a more complete description of the vocabulary and tools of Chaos(chapterII), we will examine the military

systems where one should expect to see Chaos (chapter III).

Remember: as we work through each example, the reader should gradually

come to expect and recognize Chaos in any system that changes or moves. As a

general plan for each demonstration that follows, we will:

1. Describe the physical system and answer clearly:

What is the system?

What is being measured?

2. Preview the significance of the particular system:

Why do we care about this demonstration?

Does it relate to any military system?

3. Discuss the significant dynamics and transitions.

4. Highlight those results and characteristics common to many chaotic systems.

The answers to item 1 are crucial. The confusion in many discussions about

Chaos can be traced to a failure to identify either a well-defined system or a set of

measurements. Likewise, to understand the appropriate ways to apply the insights

of Chaos, we need to use its terminology with some care. With this priority in

mind, the discussion of each demonstration will offer a first peek at the Chaos vocabulary that chapter II presents in greater detail.

Warm-ups with a Simple Pendulum

Before we exercise our imaginations with chaotic dynamics that may be entirely new, let’s first “stretch out” by examining the detailed behavior of a pendulum. The simplicity of this example makes it easy to visualize and to reconstruct

in your home or office; it gives us an indication of good questions to ask when we

observe other systems.

As a hint of things to come, an extraordinary number of complicated physical

systems behave just like a pendulum, or like several pendulums that are linked together. Picture, for instance, a mooring buoy whose base is fixed to the sea floor

but whose float on the surface (at the end of a long slack chain) is unconstrained.

Much of the buoy’s motion can be modeled as an upside-down pendulum.9

What is the pendulum’s “system,” exactly? A fixed mass, suspended at the

end of a rigid bar, swings in only two dimensions (left and right swings only, no

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additional motion). The end of the bar is fixed at a single point in space, but let us

assume there is no “ceiling,” so the pendulum is allowed to swing up over its apex

and around to the other side (figure 3). Notice that as we define the system we

must clearly state our assumptions about its components and its behavior.

What can be observed and measured in this system? Fortunately, in this example we need only two pieces of information to describe completely the physical

“state” of the system: position and velocity. The pendulum’s position is measured in

degrees; its velocity is measured in degrees per second. These two observable quantities are the only two independent variables in the system, sometimes referred to as

its degrees of freedom or phase variables. A system’s phase variables are those time-dependent quantities that determine its state at a given time. Observe that even

though the pendulum swings in a curve that sits flat in a two-dimensional plane, we

need only one variable to describe the pendulum’s position in space. Therefore, the

pendulum has only one degree of freedom in its angular position.

So, what can this pendulum do? Let’s pretend, at first, that it experiences no

friction, drag, or resistance of any kind. This ideal pendulum exhibits a rich variety of behavior. If we start it at “the bottom,” where both position and velocity are

zero, it stays there. Any state that has this property—not changing or moving

when undisturbed—is called an equilibrium, steady state, or fixed point for the system. If we displace the pendulum a few degrees to either side and just let it go, it

swings back and forth periodically, with the same amplitude, forever. In this ideal

system, we can also carefully balance the pendulum at the top of its swing, and it

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Chaos Theory
will stay put forever. This state, with position 180 degrees and velocity zero, is another equilibrium point.

Does this ideal pendulum display any other dynamics? Perhaps just one more:

we can impart enough initial velocity to the pendulum so that it swings upward over

its apex and continues to wrap around its axle, forever. This completes the list of

possible dynamics for the ideal pendulum, and it completes a first exposure to

some important terms used to describe all dynamical systems.

Now let’s get back to reality and add some resistance to the system, where the pendulum experiences “damping” due to friction and drag. This real pendulum still has

the same two equilibrium points: the precise top and bottom of its swing, with zero

velocity. A new feature we can discuss, though, is the stability of these fixed points. If we

disturb any pendulum as it hangs at rest, it eventually slows its swing and returns to

this low equilibrium. Any such fixed point, where small disturbances “die out,” and

the system always returns to its original state, is called a stable fixed point (figure 4a).

On the other hand, at the top position of 180 degrees, any perturbation to the right or

left sends the pendulum into a brisk downswing that eventually diminishes until the

pendulum hangs at rest. When a system tends to depart from a fixed point with any

minuscule disturbance, we call it an unstable fixed point (figure 4b).

We should note several other issues concerning the pendulum’s motion that

will arise when we study more complicated systems. First is the observation that

the pendulum (with friction) displays both transient and limit dynamics. The transient dynamics are all the swings, which eventually damp out due to resistance in

the environment. After all the transients die out, the system reaches its limit dynamics, which in this case is a single state: the lower fixed point, with zero position

and velocity.

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It seems we may be reaching the point where we have exhausted the possible

dynamics in this simple pendulum system. After all, even though this is a harmless way to introduce the vocabulary of fixed points, dynamics, transience, and

stability, there is only so much a pendulum can do. Right?

When the system remains undisturbed, the answer is a resounding Yes! The

reason: the motion of a simple pendulum, unforced, is a linear system whose solutions are all known. In particular, the equation of motion, for the position y, comes

from Newton’s familiar relation, force equals mass times acceleration:

my cy ky ′′ + ′ + = 0, (1)

where m is the pendulum mass, c is a measure of friction in the system, and k includes measures of gravity and bar length.

Now, the swinging motion we observe appears to be anything but linear: a pendulum swings in a curve through space, not a straight line, and the functions that

describe oscillations like these are wavy sines and cosines. However, the equation

of motion—like the system itself—is called linear because the equation consists of

only linear operations: addition, multiplication by constants, and differentiation.

When the pendulum experiences no external forces, the resulting homogeneous

equation shows a zero on the right-hand side of equation (1). What is the significance of recognizing a linear, homogeneous system? All the solutions are known;

all possible behaviors are known and predictable.

To add the last essential layer of reality and to generate some interesting motion

in the pendulum system, envision a playground swing. Once you start yourself

swinging, how do you get yourself to swing much higher? You add a relatively

small external force to the system: you kick your legs and lean forward and back in

a manner carefully timed with the larger motion of the swing itself. By pumping

your legs like this, you add a periodic force to the right side of equation (1) and you

resonate and amplify a natural frequency of the large swing.

This addition of an external kick, or forcing function, to the pendulum system

can induce interesting new dynamics. Be aware that, especially if you are pushing

someone else on the swing, you can control three different features of the perturbation: where you push, how hard, and how often. The system may respond to the

external forcing in many different ways. It may resonate with one of its natural frequencies (you may have seen the film of the Tacoma Narrows Bridge being destroyed by the violent oscillations induced by resonance with wind gusts). In

another instance, the swing may behave quite unpredictably if you push the

chains instead of the swing. You may momentarily bring the entire system to a

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Chaos Theory
halt, or cause intermittent lurches in the swing; or you may get very regular motion for a long time, only occasionally interrupted by off-cycle bumps or jostles.

The playground swing, as a system, is just like the simple pendulum. However,

when you “kick” it occasionally, you begin to observe departures from predictable

behavior. This irregular sort of behavior, characteristic of a kicked pendulum, is one

of the many traits of Chaos: behavior that is not periodic, apparently random, where

the system response is still recurrent (the pendulum still swings back and forth) but no

longer in a predictable way. In his classic work on Chaos, James Gleick correctly assertsthat, because of the rich dynamics possible in such a simple system, physicists

were unable to understand turbulence or complexity accurately until they understood pendulums. Chaos theory unites the study of different systems so that the dynamics of swings and springs broaden to bring new insights to high technologies,

from lasers to superconducting Josephson junctions, control surfaces in aircraft

and ships, chemical reactions, the beating heart, and brain wave activity.

10

As this paper continues, we will see more detailed connections between the behavior of pendulums and other more complicated systems. For now, let us move

on to our second home demonstration of Chaos, introduce some additional vocabulary, and continue to build our intuition for what we should expect to see in a chaotic system.

The Dripping Faucet

The second home demonstration can be done at the kitchen sink, or with any

spout where you can control the fluid flow and observe individual drops. This

demonstration mimics an original experiment by Robert Shaw and Peter Scott at

the University of California Santa Cruz.11 It wonderfully illustrates several features of Chaos, particularly the transitions between various dynamics, which are

common to many systems. The results are so astounding that you may want to

bring your reading to the sink right now and experiment as you read along. Otherwise, you may not believe what you read.

What is the system? To recreate the Santa Cruz experiment, we need a faucet

handle or spigot that can be set at a slow flow rate and then be left alone so we can

observe drops emerging for a few minutes. We need enough water available so the

flow continues without interruption. Finally, we need some means to detect the

time intervals between drops. We don’t need a stopwatch, exactly, but we do need a

clear view of the drops, or we need the drops to land on some surface that resounds

loudly enough for us to detect patterns and rhythms as the drops fall. Fortunately,

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we need no assumptions about the water quality or any details about the size or

material of the spout. We just need drops.

What can we observe and measure in this system? We want to have a clear

view of the drops forming; this will give us some intuition as to why the flow

makes transitions between different kinds of behavior. We want to measure the

time intervals between drops. Shaw and Scott did this very precisely with a laser

beam and computer. For us, it will suffice to watch or listen as the drops land.

What’s the significance? Because of the difficulties in modeling any fluid,

there is absolutely no hope of simulating even a single drop forming and dropping

from a faucet. However, by measuring only one simple feature of the flow, the time

between events, we can still understand many characteristics of the system dynamics. We will observe, for example, specific transitions between behaviors,

transitions that are common to many chaotic systems. We will also gain some useful metaphors that are consistent with our intuitions of human behavior; but,

much more important, we will learn some specific things to expect in chaotic systems, even when we cannot model their dynamics.

So, what kinds of things can a sequence of water drops do? If the spigot is barely

open and the flow extremely slow, you should observe a slow, regular pattern of

drips. Leave the faucet alone, and the steady, aggravating, periodic rhythm will

continue far into the night. This pattern representssteady state, periodic output for

this system. Increase the flow ever so slightly, and the drips are still periodic, but

the time interval between drips decreases, that is, the frequency increases. At the

other extreme of its behavior, with the flow rate turned much higher, the water

will come out as a steady, unbroken stream. No real excitement yet.

The big question is: What happens in between these two extreme behaviors?

How does the flow make its transition from periodic drips to a steady stream?

We’ll move step by step through the transitions in this system. Low flow rates will

generate slow, regular drips. Increased flow will produce regular drips with new

patterns. After a certain point, the drop dynamics will prevent the faucet from

dripping regularly, and we will see evidence of Chaos.

Here’s how to proceed with the experiment. Start with slow, steady dripping.

Watch, for a moment, how the drops form. A single drop sticks to the end of the

spout and begins to fill with water, like the elastic skin of a balloon (figure 5).

Eventually the drop grows large enough to overcome its surface tension; it breaks

off and falls. The water left on the spout first springs back and recovers, then it begins to fill up to form the next drop: we will see that the time it takes to do all this is

a critical feature of the system.

15

Chaos Theory
Now, very gradually, increase the flow rate. For a while, you will still see (or

hear) periodic dripping, while the frequency continues to increase. However, before too long—and before the flow forms a solid stream—you will observe a different dripping pattern: an irregular pattern of rapid dripping interspersed with

larger splats of various sizes, all falling at erratic, unpredictable time intervals.

What causes the new behavior? The drops are beginning to form so quickly that a

waiting drop does not have time to spring back and completely recover before it

fills with water and breaks off. This is chaotic flow.

This deceptively simple demonstration is essential to our intuition of Chaos, for

several reasons. First, despite the nasty fluid physics that is impossible to model in detail, we are able to make simple measurements of time intervals and learn a great deal

about the system dynamics. We learn in this experiment that we need not dismiss as

intractable the analysis of a system that seems to be too large or has “too many variables” or “too many degrees of freedom.” (One can surely imagine quite a few military systems with these imposing properties, starting with a conventional battlefield.)

The water drops give us hope: by isolating and controlling one key parameter and

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making one straightforward measurement, we can still come to understand, and

perhaps manipulate, a very complicated system.

The second common feature of Chaos illustrated by the dripping faucet is the

presence of this key control parameter—in our case, the flow rate, controlled by the

spigot. Think of a control parameter as a single knob that allows regulation of the

amount of energy in the system. Not only does this energy control provide a means to

dictate the dynamics of the dripping faucet, but the transitions between various behaviors are identical in countless, seemingly unrelated, physical systems. In the

faucet, for instance, low flow generates periodic output; an increase in flow leads

to higher-period behavior; even higher flow—more energy in the system—allows

chaotic dynamics. Moreover, the Chaos appears when the system has insufficient

time to relax and recover before the next “event” occurs. These same transitions take

place in mechanical, electrical, optical, and chemical systems. Even more surprisingly, the transitions to more complicated behavior can occur at predictable parameter values (“knob” settings), a result that will be treated in the demonstration

that follows.

The critical conclusion is that our knowledge of chaotic systems teaches us to

expect specific behaviors in a system that displays periodic behavior; to expect to

see higher periods and Chaos with more energy input; and to forecast, in some

cases, parameter values that permit these transitions.

A third common characteristic of chaotic systems highlighted here is the fact

that the system dynamics are revealed by observing time intervals between events.

The physical event—droplet formation and break-off—is impossible to simulate,

so we avoid taking difficult measurements like drop diameter, drop mass, or velocity. Instead, we note the length of time between events; if we can measure this accurately, we are able to construct a return map or first-return map that clearly

indicates the various patterns of behavior (figure 6).

On the x-axis, a return map plots the time difference between, say, drops 1 and

2, versus the y-axis, which plots the time difference between the next two—here,

drops 2 and 3. When the flow is slow and periodic, the time intervals are regular, so

the time between the first drops is equal to the time between the next pair of drops.

On the plot, that means we are plotting x-values and y-values that are always equal,

so we see a single dot on the plot (figure 6a). So, if we ever observe a return map

where all the data fall on a single point, we can conclude the system is behaving periodically.

If we consider our time-difference measurement a record of the state of our system, then any limit behavior summarized on the return map represents an attractor

for the system. An attractor is a collection of states that a system “settles” into after

17

Chaos Theory
its transient dynamics die out. For the periodic flow, the attractor is a single point

on the return map.

The next transition in drop dynamics was reported by Shaw and Scott but is

fairly difficult to perceive in our home experiment. At a specific range of flow

rates, before the onset of Chaos, they observed a rapid string of drops that fell off in

close pairs. The flow showed a different periodicity, with one short time-step followed by a longer time-step: drip-drip drip-drip drip-drip. They confirmed the

existence of this change in periodicity by using a simple model of their system, but

its presence was clear on the return map (figure 6b).12 In this case, we say the sequence of drops has period-2, that the system has undergone a period doubling, and

that the attractor is the set of two points on the plot. For the record, this system

(like many others) experiences additional period doublings to period-4, period-8,

etc., before the onset of Chaos. These transitions, however, can be difficult to detect without sensitive laboratory equipment.

Finally, chaotic flow generates time intervals with no periodicity and no apparent pattern. However, the chaotic return map does not simply fill all the available

space with a random smear of points. There is some rough boundary confining the

chaotic points, even though they appear to fill the region in an erratic, unpredictable way (figure 6c). What is most astonishing isthat thissmear of pointsis amazingly reproducible. That is, we could run the experiment anywhere, with virtually

any water source, and a very similar pattern of points would appear on the return

map for chaotic flow. The structure of the collection of points is due to the dynamics of water drops in general, not the specific experimental machinery.

The water drop experiment offers additional hope that we might control a chaotic system. (What follows is easiest to demonstrate if you use a portable water

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spout, like an empty mustard bottle, but it may work well if your kitchen spout is

sufficiently flexible.) Set the spout so you have flow that remains chaotic. Then

jiggle the spout in some regular, periodic way. You might bounce the mustard bottle up and down, or simply tap the end of your kitchen faucet with a regular beat.

You should be able to find the right strength and frequency to perturb your system

and get it to change from Chaos back to periodic drips, with a periodicity that will

match the beat of your tapping. This is not very different from kicking your legs

on the swing. However, in this case, we are starting with a chaotic system and applying a relatively small disturbance to force the system to return to more stable

periodic behavior.

Later discussion will offer more details concerning Chaos control that has been

demonstrated successfully in both theory and practice. We will also consider issues of when we might prefer Chaos to be present, or not present, in a given system. At this point, it is interesting to notice that the perturbation of the dripping

faucet can drive a chaotic system into stable (periodic) behavior, while our previous perturbation of the park swing forced it to go from stable periodicity into

Chaos.

So far we have introduced two chaotic systems whose dynamics will lend some

insight to the behavior of more complicated military systems. The first was mechanical, the second fluid. Our next demonstration involves some simple (and inexpensive) electro-optics that can be picked up at any hardware store.

Night-light

I stumbled onto this demonstration quite accidentally, when I went to plug in a

small night-light in our bathroom—one of those automatic lights, about the size

of your palm, that turns on automatically when the room is dark. I plugged it into

the socket; the room was dark. Just before I pulled my hand away from the

night-light, it flickered rapidly. I put my hand near the light again and I saw the

same flicker. What interesting dynamics are hiding in this system?

What’s the system? To reconstruct this system we need a light source of any

kind that includes an automatic sensor that cuts off the electric current when it

senseslight (figure 7). We also need a dark room and a mirror,small enough so we

can move it around near the light, and supported in a stand so we can steady the

mirror and observe the light. Now, set the mirror so it reflects light from the bulb

back onto the sensor (as my palm had for my night-light in the bathroom). By adjusting the mirror’s distance from the sensor, we can vary the delay of feedback in

the system.

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Chaos Theory
What are we observing and measuring? When the mirror is close enough to

the night-light, about four to twelve inches, you should see it flicker. What’s going

on? Quite simply, the sensor is doing its best to fulfill its mission under unusual

circumstances. Initially, the room is dark, so the sensor turns its light on; but with

the mirror in place, as soon as the light turns on, the sensor picks up the reflected

light and correctly decides to shut off. Oh dear, the sensor mutters, the room is

dark again: time to turn on, and so on. The sensor detects and responds very

quickly, so we see the night-light flicker vigorously.

What exactly should you observe in this system? Like the dripping faucet, the

output to measure here is the frequency—in this case, the flickering frequency—the time difference between events. We would probably learn even more

by also monitoring the light’s intensity, but this requires fancier equipment than

most of us keep around the house.

What transitions should we expect? To see the range of dynamics possible in

this system, start with the mirror far from the sensor, about a yard or so away.

Slowly draw the mirror closer to the sensor. The first change you’ll see is a noticeable dimming in the light. Honestly, I don’t know yet whether this is a simple

change in the light’s output or a fluctuation whose frequency exceeds our visual

resolution. Do your best to locate the farthest point from the light where the dimming begins. Let’s label this distance d0. You may find that d0 is up to a foot or two

away from the light.

As you move the mirror even closer, the next change you’ll probably see is the

first sign of flickering. Once again, try to mark the farthest place where the flicker

is noticeable and label it d1. As you continue to move the mirror toward the sensor,

you will see various ranges of distances where the flickering displays other periodicities, and you ought to see at least one region where the reflected feedback

drives the system into Chaos: irregular bursts of brightness and flickering. Mark

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the distances, as well as you’re able, where you see transitions: d2, d3, etc. If you

don’t observe any Chaos, how might you alter your system? There are several accessible control variables: try a different (cleaner?) mirror; change your reflection

angle (are you hitting the sensor efficiently?); or use a brighter light bulb.

What’s the significance? The dynamics exhibited by the night-light system

point to several critical insights that will help us apply the general results of Chaos

theory to other systems. The first new insight comes from the dynamics we can

generate by imposing feedback on a system. Of course, the use of feedback itself is

not new, but the output we observe in the night-light provides a new understanding of the dynamics that control theorists have wrestled with for decades.

The night-light demonstration also offers practical new approaches to the

study and control of a system whose output sometimes fluctuates. In particular,

once I observed periodic behavior in the system (accidental though it was), I knew

to expect several ranges of periodicity and Chaos if I varied one of the control parameters available to me. That is, my experience with Chaos gave me very specific

behaviors to expect, in addition to obvious suggestions of ways to control the dynamics. Moreover, I had some idea of the kinds of dynamics to expect without knowing

anything about the internal workings of the system!

This universality of chaotic dynamics underscores the power of understanding

the basic results of Chaos theory. Certainly, not every system in the world is capable of generating Chaos, but in many systems we can control and analyze a system

with no need for a model.

Here are two simple examples of the kind of analysis that is possible, even without a model. For this analysis we need only the list of distances (d0, d1, etc.) where

we noted transitions in system behavior. First of all, we know that the signal in our

demonstration, the light from the bulb, is traveling at a known constant,

c = 3.0 x 108 meters/second. Therefore, we can quickly assemble a list of important

time constants for this system by dividing each of our distances by the speed of

light, c. These time constants directly affect important transitions in the light’s

output; we know we can alter the system’s behavior by applying disturbances that

are faster or slower than these key time delays. Other time constants we might

consider include the frequency of the electric current and the frequency (color) of

the light.

A second numerical result gives us some hope of predicting the parameter values where transitions in dynamics should occur. Dr. Mitchell Feigenbaum of Los

Alamos National Laboratory, New Mexico, discovered that many chaotic systems

undergo transitions at predictable ranges of their parameter settings. In particu21

Chaos Theory
lar, he compared the ratio of differences between key parameter values, which for

us translates into calculating a simple ratio:

( )/ ( ) d d dd 01 1 2 − − (2)

He discovered that this ratio approaches a universal constant, approximately

4.67—now known as the Feigenbaum number—which appears in chaotic systems

where Chaos arrives via period doubling, such as in our dripping faucet. This amazing result tells us when to anticipate changes in dynamics. For instance, if our first

transition happens when the mirror is 12 inches out, and the second transition occurs

at 8 inches, we note the difference in these parameter values, 4 inches (figure 8).

Feigenbaum tells us that we ought to expect another transition (d1 - d2) in another

4/4.67 inches, or 0.85 inches from the point of the last transition.

Now, in any system where we try to make predictions this way, we may face

other complications. Our moving mirror, for example, may actually change several control parameters at once, such as brightness and focus. However, the mere

existence of the Feigenbaum constant gives us hope of being able to anticipate

critical changes in complicated systems; in fact you should find that this prediction works quite well for your measurements with your night-light system!

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This third home demonstration brings to light several key results that apply to

many chaotic systems. In particular, the demonstration illustrates: the potential

dynamics that can be generated by imposing feedback on a system; very specific

behaviors to expect in chaotic systems; suggestions of ways to control a system’s dynamics; ways to analyze and control a system with no need for a formula or model;

and how the Feigenbaum constant helps anticipate system transitions.

Other Home Demonstrations

Many other systems you see every day exhibit chaotic dynamics. Watch the

cream stir into your coffee. How does a stop sign wobble in a rough wind? Think

about the position and speed of a car along a major city’s beltway. What are the

states of all the cars traveling the beltway?13 Watch the loops and spins of a tire

swing in a park. If you are really adventurous, hook up your home video camera as

it shows a live picture on your television set, then aim the camera at the television

set and zoom in and out to generate some exciting feedback loops.

Consider how you might carefully describe those systems. What can you observe and measure in those systems? What are the important parameters? As the

control parameters increase or decrease, what transitions in behavior should you

expect?

I have summarized several home demonstrations in this chapter to develop

some intuitions and to introduce the vocabulary and tools of dynamical systems. I

hope they spark your imagination about comparable systems that interest you.

More importantly, they may represent your first understanding of chaotic systems, so you can begin to expect and anticipate Chaos in your systems. The next

chapter adds more detail to the vocabulary and ideas introduced here.

23

Chaos Theory

II

Definitions, Tools, and Key Results

Of all the possible pathways of disorder,

nature favors just a few.

14

James Gleick

The previous chapter described a few simple demonstrations so that we could

begin to develop some basic intuition for chaotic dynamics. I also used

some of the associated Chaos vocabulary in those demos in order to introduce the

definitions in the context of real systems. Detailed definitions require too much

time to present in full. However, we need to review some vocabulary with care,

since the tools for observing and exploring complex dynamics are linked closely

to the vocabulary we use to describe our observations. Rather than pore through

excruciating details of precise definitions, this chapter concentrates on the consequences of the definitions. The focus will be to answer questions such as, “What

does it mean to be an attractor?”

We will narrow the discussion to the most important issues: What is Chaos?

How can we test for it? What does it mean to me if I have Chaos in my system? By

concluding with a summary of Chaos theory’s key results, the way will be paved

for later chapters that suggest ways to apply those results.

This chapter concentrates on two classic chaotic systems: the logistic map and

Lorenz’s equations for fluid convection. These two examples reinforce some of
the lessons learned in the last chapter, and they make a nice bridge to the military

systems examined in the next chapter. In particular, I will apply common Chaos

tools to these two examples so that the reader can visualize the kind of new information Chaos theory can provide about a system’s behavior.

The Logistic Map

What is the system? Our first case looks at the work of biologist Robert May,

who in the early 1970s researched the dynamics of animal populations. He developed a simple model that allowed for growth when a population of moths, for instance, was small; his model also limited population growth to account for cases of

finite food supply.

15 Hisformula is known asthe logisticequation orthe logisticmap.

What are we observing and measuring? The logistic map approximates the

value of next year’s population, x[n+1], based on a simple quadratic formula that

uses only information about this year’s population, x[n]:

xn xn xn [ ] [ ]( [ ]) += − 1 1 λ .

The parameter λ quantifies the population growth when x[n] is small, and λ

takes on some fixed value between 0 and 4. In any year n, the population x[n] is

measured as a fraction, between 0 and 1, of the largest community possible in a

given physical system. For example, how many fish could you cram into the cavity

filled in by a given lake? The population x[n] expresses a percentage of that absolute maximum number of fish.

It is not too hard to illustrate the dynamics of the logistic map on your home

computer. Even with a spreadsheet program, you can choose a value for λ and a

starting value for x[1], and calculate the formula for x[2]. Repeated applications

of the formula indicate the changes in population for as many simulated years as

you care to iterate. Some of the dynamics and transitions you should expect to see

will be discussed in this chapter.

What’s the significance? One helpful simplification of May’s model was his

approximation of continuously changing populations in terms of discrete time intervals. Imagine, for instance, a watch hand that jerks forward second by second

instead of gliding continuously. Differential equations can describe processes

that change smoothly over time, but differential equations can be hard to compute. Simpler equations, difference equations, like the logistic map, can be used for

processes that jump from state to state. In many processes, such as budget cycles

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and military force reductions, changes from year to year are often more important

than changes on a continuum. As Gleick says, “A year-by-year facsimile produces

no more than a shadow of a system’s intricacies, but in many real applications the

shadow gives all the information a scientist needs.”16

The additional beauty of the logistic map is its simplicity. The formula includes nothing worse than an x

2 term—how badly can this model behave? Very

shortly, you will find that this simple difference equation produces every significant feature common to most chaotic systems.

The Lorenz Equations

What’s the system? Our second case began as a weather problem. Meteorologist Edward Lorenz wanted to develop a numerical model to improve weather

predictions. Focusing on a more manageable laboratory system—the convection

rolls generated in a glass of heated water—Lorenz modified a set of three fairly

simple differential equations.17

′=− +

′= − −

′=− +

x xy

y Rx y xz

z Bz xy

σ σ (3)

What are we observing and measuring? The phase variables, x, y and z combine

measurements of the flow as the heated water rises, cools, and tumbles over itself

(figure 9a). The x variable, for instance, is proportional to the intensity of the convection current; y is proportional to the temperature difference between the rising

27

Chaos Theory

18
and falling currents. The numbers σ, R and B are the system’s physical parameters,

which Lorenz set at σ = 10, R = 28, and B = 8/3. As the phase variables change in

time, they trace out fascinating patterns, like those illustrated in figure 9b.

What’s the significance? The Lorenz equations crudely model only one simple feature of fluid motion: temperature-induced convection rolls. However, even

in this simple system, Lorenz observed extreme sensitivity to initial conditions as

well as other symptoms of Chaos we will see momentarily. He clearly proved that

our inability to predict long-term weather dynamics was not due to our lack of

data. Rather, the unpredictably of fluid behavior was an immediate consequence

of the nonlinear rules that govern its dynamics.

Definitions

Now that we have two new systems to work with, along with the “experience” of

our home demonstrations, let’s highlight the vocabulary we will need to discuss

more complicated systems.

Dynamical System. Recall how we defined a system as a collection of parts along

with some recipe for how those parts move and change. We use the modifier dynamical to underscore our interest in the character of the motions and changes. In

the case of the logistic map, for example, the system is simply a population measured at regular time intervals; the logistic equation specifies how this system

changes in time.

Linear and Nonlinear. The adjective linear carries familiar geometrical connotations, contrasting the linear edge of a troop deployment, for example, with the

curved edge of a beach. In mathematics, the concept of linearity takes on broader

meaning to describe general processes. We need to understand linearity because isolated linear systems cannot be chaotic. Moreover, many published explanations of linearity make serious errors that may prevent you from grasping its significance.

Some authors condense the definition of linearity by explaining that in a linear

system the output is proportional to the input. This approach may be helpful

when we model the lethality of certain aircraft, saying that three sorties will produce three times the destruction of a single sortie. However, there is at least one

further level of insight into linearity. That insight comes from our first home

demonstration, the pendulum.

Even though a pendulum swings in a curve and we describe its motion with

sine and cosine functions, an ideal pendulum is a linearsystem! It’s linear because

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the equation that defines its motion has only linear operations: addition and multiplication by constants. Common nonlinear operations include exponents, trigonometric functions, and logarithms. The important consequence is that the

solutions to most linear systems are completely known. This may not seem

earth-shattering for a single pendulum, but many oscillating systems—such as vibrating aircraft wings, mooring buoys, and concrete structures subjected to shock

waves—behave just like a collection of coupled pendulums. Therefore, as long as

they aren’t regularly “kicked” by external forces, those real systems are just enormous linear systems whose range of possible motions is completely known.

Without delving into the subtleties of more analytical definitions, here are

some important consequences of the property of linearity:

• The solutions to linear systems are known (exponential growth, decay, or

regular oscillations), so linear systems cannot be chaotic.

• “Kicking” or forcing an otherwise linear system can suffice to drive it into

Chaos.

• If Chaos appears in a system, there must be some underlying nonlinear

process.

This discussion of linearity should serve as a wake-up call. Basically, if you

have a system more complicated than a pendulum, or if you see an equation with

nonlinear terms, you should be alert for possible transitions from stable behavior

to Chaos. This is certainly a simplification, since many systems include additional control mechanisms that stabilize their dynamics, such as feedback loops in

human muscles or in aircraft control surfaces. However, the minimum ingredients that make Chaos possible are usually present in systems like these. In the absence of any reliable control, unpredictable dynamics are not difficult to generate.

Phase Space and Trajectories. A system consists of components and their rules of

motion. To analyze a system one must decide exactly what properties of those components to measure and record. The time-dependent properties necessary to determine the system dynamics are known as the system’s phase variables. The

collection of all possible combinations of values those variables can attain is then

the phase space for our system.

Phase space is the canvas where a system’s dynamics are painted. The Lorenz

equations, for example, define the time-dependent changes of fluid flow in a

heated beaker of water. If we start at some initial state and let the system evolve in

time, we can track how the three system variables change. We can then plot that

information with a three-dimensional curve (figure 9b). Notice that the curve

does not directly illustrate the physical motion of the water. Rather, the curve indicates changes in all three phase variables; at least one of these—the temperature

29

Chaos Theory
gradient, y—quantifies changes we cannot see. The plot’s entire three- dimensional space constitutes the phase space for the Lorenz equations; we call the single curve that leaves a particular initial state a trajectory (ortrajectory in phase space)

for that initial condition.

Parameter. A parameter is a quantity that appears as a constant in the system’s equations of motion. The logistic map has only one parameter, λ, which expresses the

rate of growth for small populations. A pendulum’s parameters include its mass

and the length of its bar. Sometimes a parameter expresses a physical constant in the

system, such as the gravitational constant for the pendulum. Most important, a

system parameter often represents a control knob, a mechanism to control the

amount of energy in a system.

For instance, we saw earlier how changes in flow rate, the key parameter for the

dripping faucet, drove transitions in system output. In the following section on

Chaos Tools, we’ll see how the logistic map undergoes transitions as we increase λ

from 0 to 4. It is important to note that even in relatively simple systems like the

faucet, there are many influential parameters that are not easily controlled: spout

diameter, mineral content of the water, local humidity, spout viscosity, etc. One

crucial skill for any decision maker is the ability to identify all the parameters accessibleto external control, and to isolatethose parameters that have the greatest influence on a system.

Sensitivity to Initial Conditions (SIC). Any system “released” from its initial state

will follow its laws of motion and trace some trajectory in phase space, as we saw

with the logistic map above. However, if a system is sensitive to initial conditions

we also know that any two initial states that deviate by the slightest amount must

follow trajectoriesthat diverge from each other exponentially. Considerfigure 10.

The lower series started from an initial population only slightly greater than the

upper series; after about sixteen iterations, the two trajectories bear no resemblance to each other. This is an illustration of SIC.

We can measure how fast neighboring trajectories tend to diverge. At any given

point in phase space, a Lyapunov (lee-OP-uh-noff) exponent quantifies this rate of

divergence. This exponent has properties that come from its role as the constant k

in the exponential function e

kt. If k is negative, then small disturbances tend to get

smaller, indicating no SIC; if k is positive, small perturbations increase exponentially. With these measurements, we can estimate how “touchy” a system is, how

vulnerable the system may be to external disturbances, and how unpredictable the

consequences of our actions may be. We can often calculate an average Lyapunov

exponent for an entire region of phase space. This allows us to compare two sysThe Newport Papers

30
tems, or two scenarios, and decide which one tends to be more or less predictable.

Information like this could prove valuable for prioritizing the courses of action

available to a commander.

Many systems, as we say, are SIC, including some non-chaotic systems. For example, take the simplest case of exponential growth, where a population at any

time t is given by a recipe such ase

3t

. This system is SIC, but certainly not chaotic.

What does this mean for us? If a system is SIC, you are not guaranteed to find

Chaos; if, however, a system is not SIC, itcannot exhibit Chaos. Thus we have identified SIC as a necessary but not sufficient condition for Chaos to occur.

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Chaos Theory

19
Attractors. Despite the fact that chaotic systems are SIC, and neighboring trajectories “repel” each other, those trajectories still confine themselves to some limited

region of phase space. This bounded region will have maximum and minimum parameter values beyond which the trajectories will not wander, unless perturbed.

In the logistic equation, the population remains bounded between the values of 0

and 1, though it seems to take on every possible value in between when it behaves

chaotically.

In the Lorenz equations, the trajectories also stay within finite bounds, but the

trajectories do not cover all the possible values within those limits. Instead, a single trajectory tends to trace out a complicated, woven surface that folds over itself

in a bounded region of phase space (refer to figure 9). The collection of points on

that surface is an attractor for those dynamics; the classic Lorenz attractor is a particularly striking example.

Left to itself, a single trajectory will always return to revisit every portion of its

attractor, unless the trajectory is perturbed. All chaotic, orstrange, attractors have

this mixing property, where trajectories repeatedly pass near every point on the attractor. Envision where a single droplet of cream goes after it is poured into coffee.20 Or imagine the path of a single speck of flour as it is kneaded into a ball of

dough. If the mixing continued long enough, the small particle could be expected

to traverse every neighborhood of its space. Actually, one way to sketch a rough

image of an attractor is simply to plot a single trajectory in phase space for a very

long time.

Transient states are all the initial conditions off the attractor that are never revisited by a trajectory. If we gather together all the transient states that eventually

evolve toward a single attractor, we define the basin of attraction for that attractor.

Thus, the basin represents all the possible initial states that ultimately exhibit the

same limit dynamics on the attractor. In the Lorenz system, for instance, we might

start the system with a complicated temperature distribution by dropping an ice

cube into hot water. However, that transient extreme will die out, and after a while

the system must settle down onto the collection of temperature variations that

stay on the attractor. Because of SIC, the precise state of the Lorenz system at any

given time cannot be predicted. However, because the attractor draws dynamics

toward itself, we do know what the trends in the dynamics have to be!

When those trends are examined closely, a single trajectory will be found to

visit certain regions of the attractor more often than others. That is, if we color each

point on the attractor based on how often the trajectory passes nearby, we will

paint a richly detailed distribution of behavior on the attractor. To picture this, visualize the distribution of cars on the interstate beltway around a big city. At any

time of a given day, we could note the number of vehicles per mile and begin to

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identify patterns of higher traffic density for certain times of day. We could continue and consider the distribution of cars on whatever scale interests us: all interstates, all streets, or just side streets. Even though we cannot predict the number of

cars present on any particular street, these distributions and patterns give us crucial information on how the overall system tends to behave.

The properties of attractors are key signposts at the junction where Chaos

theory matures past a mere metaphor and offers opportunities for practical applications. Attractors provide much more information than standard statistical observations. This is because an attractor shows not only distributions of

system states but also indicates “directional” information, that is, how the system tends to change from its current state. As a result, when we construct an attractor we reconstruct an image of the system’s global dynamics—without appealing

to any model. In subsequent chapters, we will show how this reconstruction allows us to predict short-term trajectories and long-term trends, to perform pattern recognition, and to carry out sensitivity analysis to help us make strategic decisions.

Fractal. Though there are standard definitions of several types of fractals, the important consequence for us is that fractals describe the complexity, or the amount

of detail, present in objects or data sets. A well-defined line, like the y-axis on a

graph, is one-dimensional because one piece of information, the y-coordinate, suffices to pinpoint any position on the line. To get an idea of what dimension means in

a fractal sense, first imagine using a microscope to zoom in on an ideal line. However intently we zoom in, the most detail we can expect to see is a razor-thin line

cutting acrossthe field of view (figure 11a).If, as a second case, we focusthe microscope on a two-dimensional object, like a square, sooner or later the narrow field of

view will fill with an opaque image. We need two coordinates to pinpoint any place

on that image.

On the other hand, a fractal image has a non-integer dimension. An image with

dimension 1.7, for instance, has more detail than a line but too many holes to be

worthy of the title two-dimensional. Fractal images contain infinite detail when

we zoom in (figure 11b). The good newsisthat the extraordinary detail present in

fractal images can be generated by very simple recipes.

The term “fractal” refers specifically to a mathematical dimension defined by

executing this zooming process very precisely. First, assume the line in figure 11 is

a centimeter (cm) long. It only takes one circle of 1 cm diameter to completely

cover the line. If I cover it with circles 1/2 cm across, I need two. Similarly, I need

17 covering circles of diameter 1/17 cm, or 1986 circles of diameter 1/1986. Since

the number of circles needed to cover the image scales is (1/diameter) to the first

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Chaos Theory
power, we say that image has dimension “1.” This comes as some relief, since we

all survived geometry class knowing that lines are one-dimensional.

Now consider the complex fern in figure 11. If its total length is about 1 cm, a

single large circle will cover it. However, as we start to cover it with smaller and

smaller circles, we find that we need fewer circles than we would need if we were

trying to cover a solid square (of dimension 2). In fact, the number of circles

needed scales like (1/diameter) raised to the 1.7 power. We say, then, that the fern

has dimension 1.7, and since that dimension is not an integer, or fractional, we call

the image a “fractal.”

The study of fractal geometry becomes important to military applications of

Chaos in three main areas: image compression, dimension calculation, and basin

boundaries. In image compression, the infinite detail generated by simple sets of

fractal instructions allows mathematical instructions rather than pixel-by-pixel

values to be transmitted; the image can then be recreated by the receiver using the

instructions.

The second application, dimension calculation, is possible with time series as

well as with geometric figures; when we calculate the dimension of a sequence of

data points, we get an estimate of the minimum number of variables needed to model

the system from which we measured the data. Often the estimate lies very close to

the number of variables needed in a model, thus saving analysts the struggle of developing overly complex situations.

Thirdly, many systems that have two or more attractors also have two or more

basins of attraction. Very often, the boundaries between basins are not smooth

lines. Instead, the basins overlap in fractal regions where one initial condition

may lead to steady state behavior, but any nearby initial condition could lead to

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21
completely different behavior. Considerthe illustration in figure 12, the basins of

attraction for a numerical model. All the initial conditions (white areas) lead to

one kind of behavior; all the dark points lead to entirely different behavior. A

commander making decisions in such an environment will have to be

alert—small parameter changes in certain regions produce dramatic differences

in outcomes.22 For instance, the pictured decision space might simulate, on one

axis, the number of troops available for reinforcement, while the other axis indicatestime intervals between sending in fresh troops.If the combatsimulation indicates eventual victory with a black dot, and defeat with white, commanders

would need to choose reinforcement strategies with great care in orderto turn the

scenario’s outcome in their favor.

recognize the word in otherreferences.In the context of the demonstrationsthus

far, a bifurcation is simply a transition in dynamics. The faucet, for example, drips

slowly when the flow rate islow. Atsome higherflow rate,the drops come out with

period-2; we say the system has undergone a bifurcation from one kind of period35

Chaos Theory

Bifurcation. Bifurcation theory represents an entire subdiscipline in the study of

dynamical systems. I mention bifurcations here for two reasons. First, so you will

23
icity to another. A bifurcation is a transition in system dynamics due to a change

in a control parameter.

The second reason for offering this new terminology is to highlight the universality of bifurcation types. That is, when one system parameter is changed, you

may see subtle bifurcations or catastrophic ones, but a few classes of bifurcations

are common to many dynamicalsystems.24 Recall the discussion of transitionsin

the night-light demonstration. The transitions came atsmaller and smallerintervals, roughly according to patterns predicted by the Feigenbaum constant (refer

to figure 8).Feigenbaum first discovered this constantthrough hisstudy ofthe logistic map, where transitions occurin the same pattern asin the night-light. Overall, the most important consequence of Feigenbaum’s discovery is that the same

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Reproduced with the kind permission of Springer-Verlag New York, Inc., after

H.O. Peitgen/H. Jürgens/D. Saupe, Chaos and Fractals (figures 1.35 and 1.36, page

59), © 1992. Any further reproduction is strictly prohibited.

25
transitions he observed in the logistic map also appear in many diverse physical

systems.

Dense, Unstable, Periodic Orbits. Consider one last feature of the logistic map that

ultimately makes it possible for us to control chaotic systems. Chaos control will be

addressed in the next chapter; for now, we take a few steps through the dynamics

of the logistic map in order to glimpse the complicated activity on an attractor, as

illustrated in figure 13.

Suppose we set the parameter to a small value, say λ = 1.8. We can start the system with x[1] anywhere between 0 and 1, and successive iterations of the logistic

equation will always drive the value of x[n] toward 0.44, a stable, fixed point. If we

increase λ to 2.75, the system still has a stable, fixed point, but that point is now

around 2/3. Raising the control parameter produces no qualitative change in behavior. However, if we raise λ slightly above 3, the system does not settle into a

fixed point but falls into a cycle of period-2. Thus, at λ = 3 we see a bifurcation

from stable to periodic behavior.

Transitions come hand-in-hand with changes in stability. Any system might

have both stable and unstable behaviors. For instance, the equations governing a

pencil standing on its point have a good theoretical equilibrium one with the center

of gravity directly above the point—but we cannot stand a pencil on its point, because that state is unstable. That is, the slightest perturbation draws the system

away from that state. On the other hand, a marble lying at the bottom of a bowl

stays there, because if the marble is perturbed slightly in any direction, it just rolls

back.26

The important feature for us hides in the chaotic trajectory “smeared out” in

figure 13, when λ = 4. Inside that smear—the attractor for this chaotic system—many periodic cycles still exist; on paper, that is. The fixed point, for instance, still lives at the place on the graph where the parabola intersects the

diagonal. However, that point is unstable, so a trajectory can never approach it.

Similarly, we can calculate trajectories of period-2, period-3, every possible period. In fact, there are infinitely many unstable, periodic trajectories woven

through the attractor, woven thickly in a way mathematicians call dense. That

means that every area surrounding every point on the attractor is crowded with

these “repelling,” unstable, periodic trajectories.

So, on one hand, it is not useful to locate any of these periodic behaviors, because all these trajectories are unstable. On the other hand, recent experiments

have demonstrated ways to force the system to follow one of these periodic behaviors. This is the power of Chaos control; as we will see later, the density of these trajectories is the property that makes this control possible.

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Chaos Theory
So How Do We Define Chaos?

A chaotic system MUST be:

• bounded;

• nonlinear;

• non-periodic;

• sensitive to small disturbances; and,

• mixing.

This is, perhaps, not so much a definition as it is a list of necessary ingredients

for Chaos in a system. That is, without any one of these properties, a system cannot

be chaotic. I believe my list is also sufficient; therefore, if a system has all these

properties, it can be driven into Chaos.

Also, a chaotic system usually has the following observable features:

• transient and limit dynamics;

• parameters (control knobs);

• definite transitions to and from chaotic behavior; and

• attractors (often with fractal dimensions).

What is the significance of these properties? Measurements of transient and limit

dynamics in a system provide new information not available to us before the advent

of Chaos theory. Our comprehension of the role of parameters in system dynamics

offers opportunities for new courses of action, to be described in subsequent chapters. Finally, the common properties of system transitions and attractors suggest

new expectations of system behavior, as well as new strategies for coping with those

expectations. For other, more detailed characteristics of chaotic data—such as exponentially decaying correlation and broad power spectra—you can refer to any

one of the texts described in chapter V, “Suggestions for Further Reading.”

Random. You may look at the above definition of Chaos and wonder if the processes we call “random” have those same properties. For those interested in more

detail, a discussion of one definition of “random” appears in the appendix. However, I will pause here to focus on one difference between random and chaotic dynamics. Please be aware that we are ignoring some large issues debated by Chaos

analysts. Some argue, for instance, that the kind of dynamics we now call “random”—like a roulette wheel—simply come from chaotic systems, with no random variables, where we just do not know the model. In other cases, “noise,” or

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random imperfections in our measurements—like radio static—may come from

Chaos that happens on a scale we have not yet detected.For our purposes, the primary

feature distinguishing chaotic from random behavior is the presence of an attractor that

outlinesthe dynamicstowards which a systemwillevolve.

27Existence ofsuch an attractor gives us hope that these dynamics are repeatable.

In the water-drop experiment, for example, if results were random, the experiment would not be repeatable. However, if you and I both run this test and I list my

experimental parameters for you, such as nozzle diameter and flow rate, the key

features of this system’s dynamics will be replicated precisely by our two separate

systems. Slow flow is always periodic. The system undergoes period doubling (period-2, then period-4, . . .) on the way to Chaos, as we increase the flow rate. Most

important, for high flow rates, your chaotic return map for time differences between drops will produce a smear of points nearly identical to mine. If the system

were exhibiting random behavior, these global features would not be reproducible.

The Chaos “Con”

Before leaving this review of basic Chaos vocabulary, we need to examine the

common mistakes and misinterpretations that appear in many papers on the subject. The sum of these errors constitutes the Chaos “con,” the unfortunate collection of misleading publications that tend to crop up when writers investigate new

topics. The con may come from well-intentioned authors who are new to the subject but miss some key concepts because they are constrained by time. Other cons

may come in contract proposals from cash-starved analysis groups who might try

to dazzle their readers with the sheer volume of their Chaos vocabulary. It is very

important to avoid the con, both innocent and intentional, but most of all, don’t

con yourself by making any of the following common errors.

“Chaos is too difficult for you.” Don’t let anyone fool you: if you finished college, you

can follow the basics of Chaos. Be suspicious of anyone who tries to tell you that

the general concepts are beyond your grasp. Some authors will disguise this false

claim with subtle references to the “mysteries of Chaos” or “mathematical alchemy” or other vocabulary designed to intimidate their readers. Don’t believe it,

and don’t pay these folks to teach you Chaos. You can learn it—just remember to

take your time.

“Linear is. . . .”Remember that some writers will oversimplify the definition of linearity by waving their pen quickly at some phrase like “output is proportional to

input.” That comment is true only if a system’s output and input are very carefully

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Chaos Theory
defined. Never forget that pendulums, swings, and springs are all linear systems!

Make sure the author’s definition of linearity admits these three important physical systems.

Bifurcation. What exactly bifurcates? Trajectories don’t bifurcate, as some authors

have claimed. A single trajectory can do only one thing. We may have a limited capacity to predictthat behavior, but—as a light bulb can be only on or off at any fixed

time—a single system can evolve through only one state at a time. Remember that

a bifurcation is a qualitative change in system behavior that we observe as we

change parameter settings. The bifurcation, or branching, takes place on plots of

parameter values.

“Complicated systems must be chaotic.” The fact that a system is complicated or has

many components does not necessarily mean that it allows Chaos. For instance,

many large systems behave like coupled masses and springs, whose linear equations of motion are completely predictable. Indeed, an old-fashioned clock is extremely complicated—but its very essence is to be predictable. Similarly, other

large systems include reliable control mechanisms that damp out perturbations

and do not permit sensitive responses to disturbances. Such systems do not exhibit Chaos.

“We need many variables for Chaos.” Many of the same authors who claim that big

systems must be chaotic also propagate the fallacy that simple systems cannot exhibit Chaos. Nothing could be further from the truth. In fact, the power of Chaos

theory is that the simplest interactions can generate dynamics of profound complexity. Case in point: the logistic map produces every symptom of Chaos described in this paper.

“Butterflies cause hurricanes.” When Edward Lorenz presented his findings of SIC

in weather systems, he described The Butterfly Effect, the idea that the flapping

wings of a butterfly in one city will eventually influence the weather patterns in

other cities. This phenomenon is a necessary consequence of the sensitivity of

fluid systems to small disturbances. However, the butterfly effect often gets fuzzy

in the translation. Be wary of authors who suggest that a butterfly’s flap in California will become amplified somehow until it spawns a hurricane in Florida. Believe

it or not, several often-cited reports make this ridiculous claim. Make no mistake,

if a weather system has enough energy to produce a hurricane, then the storm’s

path will be influenced by butterfly aerodynamics across the globe. However, the

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system does not amplify small fluid dynamics; rather, it amplifies our inability to

predict the future of an individual trajectory in phase space.

“Chaos” versus “chaos.” One of the first signals of a weak article is when the author

inconsistently mixes comments on mathematical Chaos and social chaos. Unless

we can distinguish between the two, we cannot get past the metaphors of Chaos to

practical applications. As will be explained below, the existence of Chaos brings

guarantees and expectations of specific phenomena: attractors; complex behavior

from simple interactions; bounded, mixing dynamics; and universal transitions—from stable to erratic behavior—that make Chaos control possible.

The worst consequence of the Chaos con is that the well-intentioned reader

may not discern the important results of Chaos theory. These results highlight the

common characteristics of chaotic dynamics, a useful template for the kinds of dynamics and applications we should expect in a chaotic system. A review of the

most important results follows here; a discussion of their applications constitutes

the remaining portion of this essay.

Tools of Chaos Analysts

One of the most important outcomes of the study of Chaos theory is the extraordinary array of tools that researchers have developed in order to observe the behavior of nonlinear systems. I cannot emphasize enough that these tools are not

designed solely for simulated systems. We can calculate the same information from

experimental time series measurements when there is no model available, and often when we can measure only one variable in a multi-variable system! Moreover,

decision makers need the skills to differentiate random behavior and Chaos, because the tools that allow us to understand, predict, and control chaotic dynamics

have no counterpart in random systems.

For the military decision maker who can use these tools, two issues stand out:

What are the preferred tests for deciding if a system is chaotic?

How can we tell the difference between randomness and Chaos?

The analytical tools used by Chaos analysts answer these questions, among many

others. Our brief summary of the most basic tools begins with an important reminder.

We always need to begin our analysis by answering two questions: what is the system,

and what are we measuring? For example, recall the dripping faucet system, where

we observe the dynamics not by measuring the drops themselves but by measuring

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Chaos Theory
time intervals between events. Only after we answer those two questions should we

move on to consider some of the qualitative features of the system dynamics:

• What are the parameters? Can we control their magnitude?

• Does the system perform many repetitions of its events?

• Are there inherent nonlinearities or sources of feedback?

• Does the phase space appear to be bounded? Can we prove it?

• Do we observe mixing of the phase variables?

When we have a good grasp of the general features of a system, we can begin to

make some measurements of what we observe. We should note, however, that our

aim is not merely to passively record data emitted from an isolated system. Very

often our interest lies in controlling that system. In an article on his analysis of

brain activity, Paul Rapp summarizes:

Quantitative measures [of dynamical systems] assay different aspects of behavior, and they have different strengths and weaknesses. A common element of all

of them, however, is an attempt to use mathematics to reconstruct the system

generating the observed signal. This contrasts with the classical procedures of

signal analysis that focus exclusively on the signal itself.

28

Therefore, keep in mind that the tools presented here are not used for observation

only. They provide the means to re-create a system’s rules of motion, to predict that motion over short time scales, and to control that motion.

Depicting Data. We have already encountered most of the basic tools used for observing dynamical systems. The two simplest tools—time series plots and phase diagrams—display raw data to give a qualitative picture of the data’s bounds and

trends. A time series plot graphs a sequential string of values for one selected

phase variable, as in the plot of population variation for the logistic map in figure

10. Sequential graphs give us some intuition for long-term trends in the data and

for the system’s general tendency to behave periodically or erratically.

Phase diagrams trace the dynamics of several phase variables at the same time,

as the Lorenz attractor does in figure 9. The first piece of information apparent

from a good diagram is the nature of the system’s attractor. The attractor precisely

characterizes long-term trends in system behavior—how long the system spends

in any particular state. This information translates directly into probabilities.

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Attractors and Probabilities. As a demonstration of translating attractor dynamics

into probabilities, consider the chaotic trajectories of the logistic map shown in

figure 13. The smear of trajectories makes it obvious that the population x[n]

takes on most of the values between 0 and 1; but is the smear of valuesevenly distributed across that range? One way to find out is to build a quick histogram: divide

the interval from 0 to 1 evenly into a thousand subintervals; keep a count of every

time the evolving population x[n] visits each subinterval. Figure 14 showsthe results of such a calculation; we see from the figure that the trajectory of the logistic

equation spends more time closer to 0 and 1 than it does to other values. To illustrate, if this equation modeled the number of troops assigned to a certain outpost,

a distribution like this would tell a commander that the site tends to be fully

staffed or nearly vacant, with noticeably less probability of other incremental options.

Probability information like this has several immediate uses. First, of course,

are the probability estimates that commanders require to prioritize diverse

courses of action. Second, analysts can use this information to compare models

with real systems, to gauge how well the distribution of a simulated system relates

to real data. Third, since many simple chaotic models use non-random formulas

to generate distributions of behavior, the resulting distributions can be used in

various simulations to replace black-box random number generators. We will explore these applications in greater detail in chapter IV.

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Chaos Theory

29
Attractors and Sensitivity. As a single trajectory weaves its way through its attractor,

we can also calculate local Lyapunov exponents(see pages 30-31) atthe individual

points on the attractor as well as an average Lyapunov exponent for the entire system. This exponent measures how sensitive trajectories are to small disturbances.

Therefore, details about these exponents can guide decision makers to particular

states where a system is more or less vulnerable to perturbation. The same exponents can also be calculated for various ranges of parameter settings so that commanders can discern which variables under their control may produce more

predictable (or unpredictable) near-term outcomes.

Embedding. However directly we might calculate system features like attractors

and Lyapunov exponents, how can we apply these tools to a real system where we

have no descriptive model? Suppose we have a complicated system—like the dripping faucet—that gives us a time series with only one variable. What can we do?

The answer comes from a powerful technique known as embedding. Very simply, we can start with a sequence of numbers in a time series, and, instead of isolating them as individual pieces of data, we can group them in pairs. The resulting

list of pairs is a list of vectors that we can plot on a two-dimensional graph. We can

also start over and package the data in groups of three, creating a list of vectors we

can plot in three-dimensions, and so on. This process embeds a time series in

higher dimensions and allows us to calculate all the features of the underlying dynamics from a single time series. The suggested reading list in chapter V offers

several sources that discuss this technique in detail.

Embedding is a powerful instrument for measurement because by embedding

a time series we can calculate the fractal dimension of a data set. Since random data

have theoretically infinite dimensions, and many chaotic systems have smaller dimensions, this is one of the first tools that can help us distinguish noise from

Chaos.

Even more important, the dimension of a time series measures the amount of

detail in the underlying dynamics and actually estimates the number of independent variables driving the system. So, when Tagarev measures a fractal dimension of 2.9 for a time series of aircraft sorties (see figure 2), he presents strong

evidence that the underlying system is notrandom but that it may be driven by as

few as three key independent variables.30

Recent studies of embedded time series also have uncovered ways to use embedding as a vast, generalized grid through which we can interpolate to approximate a system’s dynamics. In this way, researchers have made tremendous strides

in predicting the short-term behavior of chaotic systems. More details of these results will be discussed in chapter IV.

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And Much, Much More.... These tools represent only a small sample of the standard analytical tools currently in use. Consult the references highlighted in chapter V to find complete discussions of these and other tools, such as return maps,

Poincaré sections, correlations, Fast Fourier Transforms, and entropy calculations. These tools constitute the primary sources of the new information that Chaos

theory brings to decision makers.

Results of Chaos Theory

Let us gather together the theoretical results scattered through these first two

chapters. First, I will summarize the common features of chaotic systems. Then, I

will review what it means for us to have Chaos in our systems.

Here is a brief snapshot of the common characteristics of Chaos, a sample of what

to expect in a chaotic system. Most of these characteristics have been highlighted in

our earlier examples. Not much is needed in a system in order for Chaos to be possible. In most physical systems, whose smooth changes in time can be described by differential equations, all that is needed are three or more independent variables and

some nonlinear interaction. In difference equations, like the logistic map, where

change occurs at discrete time intervals, all that is required is a nonlinear interaction.

Most systems have accessible parameters, system inputs we can control to adjust

the amount of energy in the system. We should expect systems to have qualitatively different behaviors over different parameter ranges.

Surprisingly common transitions, from stable equilibria to periodicity and

Chaos, occur in completely unrelated systems.

Influential dynamics occur on many different scales. For instance, the cloud

cover that concerns forces during a combat operation is affected by the activity of

butterflies across the globe. To understand the larger scale dynamics, we may need

to consider the smaller scales.

Attractors draw trajectories towards themselves. So, if an attractor exists (in an

isolated system), and the state of a system is in that attractor’s basin, the system

cannot avoid proceeding toward the attractor. Dynamics on the attractor represent global trends of the underlying system, and they set global bounds on system

behavior. The density of trajectories on the attractor also reveals the relative distribution of behavior.

Because of the trajectory mixing that takes place on attractors, the attractors

are immersed in dense weavings of unstable periodic trajectories. The presence of

these potential periodic behaviors makes Chaos control possible.

The universal nature of these properties helps us answer a somewhat bigger

question:

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Chaos Theory
What does it mean to me to have Chaos in my system?

One consequence of understanding the results of Chaos theory is that if we are

confident that a system can behave chaotically, then we know that it must have all

the properties of Chaos. Some of these properties are hard to prove, but we “get

them for free” if we know the system is chaotic. In particular, if a system is known

to be chaotic, then we know, for example, that any models of that system must include nonlinear terms. We also know we have avenues to controlthe system; that is,

any attractor for that system is densely woven with unstable periodic trajectories

toward which we can drive the system (see the discussion of Chaos control in

chapter IV).

In a 1989 Los Alamos report, David Campbell and Gottfried Mayer-Kress

summarized their “lessons of nonlinearity”:

1. Expect that nonlinear systems will exhibit bifurcationsso that small changes

in parameters can lead to qualitative transitions to new types of solutions.

2. Apparently random behavior in some nonlinear systems can in fact be described by deterministic non-random chaos.

3. Typical nonlinear systems have multiple basins of attraction, and the boundaries between different basins can have incredibly complicated fractalforms.

4. Our heightened awareness of the limits to what we can know may lead to more

care and restraint in confronting complex social issues.

5. The universality of certain nonlinear phenomena implies that we may hope

to understand many disparate systems in terms of new simple paradigms

and models.

6. The fact that Chaos follows from well-defined dynamics with no random influences means that in principle one can predict short-term behavior.

7. The dense paths oftrajectories on an attractor make Chaoscontrol possible.31

To this list I would add that a basic understanding of Chaos brings not only limits to what we can know, but also new information about the dynamics that are

possible. In the next chapter I outline some common military systems where one

can expectto seeChaos. Then, in chapterIV we will be ready to learn how to apply

these results.

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Part Two

Who Needs Chaos Theory?

Applications

Big whorls have little whorls

Which feed on their velocity,

And little whorls have lesser whorls

And so on to viscosity.

Lewis F. Richardson32

Thank heaven

For little whorls.

-not quite Maurice Chevalier

III

Expect to See Chaos

Specific Military Systems and Technology

Chaos Theory does not address every military system. However, while some

authors still treat Chaos as a fashionable collection of new cocktail vocabulary, Chaos is neither a passing fad nor a mere metaphor. The extensive

applications of Chaos to military systems make it imperative for today’s decision

makers to be familiar with the main results of the theory. This chapter is a quick

review of the typical military technologies wherein one should expect to see chaotic dynamics. The chapter is broad by intent, since many more systems appear in

chapter IV, where we start to apply Chaos results. The present discussion concludes with a necessary review of the theory’s limitations as well as a summary of

the implications of the pervasiveness of Chaos.

In the previous chapter we showed how little is needed to generate chaotic dynamics. If a system changes continuously in time—like the motion of vehicles and

missiles—only three independent variables (three degrees of freedom) and some

nonlinearity are required for chaotic dynamics to be possible. If a system changes

in discrete jumps—daily aircraft sortie rates or annual budget requests—then any

nonlinearity, as simple as the squared term in the logistic map, may provide a

route to Chaos. These minimum requirements, present in countless military systems, do not guarantee chaotic dynamics, but they are necessary conditions.

Other common characteristics that make a system prone to Chaos include delayed feedback and the presence of external perturbations, or “kicks.” An enormous number of military systems exhibit these features. One should expect Chaos

in any system that includes feedback, fluids, or flight. The power of Chaos theory

lies in its discovery of universal dynamics in such systems. As this chapter pro-
ceeds from specific systems to general technologies, the reader should be alert for

the similarities in diverse military systems.

Naval Systems. The Thompson and Stewart text on nonlinear dynamics includes a

thorough discussion of the chaotic behavior of a specific offshore structure.33 It

reports a case history in which chaotic motions were identified in a simple model

of a mooring tower affected by steady ocean waves. Mooring towers are being used

increasingly for loading oil products to tankers from deep offshore installations.

These buoys are essentially inverted pendulums, pinned to the seabed, and standing vertically in still water due to their own buoyancy. The concern in this

“kicked” pendulum system is the potentially dangerous chaotic activity that occurs when a ship strikes the mooring. The number of impacts per cycle, which can

be high, is an important factor to be considered in assessing possible damage to

the vessel.

A 1992 Office of Naval Research report summarizes a series of studies identifying

the sources of chaotic dynamics in other ocean structures: a taut, multi-point cable

mooring system; a single-anchor-leg articulated tower; an offshore component installation system; and a free-standing offshore equipment system.34 The author

identifies key nonlinearities and analytically predicts transitions and stabilities

of various structural responses. At the time of the report, experiments were still

underway to verify the analysis. Ultimately, better ways to control these systems

and to enhance current numerical models for these systems will be developed.

The naval applications of Chaos theory are not restricted, of course, to stationary structures. A recent graduate of the Naval Postgraduate School reports

the use of nonlinear dynamics tools to control the motion of marine vehicles.

35

In this interesting application of Chaos results, the system itself does not display

chaotic dynamics. However, the knowledge of common transitions away from

stable behavior allows the author to improve the trajectory control of ships and

underwater vehicles.

Information Warfare. As yet nebulously defined, the subdiscipline of military science

known as Information Warfare certainly embraces a number of electronic systems

subject to chaotic behavior. In many instances, chaotic dynamics contribute to the

design of entirely new systems with capabilities made possible by Chaos theory. One

large field of application is digital image compression. Simple equations that generate

complicated distributions allow pictures to be expressed as compact sets of instructions for reproducing those pictures.36 By transmitting the instructions instead of

all of the individual pixel values, thousands of times more information can be sent

through the same transmission channels in a given period of time.

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On large images and color images, these fractal compression techniques perform

betterthanother current compressiontechniques.37 In 1991,the decompression speed

for the fractal method was already comparable to standard industry techniques.

Even if this process does not become the new standard for real-time communication, it will probably drive the performance standards for other technology developments. Thus, this powerful technology is already making its way into military

mapmaking and transmission as well as into real-time video links to the battlefield.

Other potential applications will be discussed in the next chapter.

Two additional features of electronic Information Warfare make it ripe for

Chaos applications. First, the high volume and speed of communication through

computer networks include the best ingredients of a recipe for Chaos: modular

processes undergoing endless iteration; frequent feedback in communications

“handshaking”; and frequencies (on many scales) faster than the time it takes

most systems to recover between “events” (messages, transmissions, and backups). Second, a likely place to anticipate Chaos is anywhere the digital computer

environment approximates the smooth dynamics of real systems. Many iterated

computations have been shown to exhibit Chaos even though the associated physical systems do not.38

Assembly Lines. A recent book on practical applications of Chaos theory presents a

detailed explanation of where to expect and how to control chaotic dynamics in

automatic production lines.39 It focuses on a few subsystems: vibratory feeding,

part-orienting devices, random insertion mechanisms, and stochastic (random)

buffered flows. Possible military applications include robotic systems for aircraft

stripping and painting and automated search algorithms for hostile missiles or

ground forces.

Let us conclude this introduction to chaotic military systems by recalling the

list of technologies in the 1991 Department of Defense Critical Technologies

Plan.40 Thistime,though, we can note the mostlikely places where these technologies overlap with the results of Chaos theory:

1.Semiconductor materials and microelectronic circuits—they contain all

kinds of nonlinear interactions; semiconductor lasers provide power to numerous laser systems whose operation can destabilize easily with any optical

feedback into the semiconductor “pump” laser.

2.Software engineering—refer to the discussion of Information Warfare, with

feedback possible at unfathomable volumes and speeds.

3.High-performance computing—see items 1 and 2.

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Chaos Theory
4.Machine intelligence and robotics—these require many varieties of control

circuitry and feedback loops.

5.Simulation and modeling—chaotic dynamics are being recognized in numerical models that we have used for twenty years; look for more details in

the next chapter.

6.Photonics—laser and optical circuitry may be subject to Chaos at quantum

and classical levels of dynamics.

7.Sensitive radar—this often combines the instabilities of electronics, optics,

and feedback.

8.Passive sensors—recall our night-light experiment.

9.Signal and image processing—fractals allow new advances in image compression.

10.Signature control—stealth technology, e.g., wake reduction in fluids.

11.Weapon system environment—this will be addressed in the next chapter’s

discussion of the nonlinear battlefield and “fire ant” warfare.

12.Data fusion—attractors and Lyapunov exponents can summarize new information for military decision makers.

13.Computational fluid dynamics—fluids tend to behave chaotically.

14.Air breathing propulsion—engines consume fluids and move through

other fluids.

15.Pulsed power—power–switching requires circuitry with fast feedback.

16.Hypervelocity projectiles and propulsion—these will include guidance,

control, and other feedback systems.

17.High energy density materials—they can undergo chaotic phase transitions

during manufacture.

18.Composite materials—these pose the same manufacturing issues as item 17.

19.Superconductivity—superconductor arrays (Josephson junctions) are a

classic source of Chaos.41

20.Biotechnology—living organisms are full of fluids and electricity,

and Chaos.

21.Flexible manufacturing—this may include automated processes prone to

Chaos.

Limitations of Chaos Theory

It may seem difficult, after the previous section, to imagine a military system

where we will not encounter Chaos. Let us, then, do a brief reality check to indicate

some systems that do not seem to benefit from the results of Chaos theory.

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In general, Chaos will not appear in slow systems, i.e., where events are infrequent or where a great deal of friction dissipates energy and damps out disturbances. For instance, we should not expect Chaos theory to help us drive a jeep or

shoot a single artillery piece. (On the other hand, the theory may eventually guide

our decisions about how to directconvoys of Humvees or how to space the timing or

position of many projectile firings.) Similarly, Chaos theory offers no advice on how

to fire a pistol, though it may pertain in the design of rapid-fire weapons.

Theoretical Chaos results are seriously constrained by the need for large

amounts of preliminary data. To make any analysis of time series, for instance, we

can make reasonable comments based on as few as one hundred data points; but

the algorithms work best with a thousand or more.42 Therefore, even if we are able

to design reliable decisions tools for battlefield use, models that require hundreds

of daily reports of enemy troop movements may be useless in a thirty-day war.

While some hope remains for the prospects of increasing the speed and volume of

simulated battlefield information, the mechanisms for using such simulations for

real-time combat decisions remain to be developed.

One may encounter scenarios and systems with erratic behavior where a

source of Chaos is not immediately evident. In this event, it may be necessary to

examine different scales of behavior. For example, Chaos theory does not help

study the flight of a single bird, free to choose where and when to fly. However,

there is evidence of Chaos in how groups of birds flock and travel together.

43

Implications

The pervasiveness of chaotic dynamics in military systems forces us to be

aware of sources of instability in system designs. We need to develop capacities to

protect our own systems from unwanted fluctuations and to impose destabilizing

dynamics on enemy systems. However, the next chapter will also present ways we

can constructively exploit chaotic dynamics, to allow new flexibility in control

processes, fluid mixing, and vibration reduction. We must remain alert for new

perspectives on old data that were previously dismissed as noise. Perhaps more

importantly, the universal results of Chaos theory open the door for new strategies—ideas we will discuss in the chapter ahead.

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IV

How Can We Use the Results?

Exploiting Chaos Theory

One of the great surprises

to emerge from studies of nonlinear dynamics

has been the discovery that stable steady states

are the exception rather than the rule.

Siegfried Grossman and

Gottfried Mayer-Kress44

At this point the reader should have some intuition for the common features

of Chaos. An enormous number of systems exhibit chaotic dynamics;

many of these systems are relevant to military decision making. But how can we

use Chaos to make better decisions or design new strategies? Even if we accept the

idea that Chaos can be applied to strategic thinking, shouldn’t we leave this

high-tech brainstorming to the analysts?

Absolutely not! As Gottfried Mayer-Kress points out, if we fail to learn the basic applications of Chaos theory, our naiveté could lead to unfortunate consequences. We may, for example, fall into the trap of thinking that successful

short-term management allows total control of a system; we may have unnecessary difficulty in making a diagnosis from available short-term data; or we may ap-
ply inappropriate control mechanisms that can produce the opposite of the

desired effect.45

This chapter lays out practical results on how Chaos theory influences a wide

range of military affairs. Sections of this chapter present specific suggestions on

how to apply these results. Although the structure of each section may suggest that

each concept or technique operates independently, like an isolated item in a tool

kit, the application of Chaos theory unifies many of the previous results.

The chapter opens with a review of some Chaos results that are consistent with

past thought and with good common sense. The meat of the chapter, of course, is a

discussion of the new tools and options available to decision makers through the

results of Chaos theory. Then, an introduction to fractals begins a section on applications that take particular advantage of the fractal geometries that appear in

many chaotic systems. Finally, the chapter closes with a discussion of other issues,

including the difficulties posed by making decisions about systems that include

human input and interactions.

Common Concerns

We should pause to consider the understandable concerns and objections of

those who may be suspicious of “all this Chaos business.” It is quite tempting to

dismiss Chaos as an impractical metaphor, especially since many authors present

only the metaphors of Chaos. Some toss around the Chaos vocabulary so casually

that they leave no hope for practical applications of the results. Margaret

Wheatley, for instance, offers Chaos only as a metaphor, hiding behind the argument that “there are no recipes or formulae, no checklists or advice that describe

‘reality’ [precisely].”46 While it certainly is the case that no formula can track individual trajectories in a sensitive chaotic system, especially with human choice

involved, many patterns are evident, many means of observation and control are

available, and the trends of chaotic dynamics are sufficiently common that one

can and should expect specific classes of behaviors and transitions in chaotic systems. Additionally, and unfortunately, many well-written Chaos texts target a

highly technical readership; their useful results are not adequately deciphered for

a larger community of potential users.

All the same, we already know that human activity is sensitive to small disturbances, that small decisions today can have drastic consequences next week, and

that troops—like water drops—need rest between events. It is simply not obvious

that there is anything new in the Chaos field. Why is it worth everybody’s time just

to learn a new vocabulary to describe the same old thing we have been doing for

decades, or in some cases for centuries? Moreover, suppose we agree that there is

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something new here. How can we useChaos results? How can Chaos help us prioritize our budget or defeat an enemy?

Peter Tarpgaard offers a fine analogy that answers some of these concerns and

offers a glimpse of the insight that Chaos theory brings to decision makers. Imagine what Galileo’s contemporaries commented when they saw him depart for Pisa

with a small ball and a large ball in his bag. “What’s the use? You’re going to climb

the Leaning Tower, and drop the things, and they’re going to fall. We know that already! You’re not showing us anything new. Besides, even if it is new, how can we

use it?”

Now consider the advance in knowledge when Newton derived precise expressions for the force of gravity. Among other things, Newton’s laws of motion identified specific behaviors to expect when various objects are subjected to gravity’s

influence. By describing gravity’s effects, Newton gave us the power to model

them—if only approximately—and to assess their impact on various systems. In

particular, we now know exactly how fast an object will fall, and we can figure out

when it will land. With this knowledge, we can also predict and control certain

systems.

Chaos theory brings comparable advances to decision makers. A number of researchers have developed techniques and tools that allow us to apply Chaos theory

in physical and human systems; but these efforts are very recent, and a great deal

of thought and study remains to be done. Enormous research questions are now

open; several of these are mentioned in the following pages.

Something Old, Something New

Various consequences of Chaos theory were recognized long before Lorenz uncovered the influence of nonlinearity in fluid dynamics. This lends some credibility to the results; as Clausewitz tells us, we need to compare new theories with past

results to ensure their consistency and relevance. Many familiar topics in military

thought disclose a relationship with Chaos theory. For example, the U.S. Army

Manual FM 100–5 holds: “In the attack, initiative implies never allowing the enemy to recover from the initial shock of the attack.”47 This general strategy follows naturally from our observation of dripping faucets: Chaos results when the

system is not allowed to relax between events. Similarly, Marine Corps doctrine

specifically discusses the advantage of getting “inside” an opponent’s “OODA”

(Observe-Orient-Decide-Act) loops in order to decrease the appropriateness—and therefore the effectiveness—of the enemy’s acts. The Marine Corps

manual titled Warfighting (FMFM–1) involves many references to the consequences of sensitivity to current states and the unreliability of plans:

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Chaos Theory
We have already concluded that war is inherently disorderly, and we cannot expect to

shape its terms with any sort of precision. We must not become slaves to a plan.

Rather, we attempt to shape the general conditions of war; we try to achieve a certain

measure of ordered disorder. Examples include:

. . . [channeling] enemy movement in a desired direction, blocking or delaying

enemy reinforcements so that we can fight a piecemealed enemy rather than a

concentrated one, shaping enemy expectations through deception so that we

can exploit those expectations....

We should also try to shape events in such a way that allows us several options so that

by the time the moment of encounter arrives we have not restricted ourselves to only

one course of action.48

Likewise, as Michael Handel observed about the analysis of

counterfactuals—alternative histories that might have occurred if key figures had

made different choices—an important question is: how far can we carry an analysis of alternatives that were not actually pursued? He argued that the further

ahead we consider, the less precision we should attempt to impose. In other words,

the further we carry our counterfactual musings, the less reliable we render our

analysis.49 This is an expression of sensitivity to initial conditions, correctly applied to historical analysis.

We can see, then, that some of the consequences of Chaos theory do not present

new findings for strategic thought. However, it is reassuring that these preliminary observations of Chaos theory are consistent with educated common sense

and the conclusions of earlier researchers and thinkers. The mark of a good scientific hypothesis is that it adequately explains well understood phenomena and, additionally, it accounts for phenomena that was anomalous in (or unanticipated by)

the hypothesis it is superseding.

So What’s New?

The applications presented in this chapter concentrate on methods, results,

tools, and traits of dynamical systems that were not recognized, or even feasible,

only thirty years ago.

The fact that deceptively simple-looking functions and interactions can produce rich, complicated dynamics constitutes a genuinely new insight. This insight grew in one case from the work of biologists’ simple population models, like

logistic maps, which were analyzed in greater detail by mathematicians. As a result, it was discovered that complex dynamics and outcomes do not have to come

from complex systems. Apparent randomness and distributions of behavior can

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be produced by very simple interactions and models. In another case, Edward

Lorenz discovered that our difficulty in predicting weather (and many other

chaotic systems) is not so much a matter of the resolution of the measurements

as it is of the vulnerability of the system itself to small perturbations. In fact,

global weather is so sensitive that even with a constellation of satellites measuring atmospheric data at one-kilometer increments across the entire globe,

we could improve our long-range weather forecasts only from five days to fourteen days.

50

So don’t fire your meteorologists or your analysts! Simply to expect and recognize Chaos in so many real systems is progress enough. The best news is that many

tools are available to understand and control chaotic systems. The tools of Chaos

theory offer hope for discerning the key processes that drive erratic patterns such

as the aircraft loss data shown in figure 2. J.P. Crutchfield highlights the importance of nonlinearity in developing those tools:

[The] problem of nonlinear modeling is: Have we discovered something in our data

or have we projected the new-found structure onto it? . . . The role of nonlinearity in

all of this . . . is much more fundamental than simply providing an additional and

more difficult exercise in building good models and formalizing what is seen. Rather

it goes to the very heart of genuine discovery.

51

A system’s sensitivity often can be quantified and an estimate offered about

how long predictions are valid. Only very recent advances in computers allow repeated measurements of such quantities as fractal dimensions, bifurcations,

embeddings, phase spaces, and attractors. The results of these measurements are

the information needed to apply the theoretical results. In this way, dynamical

systems animate innumerable phenomena that have gone unmeasured until now;

decision makers who are aware of the tools available to them can better discern the

behavior of military systems.52

HOW TO APPLY

While the results of Chaos theory improve our perspective of dynamics in military systems, the practical applications of Chaos go well beyond simple analogy.

To highlight this point, the discussion of Chaos metaphors is postponed to the end

of this chapter. The chapter focuses initially on specific processes, examples, and

cases, with suggested insights and uses for the analytical tools presented earlier.

Considering the applications of these results in one’s own systems, it should be remembered that sometimes chaotic dynamics may be desirable, while at other

times periodicity or stable steady states may be sought. In other instances, one

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Chaos Theory
may want simply to influence the unpredictability in a system: increasing it in the

adversary’s system, decreasing it in one’s own.

Feedback

The results of Chaos theory help us to:

• know what transitions to expect when we add feedback to a system;

• suggest ways to adjust feedback;

• appreciate the wide range of dynamics generated by feedback in real systems.

There is nothing new about a call for awareness of feedback in physical and social systems. Many commentators, for instance, have remarked on the impact of

real-time media reporting of combat events faster than DOD decision loops can

operate. Similarly, one may consider the feedback imposed on an organization by

requirements for meetings and reports. How often do these diagnostics “pulse” an

organization? Yearly, monthly, weekly, daily? Do supervisors require periodic

feedback, or do they allow it to filter up at will? Is the feedback in the organization

scheduled, formatted, free-flowing, “open door,” or a mixture of these? How intense is this occasional “perturbation”?

These are familiar issues for managers and commanders, but a grasp of chaotic dynamics prompts one to answer these questions with other equally important questions. What mixture of structured and free-form feedback works

best in a particular system? What would happen if the frequency of meetings

and reports were increased or decreased? What transitions in system performance should be expected? At what point, for instance, do too many meetings of

an office staff generate instabilities in the organization? Or, in a crisis situation—theater warfare, rescue, natural disaster—what characteristics of the “system” make it more appropriate to assess the system every day, or every hour? This

kind of idea was explored during a series of Naval War College war games. In these

games, one out of every three messages was arbitrarily withheld from the commanders, without their knowledge. As a result, observers noted better overall performance in command and control processes.53

An awareness of the need for, and the sensitivity of, feedback in a system will

make one more alert to the possible consequences of altering the feedback. Here,

the biggest benefit of Chaos theory seems to be transitions that should be expected

as various parameters of system feedback are adjusted. (Of course, this may or may

not have validity in the real world.)

For example, if meetings or reports cause stress on an organization, several obvious parameters—frequency of feedback, length of reports, amount of detail or

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structure required in those reports, length of meetings, number of people involved in the meetings, and so forth—can be adjusted. Some experience with dynamical systems suggests that small changes or careful control of these parameters

may suffice to stabilize some aspect of the system’s performance. One new expectation we learn from chaotic systems is that small changes in control parameters

can lead to disproportionate changes in behavior. Again, the idea of manipulating

meeting schedules and reporting cycles is not new. However, the expectations for

ranges of behavior and transitions between behaviors are new.

As a hypothetical illustration, suppose you observe changes in an adversary’s

behavior based on how often your surface vessels patrol near his territorial waters.

Let us assume that your adversary bases no forces along the coast when you leave

him alone, but he sets up temporary defenses when you make some show of

force—say, an annual open–water “forward patrol” exercise. Assume, further, that

when you double the frequency of your exercises to twice a year, you note a substantial change in your adversary’s behavior. Maybe he establishes permanent

coastal defenses or increases diplomatic and political pressures against you. You

have cut the time difference between significant events (in this case, military exercise) in half and you observe a transition in the system. Now, it would be a silly

idea to attempt to apply Feigenbaum’s constant in this scenario and predict that

the next transition in the adversary’s behavior will come if you decrease the time

interval by only 38 days. (Six months divided by Feigenbaum’s constant, 4.67,

equals 38.5 days.) On the other hand, the common features of chaotic systems suggest that—even though we have no model for the system—we should at least be

alert that the next transition in this system could come if we increase the frequency

of our exercises by only a small amount.

There may be few cases where one can afford the risk of testing such a hypothesis on a real adversary, though force-on-force dynamics like these could be simulated or gamed to reach significant, practical conclusions. We might consider, for

instance, whether Saddam Hussein was playing a game just like this when he

posted substantial forces along his border with Kuwait in 1994, while the United

states military was busy with events in Haiti. Was he determining the increments

of force size and timing that are necessary to provoke a U.S. military response?

Perhaps Hussein was not applying Chaos theory to his strategic decisions, but we

might analyze and game our own dynamics to see what increments of Iraqi force

disposition would compel us to react. An understanding of chaotic dynamics

ought to help us understand and control our response, selected from a flexible

range of options, because knowledge of Chaos helps us foresee the likely transitions when we change a system’s control parameters.

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Chaos Theory
Any one of the following questions would require a complete study in itself.

However, they are presented to stimulate thought about the role of feedback, and

transitions between behaviors.

The increasing availability of real-time information to decision makers amplifies concerns about information overload. How much detail does a leader require?

How often? How much intelligence data does it take to saturate commanders and

diminish their capacity for making effective decisions? What are the best ways to

organize and channel a literal flood of information? The common transitions of

chaotic systems suggest that it may be possible to learn how to control the flood by

studying the effects of incremental changes in key parameters such as: volume of information, frequency of reports, number of sources involved in generating the

data, and time allotted for decision making. Understanding the transitions from

reasonable decision making to ineffective performance may help one tailor intelligence fusion systems for the benefit of commanders.

The relative timing of an incursion on an adversary’s decision cycle may be

more important than the magnitude of the interruption. Many successful strategies hinge on “getting inside the decision cycle” of the enemy. The idea, of course,

is to take some action and then move with such agility as to make a subsequent

move before an opponent has time to orient, observe, decide, and act in response

to the first action. Chaos theory offers an important new insight into this basic

strategy: we should expect ranges of different responses depending on how

“tightly” we approach the duration of an OODA loop. That is, to outpace an enemy who operates on a twenty-four-hour decision cycle, revising the Air Tasking

Order every eighteen hours may produce the same disorientation and disruption

of the enemy as does revision on a twelve-hour or six-hour cycle. The planning

timetable could then be selected on the basis of other objectives, such as speed,

economy of force, efficiency, increased monitoring of combat effectiveness, or resupply requirements. The idea is that we should expect ranges of control parameter values where the system behavior is relatively consistent; but we also should

note parameter ranges where small adjustments produce drastic changes in system response. This phenomenon is not sensitivity to initial conditions. Rather, it

relates the sensitivity of the system structure and changes in parameters, or adjustments to the control knobs, if you will.

One final application to consider, in another area of the decision cycle: coordinating interactions with the news media during crises. It may be found that by adjusting the time intervals of wartime press conferences, for example, the effects of

media feedback in our own decision loops may be mitigated without having to resort to outright censorship. Periodic feedback, carefully timed, could contribute

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to desired behaviors in domestic systems, like channels of public support or an adversary’s systems that tune in to American television for intelligence updates.

Predictability

How does Chaos theory explain, illuminate, reduce, or increase predictability?

Earlier sections of this paper refer to the unpredictable nature of chaotic systems:

the irregular patterns in dripping faucets, rocking buoys, flickering lasers. Now

we will consider the results that help us understand a chaotic system’s erratic behavior.

While the paths of individual chaotic trajectories can never be accurately predicted for very long, knowledge of a system’s attractors offers practical information about the long-term trends in system behavior. This section begins with a

summary of powerful results that allow prediction of the short-term behavior of chaotic systems, even with no model. The section concludes with an explanation of

the usefulness of attractors for assessing long-term system trends.

Time Series Predictions. We record—and sometimes analyze—large quantities of

data at regular time intervals: daily closing levels of the Dow Jones Industrials,

monthly inventory reports, annual defense expenditures. A list of measured data

like this, along with some index of its time intervals, is called a time series. It may appear as a long printout of numbers, organized in a table or graph, indexed in time.

Now, if part of the list is missing, we might interpolate by various means to estimate the information we need. For instance, if we know a country’s tank production was thirty vehicles three years ago, and thirty-two vehicles last year, we might

guess that the production two years ago was about thirty tanks. To make this estimate we should first feel confident in the data we have on hand. We also should

have some idea that industrial activity over the last few years was fairly constant.

Further, there should be some reason to believe the production cycle is annual and

not biennial. Finally, we should, perhaps, have access to a model that approximates this nation’s production habits.

More often than not, though, we are concerned with forecasting issues such as

how many tanks will a country produce next year? For such questions we mustextrapolate and make some future prediction based on previous behavior. This is a

perilous activity for any analyst, because the assumptions on which any models

are made remain valid only within the time span of the original set of data. At any

point in the future, all those assumptions may be useless.

Unfortunately, predictions of behaviors and probabilities are an essential activity for any military decision maker; we have to muddle through decisions on

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Chaos Theory
budgets, policies, strategies, and operations with the best available information.

Notably, however, the results of Chaos theory provide a powerful new means to

predict the short-term behavior of erratic time series that we would otherwise dismiss as completely random behavior. Very briefly, here is the basic idea. If there

were a time series with an obvious pattern, 2 5 7 2 5 7 2 5 7 . . ., the next entry in the

list could be predicted with some confidence. On the other hand, if the time series

displayed erratic fluctuations, as in figure 15, how could it be known whether

there were discernible patterns to project into the future? Through the embedding process, Chaos analysts can uncover patterns and subpatterns that are not apparent to the naked eye and use that information to project the near-term behavior

of irregular dynamics.In figure 15, forinstance, where the time series approaches

periodic behavior for a few cycles, embedding methods identify the places in

phase space where these dynamics are most likely. This technique has been applied to several complex fluids and thermalsystems with tremendoussuccess.54

The embedding technique, of course, does not work for all time series, and the

predictions may hold for only a few cycles past the given data set. However, modern decision makers need to be aware of this tool for two reasons. First, without

any help from Chaos theory, a wise person would not dream of trying to predict a

single step of the wild dynamics illustrated in figure 15. The theoretical results

give hope that one could make reasonable projections in systems previously dismissed as being beyond analysis. However, figure 16 includes samples of the kind

of predictions possible with embedding methods. Given a thousand data points

from which to “learn” the system’s dynamics, the algorithm used here was able to

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Chaos Theory

Reproduced with the kind permission of the Addison-Wesley Longman Publishing Company, Inc., from A.S. Weigend/N.A. Gershenfeld, Time Series Prediction (figure 5, page 191), © 1994.

55
predict fairly erratic fluctuations for as many as two hundred additional time

steps.

In addition, embedding methods include estimates of the error induced by extrapolating the data, giving the decision maker an idea of how long the projections

may be useful. (For detailed presentations of this technique, see, for instance, the

notesfrom a 1992 summer workshop atthe SantaFe Institute.56 Additional explanations also appear in a recent article by M. Casdagli, “Nonlinear Forecasting,

Chaos and Statistics.”57 Both references outline the algorithmsfor near-term and

global statistical predictions of chaotic time series.) Still other researchers have

successfully applied similar methods to enhance short-term predictions by separating background noise from chaotic signals; this list includes Ott, Sauer, and

Yorke,58 J.D. Farmer,

59 and William Taylor.60

Attractors and Trends. It cannot be overemphasized that the sensitive character of

chaotic dynamics denies any hope of predicting the long-term behavior of a system, regardless of how accurately its current state can be measured. On the other

hand, any knowledge of a system’s attractors gives considerable useful information to predict long-term trends in the system. For example, based on a glance outside we can probably tell whether we will need an umbrella to cross the street. We

may even have enough information to make reasonable short-term decisions—like if we should go to the park this afternoon—even though the long-term

weather remains unpredictable. On a larger scale, we can tell the difference in how

to pack for a vacation in Hawaii versus a trip to Moscow, without any current

weather information at all.61 This is why it is fortunate that the weather behaves

chaotically and not randomly. Otherwise, there could be no hope of making even

short-term forecasts.

These simple examples illustrate how decisions can be based on some knowledge of system trends. The attractors of a dynamical system provide precisely that

information. Whether an attractor is constructed from measured data or from extensive simulations, a system’s attractor can illustrate trends that are not as intuitive as the simple weather examples above. Moreover, a well-drawn picture of an

attractor vividly displays the relative amount of time the system spends in certain

regions of its phase space.

Now, the kind of information discussed up to this point was available even before the advent of Chaos theory. However, the theory brings us several new results

when we are confident an erratic system is truly chaotic. First of all, simply by recognizing an attractor we regain some hope that we can understand and manipulate our system. After all, the attractor gives form and structure to behavior we

otherwise would dismiss as random. Thompson and Stewart advise:

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Analysts and experimentalists should be vitally aware that such apparently random

non-periodic outputs may be the correct answer, and should not be attributed to bad

technique and assigned to the wastepaper basket, as has undoubtedly happened in

the past. They should familiarize themselves with the techniques presented here for

positively identifying a genuine chaotic attractor.

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Many practical pieces of information can be derived from our knowledge of a

system’s attractor. First, the relative amount of time the system spends on various

portions of the attractor constitutes a probability distribution; an attractor could

provide key probability information to a military decision maker in many scenarios. Secondly, if we find an attractor for a system, then any disturbances to the system’s current state will still render its particular evolution unpredictable

(envision a tire-swing or a vibrating space station). However, any transient behavior must die out, and the global trends of system behavior must be unchanged.

That is exactly what the attractor describes: regions of phase space that attract system dynamics. Third, we have some hope of being able to predict or recognize the

basins of attraction in a given system.63 If we can prepare a battlefield or a negotiation scenario to our liking, we have some hope we can set up its initial state so the

system proceeds under its own dynamics toward the trends of the attractor we desire.

Visualization of attractors also makes system transitions more apparent as we

change control parameters. Recall, for instance, the return maps sketched for the

dripping faucet (figure 6). It is important to notice that when the period-2 behavior first occurs, the pair of points in the attractor “break off ” from where the single

point used to be. A bifurcation occurs here; we find that the periods of these initial

period-2 cycles are very close to the previous period-1 intervals. Thus, by tracking

the attractors for various parameter settings, we not only observe the individual

dynamics, but also discern additional information about the transitions between

those behaviors.

Unfortunately, most real dynamical systems are not simple enough to collapse

onto a single attractor in phase space. How can we understand and exploit multiple

attractors in a single system? Here’s an analogy: when my ’85 Chevette starts up in

the morning, it warms up at a relatively fast idle speed. This is one periodic

(non-chaotic) attractor for the operation of my car engine with some fixed set of

parameters. A few minutes later, when I tap the accelerator to release the choke,

the engine idles, but much more slowly. The system output has fallen onto a second periodic attractor. The system is the same, but an external perturbation

“bumped” the system to a new, bounded, collection of states.

One may now wonder, is there any chance of exploiting the existence and proximity of two attractors in a system? Assume that the system of interest is the dispo67

Chaos Theory
sition of an enemy force, and suppose the current set of control parameters allows

that system to evolve along either of two attractors, one of which is more to our advantage. Is it possible, by adjusting the control parameters available to us, to manipulate the transitions between these attractors, joining them, breaking them,

building or destroying links between them? These questions may at first glance

appear too metaphorical; but as one’s facility with models and intelligence data

increases, sometimes one finds that the answers to these questions bring extremely practical strategies to the table.

Chaos theory offers practical guidance for system predictability. Techniques

like embedding make short-term prediction possible in chaotic systems. Also,

these techniques quantify the short-term reliability of a given forecast. Attractors describe the long-term recurrent behavior of a system. The relative time spent in various states on the attractor defines useful probabilities. Images of attractors give

indicators of the features of system transitions. And, finally, the presence of multiple

attractors indicates the possibility of certain kinds of strategic options, although

usually not their precise form.

CONTROL OF CHAOS

One of the most powerful consequences of Chaos theory is that a chaotic system—whose behavior previously had been dismissed as random—can be influenced so that it becomes stable. Moreover, this is often possible without the aid of

any underlying model. This capability has no counterpart in non-chaotic systems. Researchers have successfully controlled chaotic behavior in a surprising

number of physical systems.

Three basic approaches have been demonstrated for Chaos control: regular periodic disturbances, proportional inputs based on real-time feedback, and trajectory

“steering” based on models or approximations of the dynamics on an attractor.

The first control technique was demonstrated earlier: periodic output was induced in the chaotic dripping faucet by tapping a rhythm on the spout. In some respects, this technique is consistent with standard results of resonance theory that

describe how external vibrations can excite certain natural frequencies in the system. However, in a chaotic system, infinitely many different periodic behaviors, not

just combinations of the natural modes of system, are guaranteed to be possible.

The second control method, on the other hand, requires real-time measurements of the system’s output in order to determine how far to adjust the selected

control parameter. This is a generalization of the way you balance a long stick on

the palm of your hand: you move your hand just enough, based on how you feel the

stick leaning, and you manage to keep the stick upright. This method has the disThe Newport Papers

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advantage of requiring a reliable feedback-driven control loop. The obvious advantage, though, is that stable output is achieved intentionally, not in the

hit-or-miss fashion that sometimes characterizes control experiments of the first

type.

The third control method was recently developed at the Massachusetts Institute of Technology (MIT). It requires extensive calculations in order to develop

approximations to the dynamics on a system’s attractor. Based on these approximations, the system parameters can be adjusted to guide a trajectory toward preferred regions of phase space. It has not been reported in any further experiments

yet, but it is included to provide a peek at recent results.

These three techniques are the most practical means available to control systems that would otherwise exhibit Chaos; the methods allow imposition of different types of stability, depending on the application. For example, the stability

generated may be a stable steady state (like balancing the stick), or it may be a stable periodic state (often desirable in laser systems). One also may entirely eliminate the possibility of Chaos by modifying the system in some way (see the

discussion below on process). The key observation in all three techniques is that a

chaotic attractor typically has kneaded into it an infinite number of unstable periodic orbits. Chaos control, then, comes from locking on to one of the infinitely

many unstable periodic trajectories densely woven on an attractor.

Chaos control techniques offer many benefits. A chaotic system can be converted into one of many possible attracting periodic motions by making only

small perturbations of an available system parameter. Better still, one method uses

information from previous system dynamics, so it can be applied to experimental

(real-world) situations in which no model is available for the system. Thus, control becomes possible where otherwise large and costly alterations to the system

may be unacceptable or impossible.64

Several references describe the analytical details needed to implement these

control algorithms. Ott, Grebogi, and Yorke perfected the technique that uses

real-time feedback; current publications refer to this method by the authors’ initials, as the “OGY method.”65 Since their initial report, they (and many others)

have applied the OGY method to numerous systems, from classic chaotic systems,

like Lorenz’s weather model and the logistic map, to physical systems such as

thermal convection loops, cardiac rhythms, and lasers. For example, figure 17

showsthe stable steady state imposed on the logistic map compared to its usual irregular dynamics.66 The OGY team has also applied this method of Chaos control

to reduce and filter noise that is present in measured data.67

The other control technique, which is computation-intensive, was developed

by Elizabeth Bradley at MIT.

68 Like the OGY method, this approach actively ex69

Chaos Theory
ploits chaotic behavior to accomplish otherwise impossible control tasks.

Bradley’s method, though, is more like a numerical interpolation. She successfully demonstrated her method on the Lorenz equations. Though it is not yet fully

automated and requires a tremendous amount of data or a complete model, the

technique shows great promise.

Applications of Chaos Control

Thin Metal Strip. Early applications of the OGY method stabilized vibrations in a

thin metal strip. Based on real-time measurements of the strip’s position, the apThe Newport Papers

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Reproduced with the kind permission of Macmillan Magazines Limited, from

Nature, Volume 363, 3 June 1993 (figure 4, page 415), © 1993.

69
paratus automatically adjusted the frequency and amplitude of input vibrations.

This simple experiment confirmed the validity of Chaos control theory, stabilizing period–1 and period–2 behavior and switching between the two at will. These

early successes highlighted the important consequences of Chaos control:

• no model was needed;

• minimal computations were required;

• parameter adjustments were quite small;

• different periodic behaviors were stabilized for the same system;

• control was possible even with feedback based on imprecise measurements.70

Most important, this method is clearly not restricted to idealized laboratory systems.

Engine Vibrations. Henry Abarbanel summarizes the results of several vibration

control studies for beams, railroads, and automobiles.71 He describes the use of

automated software to discover the domains of regular and irregular motions in

beams driven by external vibrations. This information is important to the study

of lateral railcar vibrations, known as “hunting,” which deform and destroy railroad beds. The hunting phenomenon—recognized for decades but never traced to

its source—was shown to arise through the same period doubling transitions we

saw in our dripping faucet and the logistic map! Understanding the source of

these oscillations should lead to ways of mitigating the vibrations, saving significant costs in safety and maintenance.

In another case, S.W. Shaw’s vibration absorber for rotating machinery successfully removed unwanted oscillations by prescribing paths for counterrotating

dynamical elements. The induced motions precisely canceled vibrations in helicopter and automotive machinery. These nonlinear absorbers may appear soon in

products of the Ford Motor Company, which sponsored the work.

Helicopter Vibrations. Chaos theory was applied recently, for the first time, to study

flight test data from OH-6A higher harmonic control (HHC) test aircraft. The

HHC is an active system used to suppress helicopter vibrations. Most vibrations in

the system are periodic, but evidence of Chaos was found. The presence of Chaos

limits the ability to predict and control vibrations using conventional active control systems; but here, control techniques take advantage of the chaotic dynamics.

Like the simple metal strip experiment, this approach uses only experimental

data—no models. By extracting information from time series, one can find the

limits of possible vibration reduction, determine the best control mode for the

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controlling system, and get vibrations under control using only a few minutes of

flight data. These powerful analytical results reduced flight test requirements for

the HHC;the same methods can be applied to other vibration controlsystems.72

Mixing. A South Korean company builds washing machines that reportedly exploit Chaos theory to produce irregular oscillations in the water, leading to

cleaner, lesstangled clothes.73 Whether or not we believe this particular claim, we

ought to consider military systems where effective mixing might be enhanced by

Chaos control—for example, in the combustion of fuel vapors in various engines.

Flickering Laser. In a low-power laser at the Georgia Institute of Technology, Professor Raj Roy controlled the chaotic output of a laser by manipulating the laser’s

power source. Very slight but periodic modulations of the input power forced the

laserinto similar periodicity.

74 In this case, Chaos control was possible without the

use of feedback. While the laser output was not driven to any specific target behavior, repeatable transitions were observed, from Chaos to periodicity, when Roy

modulated a single control parameter.

Chaos control also finds a number of applications in circuits and signals.

Ciphers. In cryptography, as well as in many simulation applications, it often is

necessary to produce large lists of pseudorandom numbers quickly and with specific statistical features. Chaotic dynamical systems appear to offer an interesting

alternative to creating number lists like these, although sometimes more work is

necessary.

75 Unfortunately,the same embedding techniquesthat allow usto make

short-term predictions of chaotic behavior also make it easier to decode random-looking sequences. However, Chaos has other applications for secure communications.

Synchronized Circuits. Even the simplest circuits can exhibit sensitive, unpredictable long-term chaotic behavior. Yet with the correct amount of feedback, two different circuits can be synchronized to output identical chaotic signals. This

extraordinary result could prove useful for securing communications by synchronizing chaotic transmitters and receivers.76

Taming Chaotic Circuits. Elizabeth Bradley has completed software that takes a differential equation, a control parameter, and a target point in phase space, and approximates the system dynamics in order to drive a trajectory to a desired target

point.77 While computationally intensive, her approach has had good success

controlling the Chaos in nonlinear electrical circuits. It takes information about

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dynamics on the attractor and translates that information into approximate dynamics that allow control of individual trajectories. As a result, this technique provides a more global approach to control processes.

Human systems? I have not yet seen Chaos control knowingly attempted on social

systems, but consider, for instance, the options available for controlling the periodic dissemination of information to decision makers, both friendly and adversary. On the operational and tactical scales, we can envision many ways to apply

periodic perturbations to a combat environment through action, inaction, deception, and information control. From a more strategic perspective, we can consider

how regular negotiations and diplomatic overtures tend to stabilize international

relations, while the absence of such measures allows relations to degenerate unpredictably. Depending on how such a system is defined, one might observe truly

chaotic dynamics and new opportunities to control these dynamics. Of course, optimism must be tempered by emphasizing that active human participants can

adapt unpredictably to their environments. However, a discussion follows shortly

on the evidence of Chaos in human systems, offering some hope for applications.

The central idea is this: if a system is known to be (potentially) chaotic, then its

attractor must contain an infinite number of unstable periodic trajectories. The

presence of all these densely packed periodicities makes Chaos control possible.

There are further implications for system design, since it is possible not only to

modify a chaotic system very efficiently with small control inputs but also to

choose from a range of desired stable behaviors. Therefore, novel system designs

are possible: we may be able to design a single system to perform in several dissimilar modes—like a guided weapon with several selectable detonation schemes, or a

communications node with diverse options for information flow control. Current

designs of systems like these usually require parallel components or entire duplicate systems in order to have this kind of flexibility. However, knowing that Chaos

is controllable, we now can consider new system designs with Chaos built in, so that

various stable behaviors can be elicited from the exact same system through small,

efficient perturbations of a few control parameters.78

Chaos and Models

Why bother with applying Chaos to modeling? Some concerns are common to

any debate about the utility of modeling. For instance, to increase doctrine’s emphasis on the human aspects of war, Air Force Manual (AFM) 1–1 argues in detail

that war must not be treated like an engineering project.79 Also, there will always

be trade-offs between the detail one would like in a model and the detail really

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needed. Gleick summarizes nicely: “Only the most naive scientist believes that

the perfect model is the one that perfectly represents reality. Such a model would

have the same drawbacks as a map as large and detailed as the city it represents, a

map depicting every park, every street, every building, every tree, every pothole,

every inhabitant, and every map.... Mapmakers highlight [only] such features as

their clients choose.”80 And sometimes, even when good models are available,initial states can not be known (regardless of desired precision). For example, what

initial conditions should be assumed for a complex model of the atmosphere, or an

oil rig at sea in a developing storm? How can we hope to explore the responses

from all possible starts?81

Sensitivity to initial conditions (SIC), of course, brings into question whether

there is any utility at all in trying to run a computer model of a chaotic system.

Why bother, if we know that any initial condition we start with must be an approximation of reality, and that SIC will render that error exponentially influential on

our results as we move forward in time? Wheatley, among others, maintains a grim

outlook on the whole modeling business in the face of SIC.

82 Yorke, however, has

proven that even though a numerical chaotic trajectory will never be exactly the

trajectory we want, it will be arbitrarily close to some real trajectory actually exhibited by the model itself.83

There are other reasons why we should struggle to understand the role of Chaos

in modeling and simulation. The calculation of a time series’ fractal dimensions is

a means of assessing the number of effective independent variables determining

the long-term behavior of a motion.84 Simple computer models can be used to

study general trends and counterintuitive consequences of decisions that otherwise appear to be good solutions. The results of even simple models will broaden

our perspective of what can occur, as much as what is likely to occur.

85 Finally,

Chaos results can help validate the behavior of models whose output appears erratic. When we cannot match an individual time series, we can often match the

distribution of behavior on an entire attractor.

Chaos in the Simplest Models. Even a brief survey of recent military models will reveal the importance of expecting Chaos in models and simulation. Ralph Abraham, for instance, gives a detailed analysis of what happens in his model of public

opinion formation; his numerical exploration is a good demonstration of the process of wringing out a model. Chaos appears as he models the interaction of two

hostile nations responding to the relative political influence of various social subgroups.86 Other researchers at Oak Ridge National Laboratory have demonstrated a range of dynamical behavior, including Chaos, in a unique, competitive

combat model derived from differential equations.87

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RecentRAND research has uncovered certain classes of combat models that behave much like chaotic pendulums, and chaotic behavior appeared in the outcomes of a very simple computerized combat model. Preliminary studies offer

ideas to better understand non-intuitive results and to improve the behavior of

combat models.88 For example, war game scenarios often produce situations

where an improvement in the capability of one side leads to a less-favorable result

for that side. Results like these have often been dismissed as coding errors. The

correct insight, of course, is that non-monotonic behavior is caused by nonlinear

interactions in the model. In the simple RAND model, reinforcement decisions

were based on the state of the battle, and the resulting nonlinearities led to chaotic

behavior in the system’s output. The RAND team drew some interesting conclusions

from their simulations:

• While models may not be predictive of outcomes, they are useful for understanding changes of outcomes based on incremental adjustments to control

parameters.

• Scripting the addition of battlefield reinforcements (i.e., basing their input

on time only, not on the state of the battle) eliminated chaotic behavior. This

may not be a realistic combat option, but it is valuable information regarding

the battle’s dynamics.

• It is sometimes possible to identify the input parameters figuring most importantly in the behavior of the non-monotonicities (in this case, they were

the size of the reinforcement blocks and the total number of reinforcements

available to each side).

• Lyapunov exponents are useful for evaluating a model’s sensitivity to perturbation.

In general, the RAND report concludes, “for an important class of realistic combat

phenomena—decisions based on the state of the battle—we have shown that modeling this behavior can introduce nonlinearities that lead to chaotic behavior in

the dynamics of computerized combat models.”89

John Dockery and A.E.R. Woodcock, in their detailed book,The Military Landscape, provide an exceptionally thorough analysis of several models and their consequences, viewed through the lenses of catastrophe theory and Chaos. New

perspectives of combat dynamics and international competition arise through extensive discussions of strategy, posturing, and negotiation scenarios. They uncover chaotic dynamics in classic Lanchester equations for battlefield combat

with reinforcements. They also demonstrate the use of many Chaos tools, such as

Lyapunov exponents, fractals, and embedding.

90

Dockery and Woodcock appeal to early models of population dynamics—predator-prey models—to model interactions between military and insurgent forces.

75

Chaos Theory
The predator-prey problem is a classic demonstration of chaotic dynamics; the

authors use common features of this model to simulate the recruitment, disaffection, and tactical control of insurgents. The analogy goes a long way and eventually leads to interesting strategic and tactical conclusions, illustrating conditions

that tend to result in periodic oscillation of insurgent force sizes; effects of a limited pool of individuals available for recruitment; various conditions that lead to

steady state, sustained stable oscillations, and chaotic fluctuations in force sizes;

and the extreme sensitivity of simulated force strengths to small changes in the

rates of recruitment, disaffection, and combat attrition.

In one of the many in-depth cases presented inThe Military Landscape, patterns

of dynamics in the simulation suggest candidate strategies to counter the

strengths of insurgent forces. The model is admittedly crude and operates in isolation, since it can not account for the adaptability of human actors. However, the

model does point to some non-intuitive strategies worth considering. For example, cyclic oscillations in the relative strengths of national and insurgent forces

can result in recurring periods where the government forces are weak while the insurgents are at their peak strength. If the government finds itself at this relative

disadvantage, and adds too many additional resources to strengthen its own

forces, the model indicates that the cyclic behavior tends to become unstable (due

to added opportunities for disaffected troops to join the insurgent camps) and paradoxically weakens the government’s position. Instead, the chaotic model’s behavior suggests carrying out moderately low levels of military or security activity

to contain the insurgents at their peak strength, and await the weak point in their

cycle before attempting all-out attacks to destroy the insurgent forces completely.

91

Process

Since many approaches to Chaos theory remain uncharted, we often find in reports of experiments and analyses that the processesfollowed are as instructional as

the results. The laser system I studied at Georgia Tech with Professor Raj Roy is a

good example.92We started with a low-powerlaser with outputintensity thatfluctuated irregularly when we inserted a particular optical crystal into the cavity. The

crystal converts a portion of the available infrared light into a visible green beam,

which is useful for many practical applications. Even though a previous set of

equations described some of the laser’s operation, no one had yet discovered the

source of the fluctuations. Alternating between output from numerical models

and the real laser, we modified the model, using reasonable basic physics, until the

numerical results displayed Chaos. As a result, we identified the specific source of

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Chaos, and we were able to eliminate the chaotic fluctuations. This is one approach to consider for analyzing a system when a system exhibits Chaos but its

model does not.

If, on the other hand, a model behaves chaotically but the real system does not,

there are a few options. There may be, of course, fundamental mistakes in the

model. A more subtle possibility is that one of the parameter values needs to be reduced (i.e., decrease the “energy” in the model) until the model matches reality. A

third option, given confidence in the model, is to be alert for conditions when the

real system might have different parameters. Expect Chaos!

If both the system and its model show Chaos, one should at least compare attractors, the distributions of the measurable output, like the histogram we drew in

chapter II. Are the bounds on the attractors comparable? Do the densities of

points on the attractors correspond? Once confidence in the model is developed,

one may seek to draw explicit connections from model parametersto quantities that

can be measured in the system. This is how to get control of the Chaos in a system.

These approaches have many potential applications, such as generating distributions for use in war-gaming models. If we can replace random algorithms in

war-game models with simple chaotic equations that produce comparable distributions, we should find clues leading to the parameters that play the greatest role

in the dynamics of given scenarios.

Exploit Chaos for Strategies and Decisions

What is new about the application of Chaos results to strategic thinking? In

general, our awareness of the new possibilities of how systems can behave brings

us definite advantages. Sometimes we will want Chaos. Perhaps an adversary’s

system will be easier to defeat if it is somehow destabilized. Cryptologists may prefer chaotic dynamics to secure their communications. On the other hand, many

systems—signal transmissions, long-range laser sensors, and regular, predictable

international relations—function better in stable, periodic conditions. Fortunately, Chaos theory also teaches us new ways to assure system stability through

careful control of feedback.

Alan Saperstein pinpoints several ideas that Chaos theory brings to the strategic planner. First, many previous attempts to analyze international relations included notions of stability and instability that are not new in the Chaos results.

However, previous models do not account for or produce extreme sensitivity to

small changes in input or model parameters. Second, models have proven to be

very useful in identifying trends, transitions, and parameter ranges where stability is prevalent. It follows that if incomplete models of international conflict show

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instability in given regions of parameter space, then more complete, “realistic”

models are also likely to be unstable in larger regions of the parameter space, i.e.,

harder to stabilize. However, the converse is not true: if a given model representing a system is stable, then a more complex, more realistic model of the same system may still be unstable.93

The ideas in this section overlap somewhat with the previous sections on

Chaos applications. The focus, though, is to assemble specific insights, options,

and techniques available to military decisions makers and strategic planners. The

examples proceed from specific results to general approaches. Among the many

efforts to apply Chaos theory lie connections to military activities.

Decision Making Tools. Let us recapitulate some of the Chaos analysis tools available to military decision makers. These tools have surfaced throughout previous

chapters in various examples and discussions:

• Given sufficient data, time series analysis allows short-term predictions,

even in chaotic systems.

• Lyapunov exponents help to quantify the limits of predictions and measure a

system’s sensitivity to small disturbances. This information can help to prioritize various strategic options according to the relative unpredictability of

their outcomes.

• Knowledge of common transitions in chaotic systems can suggest ideas for

protecting and attacking military systems.

• Calculations of attractors depict distributions of outcomes, providing probability information to decision makers.

• Calculations of information dimension indicate the minimum number of

variables needed to model a system. Moreover, a small value for dimension

also represents strong evidence that the underlying dynamics are not random. A system with a non-integer dimension must contain nonlinearities

(i.e., any previous models that are strictly linear must be incomplete).94

Pattern Recognition.In recent research at the Air Force Institute of Technology, the

theory of embedded time series allowed James Stright to automate the process of

identifying military vehicles from a few measurements of vehicle position and velocity. He also determined how long a data sequence is needed in order to classify

accurately these moving objects. We can visualize the basic concept: the position

of a drone aircraft with locked controls, for instance, should be far easier to predict

than the position of a piloted aircraft conducting evasive maneuvers. So Stright

generalized the idea of tracking objects as they move. At regular intervals, he

noted a vehicle’s position and velocity and logged that information in a vector.

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Evolution of these vectors constitutes an embedded time series; the patterns evident in this embedding allow characterization of typical vehicle behaviors.

Stright verified his technique, correctly distinguishing the motions of five kinds

of military vehicles.95

Feedback Revisited. Earlier, this paper discussed the role of feedback in chaotic

military systems. Chaos theory brings new insights and options to strategies that

include “pinging” an enemy system to see how it responds. Various parameters

can be controlled to perturb an adversary’s system—a large ground force, for instance. We can strike it periodically or unpredictably. We can change the magnitude (firepower), character (area versus directed fire), and frequency of our

assaults. We can attempt to induce or reduce chaotic responses. We can reduce the

amount of feedback in the system through operations security and information

control. One might also envision particular attack strategies that apply our study

of night-light dynamics to long-range perturbation of various enemy sensors.

Again, suppose we are forced to close a base or a port and replace our “forward

presence” there with a “forward patrol” or “frequent exercise” or some periodic

military presence. Chaos theory highlights relevant parameters that should be

considered in strategic planning, such as the size of patrolling forces, the distances to the areas of interest, and the frequency of patrolling activities. Further,

the dynamics common to chaotic systems warn of specific transitions to expect in

an adversary’s response as we vary any of those key parameters.

Fire Ants. Chaos applications in future strategies will follow in the wake of numerous revolutions in military technology. One such revolution may come in the form

of “fire ant” warfare—combat of the small and numerous. It projects a battlefield

covered with millions of sensors (the size of bottle caps), emitters (like pencils),

microbots (like mobile computer chips), and micro-missiles (like soda bottles).

These swarms will be deployed by a combination of pre-positioning, burial, air

drops, artillery rounds, or missiles, and will saturate regions of the battlefield terrain.96 Understanding the dynamics of weather systems and clouds suddenly becomes more than an academic exercise, because “fire ant” warfare produces a new

combat climate: battlefields filled with new clouds that carry lethal capabilities.

Anyone designing an enormous autonomous system like this, with millions of

nonlinear interactions, had better be familiar with the complete range of possible

dynamics as well as with the means to control and defeat such a system.

SDI Policy. Saperstein describes another use of Chaos in a numerical model to

guide policy and strategy, carefully qualifying his findings in an intelligent nu79

Chaos Theory
merical exploration and appropriately cautious use of modeling. The policy question was whether implementation of the Strategic Defense Initiative would tend

to destabilize an arms race between the two superpowers. In this case, he relied on

a nonlinear model to predict the outcomes of various options to help guide policy-making. Saperstein emphasizes that his model is a procurement model (not a

force-on-force simulation) that includes inventories and production rates of various types of weapons. Among his conclusions were that a bigger qualitative

change in the opponent’s behavior comes with the introduction of defensive

weapons, more so than with even drastic increases in annual ICBM production.

Also, beyond his specific findings, his work exemplifies the delicate process of using models to guide decision making.

97

Operational Art. Four fundamental questions face the commander of forces at the

operational level of war. First, what military condition must be produced in the

theater of operations to achieve the strategic goal? Second, what sequence of actions is most likely to produce that condition? Third, how should the resources of

the force be applied to accomplish the desired sequence of actions? Fourth, what

are the costs and risks of performing that sequence of actions?

The operational commander, of course, has access to the same tools available to

any decision maker. Using these tools, the most direct applications of Chaos results

are likely to be in answers to the second question, where Chaos tools can provide information about probabilities of outcomes. Notice, too, that when such information

is provided to a commander, it also represents feedback in his decision process, feedback that can produce transitions in his force’s performance.

The second most likely use of Chaos will come in answers to question four,

where the costs and benefits of various courses of action must be weighed. This paper proposes the use of Lyapunov exponents to help prioritize options based on

the relative unpredictability of actions (see p. 30). No simulations or computer

programs have yet been developed to implement this idea.

Moreover, Chaos theory may also address issues raised in question three, developing options for force application when one of the following conditions holds:

• We have access to enough well-synthesized data on an adversary’s behavior to

allow accurate near-term predictions of enemy actions;

• The opponent uses sensors or electronics that allow us to control enemy systems through feedback techniques;

• We face a large force, where we can exploit our knowledge of the distribution

of behaviors in large interacting systems; or

• We engage in prolonged combat, with sufficient time for our observations of

enemy behavior to reveal trends and patterns in enemy responses.

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Exploiting Chaos. Overall, we need to anticipate chaotic dynamics so we can exploit them in our own systems as well as in enemy systems. A final caveat: besides

the necessary reminder that combat participants can adapt in surprising ways, one

should also remember that unpredictable changes in enemy dispositions can turn

in the enemy’s favor. In 1941, for instance, Japan managed to destabilize America’s

isolationist position by bombing Pearl Harbor. That this destabilization worked

against Japanese hopes underscores the fact that the uncertainty produced by arbitrary disruption can lead to many unpredictable results, sometimes for better,

sometimes for worse. Fortunately, the results of Chaos theory discussed above offer many strategic options beyond the mere arbitrary disruption of enemy systems.

Information Warfare Revisited

Earlier we noted the vulnerability of communications systems to Chaos. Vast

numbers of coupled electrical systems, many of which are controlled with feedback mechanisms, process immense quantities of information, all at the speed of

light, with frequent iterations. Without the details of a given system, we cannot

guarantee the onset of Chaos, but we definitely should expect chaotic dynamics in

systems with those characteristics.

So far, we have identified the potential implications of enhanced data compression for Information Warfare, and the need to be aware of the numerical Chaos

sometimes present in digital computations. I mention Information Warfare again

in this section to tie together a few other applications discussed above. For one,

Chaos applications in secure communications, in encryption, and in synchronized circuits will certainly play a part in Information Warfare. Also, Stright’s automated algorithm for pattern recognition could eventually be applied to identify

information “targets” just as it identifies physical targets.

Fractals

Fractals have many more applications than merely serving as identifiers for

time series with non-integer dimensions. Fractals play important roles in system

scaling and in other image compression applications. First, we will examine some

consequences of the multiple scales of dynamics present in real systems. Then we

will see how researchers take advantage of these multiple scales to compress images with fractal transformations.

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Scaling. We can gain new perspectives of military systems by considering

dynamics on various physical scales, scales that become evident through the study

of fractals. For instance, the reader can probably see Chaos right now in a system

somewhere nearby: in the traffic patterns outside the building, in a stop sign wobbling in the wind, in the light flickering overhead, or on a computer display. However, and more certainly, there are many nearby chaotic dynamics occurring on

physical scales that you probably don’t care about, such as quantum fluctuations,

or irregularities in the power output from a watch battery. The important idea is

that we may sometime encounter system behavior we cannot explain because

there may be key nonlinearities on a scale we have not yet considered.

Once we develop an awareness of the universality of many chaotic dynamics, we realize that some dynamics and physical properties occur on all scales in many systems,

both natural and artificial. Gleick expresses this idea quite eloquently, guiding us to

cases where we should expect to see scale-independent structures and dynamics:

How big is it? How long does it last?These are the most basic questions a scientist can ask

about a thing.... They suggest that size and duration, qualities that depend on scale,

are qualities with meaning, qualities that can help describe an object or classify it....

The physics of earthquake behavior is mostly independent of scale. A large earthquake is just a scaled-up version of a small earthquake. That distinguishes earthquakes from animals, for example—a ten-inch animal must be structured quite

differently from a one-inch animal, and a hundred-inch animal needs a different architecture still, if its bones are not to snap under the increased mass. Clouds, on the

other hand, are scaling phenomena like earthquakes. Their characteristic irregularity—describable in terms of fractal dimension—changes not at all as they are observed on different scales.... Indeed, analysis of satellite pictures has shown an

invariant fractal dimension in clouds observed from hundreds of miles away.

98

Many other common systems exhibit the same dynamics on virtually any scale:

hurricanes, fluid flow, airplane wings and ship propellers, wind tunnel experiments, storms, and blood vessels, to name only a few.

How does awareness of scaling properties broaden our perspective of military affairs? Just as we can conserve time and money by experimenting with scale models,

we can sometimes resolve questions about a system’s behavior by examining one of its

components on a more accessible scale. For example, the electronic architectures of

our war-game facilities nationwide are being configured to network as many sites as

possible to conduct large-scale simulations. Unfortunately, the combat dynamics

that are simulated at different facilities operate on different scales of combat: some are

tactical simulations, some operational, and others strategic. War-game designers are

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mation among these participants on differing scales. The answer may eventually lie

in a network based on fractal scaling of some kind.99

Fractal Image Compression. The need for data compression grows more apparent

daily, as ships at sea saturate their available communication links, and users

worldwide crowd a limited number ofsatellites and frequency bands.

100 Otherrequirements for information compression arise in large modeling problems, where

physicists, for example, try to model cloud dynamics in simulations of laser propagation. One recent breakthrough in image compression came from Michael

Barnsley’s ingenious manipulation of fractals, leading to a process defined in his

College Theorem.101

To compress an image of a leaf, for instance, Barnsley makes several smaller

copies of the original image, and then he covers the original with the smaller copies. He tabulates all the transformations necessary to shrink, rotate, and translate

those copies in order to cover the original leaf. That list of transformations is the

only information necessary to reproduce the original image. Now, rather than

transmit a picture of a leaf via pixel-by-pixel arrays of hue and brightness, we can

transmit a brief set of instructions that allow the receiver to redraw the leaf very

efficiently. By transmitting these short instruction sets, Barnsley’s process compresses large color images by ratios in excess of 250:1. Not only has Barnsley demonstrated this process with simple images, but he has proven that one can derive

transformations for any image, up to the best resolution of a sensor.

The tremendous compression ratios by these fractal compression techniques

make possible new applications in digitized maps for numerous uses, including

devices for digitized battlefield equipment and avionics displays. Moreover, the

end-product of this transmission process is, in fact, an attractor of a chaotic system, so it contains density information about how often a given pixel is illuminated by the receiver’s redrawing program. Among other uses, this local density

information translates into useful data for the physicist interested in propagating

lasers through clouds.

Barnsley’s company, Iterated Systems, Inc., has already won several Army and

Navy research contracts to make further advances with this compression technique.102 One of the resulting products was a patented algorithm for pattern recognition, with the potential to develop automated means to prioritize multiple

targets for a weapon system. Iterated Systems has also used fractal compression to

transmit live motion video across standard telephone lines, a capability with numerous operational applications.103

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Metaphor

You don’t see something

until you have the right metaphor to let you

perceive it.

Robert Shaw104

This section deliberately is short. Chaos does offer powerful metaphors that

lend genuinely new perspectives to military affairs, but since we have access to so

many practical applications that flow from Chaos theory, I will minimize this

brief digression. The main idea is that the metaphors of Chaos bring a fresh perspective—not just a new vocabulary for old ideas. This perspective comes with an

awareness of new possibilities: new information (fractal dimensions, Lyapunov

exponents), new actions (feedback options, Chaos control), and new expectations

(stability, instability, transitions to Chaos).

In a recent attempt to use Chaos metaphors for new historical perspectives,

Theodore Mueller of the Army War College depicted the Mayaguez crisis as the result of a system destabilized due to its sensitivity to small disturbances. He used

the image of an attractor to describe departures from the “range of expected behavior” for an adversary.

105 In another case, a Santa Fe Institute study generalized the results of classic predator-prey equations and drew interesting

politico-military analogies from simple models. The study made a rough comparison of how the onset of epidemics, modeled in these equations, compares to social

dynamics that may spark political revolutions.106 More case studies applying

Chaos metaphors are likely to follow, as the military community grows familiar

with the theory’s more practical results.

The Human Element—Chance, Choice, and Chaos

Problems. Certainly, Chaos theory can boast an impressive record in mechanical

and numerical applications, but can we, and should we, use these results in systems that include human input? How do we reconcile Chaos results with the apparently random dynamics of unpredictable human decisions, the transient

nature of social systems, or the Clausewitzian interaction of adversaries in combat?

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Some of these questions necessarily arise in any debate over the utility of modeling a system that includes human decisions or responses. We have cause for suspicion, in particular because any analysis of social systems assumes we are able to

recognize and predict trends in human behavior. If such predictions are possible,

where does that leave our perspective of choice and free will?

Even if we suspend our disbelief long enough to explore candidate models for

human behavior, we face significant obstacles to executing our analysis. Aggregate data sufficient for strong empirical tests simply do not exist for many important social systems. Social systems are not easily isolated from their environment.

These systems encompass huge scales in time and space, vast numbers of actors,

cost variables, and ethical influences. The laws of human behavior are not as stable as the laws of physics.107

This section argues that Chaos theory does shed light on human behavior that

is relevant to military affairs. Certainly, Chaos is only one of the many rich dynamics we can observe in human behavior. However, we will focus on some of the constraints on human behavior that give us reason to look for insight from chaotic

modeling and simulation efforts. Next, we will present recent evidence of the

presence of Chaos in human behavior. Finally, we will offer some preliminary

ideas on how additional Chaos results can be applied to military affairs.

Hope. Let us look at some sources of hope for understanding human systems with

the help of Chaos theory. First of all, despite our seemingly unlimited capacity for

creativity, we will always make decisions within constraints imposed by limited

resources, limited time, personal habits, and external pressures such as policy and

opinion. Some of our constraints stem from periodic cycles in our environment,

both natural and fabricated: twenty-four-hour days, human physical endurance,

seasonal changes, planetary motion, tides, revisit times for a satellite with a small

footprint, equipment reliability and maintenance, replenishment and resupply,

time cycles necessary to conduct battle damage assessment, budget cycles, and periodic elections. This list is not intended, of course, to promote astrological applications in strategic planning. However, we have seen numerous examples where

periodic perturbations can drastically alter a physical system’s dynamics, causing

significant shifts toward or away from stable behavior. The pervasiveness of these

constraints—often periodic constraints—gives us cause to expect chaotic dynamics even in systems influenced by human decisions and responses.

Another reason to be optimistic about Chaos applications in human behavior

comes from the very nature of attractors: within an attractor’s basin, transient behavior will die out and a system will be found only in states that lie on the attractor.

Even if the system is perturbed at a later time, it must return to the attractor. Evi85

Chaos Theory
dence exists that points to the occurrence of non-random chaotic dynamics in human systems. Those dynamics, in turn, imply the presence of attractors for those

systems. This does not imply that there is no influence of choice and chance in

these systems. Rather, in these cases, human decisions represent one of the following influences: perturbations of behavior which would otherwise remain on an attractor; changes in the distribution of behavior, i.e., tendencies of the system to stay

on any particular portion of the attractor; or choices among multiple attractors that

exist in a single system.

A personal guess is that we will eventually find phase spaces with multiple attractors to serve as the model for the various options available to us or to an adversary. As a playful analogy, think about the possible “state” of your mind as you

read this essay; suppose we can somehow characterize that state by measuring

your thoughts. Is there any hope of controlling or manipulating that system? If

you think not, consider what happens to your thoughts when I tell you, “DON’T

think of a pink elephant.” Whatever attractor your mind was wandering on before, did your thoughts pass through my “pink elephant” attractor, even momentarily? I contend that we have hope of modeling, understanding, and perhaps

controlling some features of human influences in military affairs, perhaps only

briefly, but long enough to enhance the planning and execution of numerous military activities from acquisition to combat.

In a study of two species of ants, whose social dynamics are much easier to observe than human ones in a controlled environment, Nobel Prize winners

Gregoire Nicolis and Ilya Prigogine give us some additional hope for making

analyses of human systems.

What is most striking in many insect societies is the existence of two scales: one at

the level of the individual, characterized by a pronounced probabilistic behavior,

and another at the level of the society as a whole, where, despite the inefficiency and

unpredictability of the individuals, coherent patterns characteristic of the species

develop at the scale of an entire colony.

108

While they draw no premature conclusions about the immediate consequences of

these results for human behavior, Nicolis and Prigogine offer this evidence as reason to be optimistic about the possibility of analyzing and controlling group dynamics. Ralph Abraham also reminds us that we can study human decisions

through game theory, where chaotic dynamics have surfaced in the conduct of different games. A number of complex models are already making significant progress in explaining the actions of, and reactions among, multiple players.109

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Evidence of Chaos. Is there evidence of chaotic behavior in human systems? The

sort of symptoms one should be looking for are: a well-defined system, a clear list

of observables to measure, aperiodic changes in those observables, bounded output, sensitivity to small disturbances, evidence or knowledge of nonlinear forces

or interactions, attractors with fractal dimension, and small, non-integer information dimension. Several research papers report findings of many of these

symptoms in historical data as well as in simulations using models that correspond well with observed human behavior.

Robert Axelrod, for example, has created a model that predicts how elements in

a system group themselves into patterns of compatible and incompatible elements. He modeled nonlinear interactions with basins of attraction that predict

how multiple actors in a scenario form opposing alliances. Typical aggregation

problems where his results may apply include international alignments and treaties, alliances of business firms, coalitions of political parties in parliaments, social networks, and social cleavages in democracies and organizational structures.

The basic inputs to his model are a set of actors, the size of each nation-actor, their

propensity to cooperate with each other, partitions (physical and otherwise), the

distance between each pair, and a measure of “frustration” (how well a given configuration satisfies the propensities of a country to be near or far from each other

actor). Axelrod’s theory correctly predicts the alignment of nations prior to World

War II, with the exception that Poland and Portugal were mistakenly placed on

the German side. He also had comparable success predicting how computer businesses would align behind various market standards, such as the selection of operating systems. His prediction correctly accounted for 97 percent of the total

number of firms in the sample.110

In another discovery of Chaos in social systems, Diana Richards presented several examples of experimental and empirical evidence in strategic decision making. In one example, she expanded a simulated prisoner’s dilemma game to

illustrate possible dynamics in collective decision making in politics and economics. In this model, nonlinear interactions arose because the players’ decisions

depended on their responses to actions in previous steps. She allowed each of two

simulated participants to choose from a hundred options; various stable and chaotic dynamics resulted when she iterated the model.

On one hand, Richards emphasizes the difficulties in verifying such a model

because of the problem of collecting real data over as many repetitions as she can

easily simulate numerically. On the other hand, she was able to apply time series

analysis to uncover chaotic dynamics in historical data. In particular, she discovered evidence of Chaos in U.S. defense spending (as a percentage of total federal

spending) between 1885–1985, and in the number of written communications per

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day (between and within governments) during and following the Cuban missile

crisis, October 1962 to January 1963.111 Again, the presence of Chaosin these systems does not indicate that their behavior is completely predictable; but the number of variables which drive their dynamics may be much smaller than our

intuition might suggest, and we may have a better chance of modeling, understanding, and controlling these situations than previously thought possible.

A significant study of historical military data was completed by a team of students at the Air Command and Staff College (ACSC) in 1994. Their report appears

to be the most thorough research to date that examines historical data with the

tools of Chaos theory. Their calculations of fractal dimensions and return maps

present conclusive evidence of Chaos in tactical, operational, and strategic dynamics of military activity, as shown in aircraft loss data for the entire Vietnam

War(see figure 2), Allied casualty data during their advance through western Europe in World War II, and historical U.S. defense spending (with results consistent with the Richards report mentioned above).112

Recent investigations of well-known models in system dynamics have revealed

previously unsuspected regimes of deterministic Chaos. One outstanding example is John Sterman’s comparison of two numerical models to controlled tests with

human players. The first scenario is a production-distribution model of the Beer

Distribution Game, where subjects are asked to manage a product inventory in

the face of losses, delays in acquiring new units, multiple feedbacks, and other environmental disturbances. Despite the difficulties of conducting controlled experiments, Sterman found that the human subjects’ behavior is described fairly

well by the model dynamics. This direct experimental evidence that Chaos can be

produced by the decision-making behavior of real people has important implicationsforthe formulation, analysis, and testing of models of human behavior.

113

Sterman’s second scenario simulates a long economic wave in which players

adjust inventory orders in response to long-term indicators of supply and demand. The simulated business begins in equilibrium; an optimal response to the

provided indicators actually returns the system to equilibrium within six annual

cycles. However, of the forty-nine subjects tested, none discovered the optimal

behavior, and the vast majority of subjects produced significant oscillations,

many of which showed evidence of Chaos.114

Further practical evidence of Chaos in individual behavior is discussed in recent NASA-sponsored research. In lab tests, researchers took electroencephalogram (EEG) measurements of a human in efforts to characterize the “error prone

state” of, say, a tired pilot. Are some individuals more prone to enter these states

than others? What is the EEG signature of such a “hazardous state of awareness”?

They found that standard statistical tools could not distinguish the EEG signal of

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an individual engaged in various activities from mental arithmetic to image identification. However, the average point-wise (fractal) dimension of the EEG did distinguish the different types of activity. This work has the potential to develop

automated monitoring of pilots in flight to warn them of decreased alertness.

More generally, this gives hope of applying Chaos results in order to understand

the dynamics of human behavior.

115

Implications. There are still very few documented attempts to apply Chaos results

to social systems, due partly to the novelty of Chaos theory, and partly to the practical problems discussed above. However, many authors have noted important

implications of the evidence of Chaos in social systems. Hal Gregersen and Lee

Sailer, for instance, draw two principal conclusions. First, social studies rely too

much on single measurements of population cross-section; we need to focus instead on data taken incrementally over long periods of time. Second, in addition

to standard statistical analysis, we need to recognize Chaos and use the new tools

of dynamical systems.116

The ACSC research team also offered a good summary of the implications of

chaotic dynamics in the data they studied:

• Many erratic systems are at least partly deterministic, so do not throw out

data that appears to be noisy.

• The presence of Chaos requires models that include nonlinear interactions.

• The inclusion of nonlinearity implies that models are likely to have no analytical solution, so do not throw out the computers (or the analysts)!

• Fractal dimensions estimate the minimum number of variables needed to

build models.

• Some regions of phase space are more sensitive than others; Chaos tools can

help identify those different regions.

• Tracking the patterns in attractors also helps identify excluded regions of behavior.

117

How to Apply the Results. Ultimately, we will need to verify any theoretical claims

by comparing them with real systems. In light of the problems of matching numerical models to human behavior, we are left with two basic options. We can construct and analyze formal models only, comparing model results to historical data;

or we can develop lab experiments with human subjects interacting with computer-simulated social systems, or “microworlds.”118

These two options still leave much room to apply Chaos theory to the study of

social systems. For instance, Gottfried Mayer-Kress set up a simple model of a superpower arms race and discussed several immediate consequences of his simu89

Chaos Theory
lated results. Surprisingly, the model gave little or no warning of the onset of

politicalinstability via the usualtransitionstoChaos.119 Thus,the use of a chaotic

model can indicate uncommon transitions to unstable behaviors, providing new

insight to what can happen in reality, despite the crudeness of the model.

How might we specifically adapt Chaos results to organizational behavior? A

recent article discusses The Conference ModelTM, a series of conferences structured to help a large group implement effective reorganization.120 The process entails several carefully structured steps, involving a large number of group

members, that encourage “ownership” of the process—comparable to current

DOD Total Quality policies and processes. The authors report significant success

with their process; it can be couched in terms of Chaos theory to shed light on outcomes to expect from their suggestions for further research.

To begin, the researchers define their system well: basically, it is an organization with fixed membership, divided into subgroups of managers and employees,

planners, and doers. The key parameters are the number of people of the various

groups involved in the planning activities, the number of meetings, the number

and timing of follow-up activities. The measures of effectiveness include the time

required to design the organization’s plan for change and the time taken to implement the changes.

One of the issues raised in this study is, what is the outside limit on the number

of people who can attend a conference? This question could be recast as an issue

about the ranges of possible dynamics as the key parameters are changed. For instance, what transitions are likely as the number of participants involved in the

planning process gradually decreases from 100 percent of the organization? At

what point do we note a substantial decrease in the effectiveness of the plan’s implementation? The universal results of chaotic dynamics suggest we should expect specific transitions (e.g., oscillations of some type) sometime before we reach

the point of total failure of the planing process.

John Sterman’s conclusions about his lab experiments provide a good summary of both the tremendous potential and the unresolved issues of applying

Chaos to human systems. Test results, he notes, show that participants’ behavior

can be modeled with a high degree of accuracy by time-tested decision rules. New

chaotic dynamics have been observed, in well-accepted models, for reasonable parameter ranges. The evidence strengthens the arguments for the universality of

these phenomena. However, the short time scales of important social phenomena

often render the utility of Chaos questionable. The role of learning is difficult to

gauge, e.g., in the experiments discussed here, thousands of cycles are simulated;

however, evidence shows that subjects began learning after only a few cycles. Most

important, the results demonstrate the feasibility of subjecting theories of human

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behavior to experimental test in spite of the practical difficulties.

121 Chaotic dynamics will continue to surface in future investigations of human systems. We

need to be prepared to recognize those dynamics when they occur.

Chaos and Military Art

This chapter compiles substantial evidence of predictable, controllable

dynamics governing many aspects of military affairs. Does it say there is no room

left for military art? Quite the contrary: while chaotic dynamics are sufficiently

universal to revolutionize our profession, Chaos theory is only one of many necessary tools. Where is the individual art of the commander still evident? A good simulation, for instance, or a good summary of intelligence estimates may draw a clear

picture of an adversary’s attractor. Perhaps the image displays trends in force deployment, in aircraft ground tracks or in satellite footprints. However, an attractor only helps express probabilities; the commander still requires a sense of

operational art to evaluate those probabilities in various courses of action, assess

the risks of diverse options, and choose a single course of action.

What Do You Want Us To DO?

This nontrivial question was posed by a concerned audience member after I

presented an introduction to Chaos at ACSC. I am convinced we must not leave

Chaos to the analysts and wait a few years for more results. I encourage you to gain

confidence that you can learn the essential material from good readings and patient thought. You can discern good sources from bad, using the “Chaos con” tips

and good sense. You can build better intuition for what to expect, what Chaos can

do for you, when you need to consult your in-house analysts, when you need to pay

a contractor to do more research, and when you should tell the contractors to go to

the library and do their own homework on their own money. You should develop

an expectation of, an anticipation for, chaotic dynamics in the motion and

changes you observe daily.

Read confidently. When you write, use the vocabulary with care, and at least

avoid the pitfalls outlined in my section on the Chaos con! However, do write.

Publish your progress and successful problem-solving and models to show others

your process for applying the results of Chaos theory. Above all, be aware of the avenues that are opening due to the far-reaching results of Chaos theory.

David Andersen outlines several additional points he feels should be highlighted when we teach anyone about chaotic dynamics. These points certainly offer good advice for any decision maker considering the application of Chaos to

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military affairs.122 Andersen urges us to understand phase plots in order to develop an intuition for Chaos. We should learn to distinguish between transient

and steady-state dynamics. We must be ready to spend time computing. He recommends that we take the time to get some theoretical background. Most significantly, we should learn to recognize when Chaos might be near and how to

diagnose it when it does appear.

Chapter Summary

Tremendous opportunities await us in the numerous realms of Chaos applications. We have access to insights and strategic options that were unimagined only

twenty years ago: universal transitions in system behavior through the careful

control of system feedback; new capabilities to predict short-term dynamics and

long-term trends; options for controlling erratic systems previously dismissed as

random; extraordinary advances in computations that enhance our communications capacity and improve our simulations. In the end, despite reasonable concerns about the utility of modeling, in general—and the analysis of human

systems, in particular—we find a wealth of new information, actions and expectations made possible due to the continuing advances in the understanding of

Chaos theory.

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Part Three

What Next?

A Road Map to More Chaos

V

Suggestions for Further Reading

This chapter summarizes the best resources I encountered during my research. Many Chaos books have appeared in just the last four years; this

review only scratches the surface of this pool of published resources, not to mention numerous videos and software. My aim is to offer some guidance to

instructors on sources to recommend for additional reading, to students on the

best leads for more detail, and to all readers curious about the individuals and organizations who are researching and writing in diverse areas.

The focus of this paper has been to build a bridge from Chaos theory to your areas of interest; the following books and periodicals offer interesting destinations

for you to consider. The most thorough, well-developed readings came from

Gottfried Mayer-Kress (numerous articles), Woodcock and Dockery (The Military

Landscape), John D. Sterman (writing in a special issue of System Dynamics Review), James Gleick’s classic, Chaos, and a special issue of Naval Research Review

devoted to Chaos research sponsored by the Office of Naval Research. Further discussion of these and other references follows.

James Gleick, Chaos: Making a New Science (New York: Viking Penguin, 1987).

Gleick composes vivid descriptions of the people and places at the roots of

Chaos theory. He interlaces narratives with detailed personal interviews. This

book is very readable, and it assumes no technical background. It is not the best

place to learn the details of Chaos—the concepts presented are very general—but it is a pleasant exposition of the wonder of discovery, the universality

of Chaos, and its range of applications. Take the time to read all the endnotes

where Gleick hides additional interesting facts. A great piece of storytelling.
Heinz-Otto Peitgen, Hartmut Jürgens, and Dietmar Saupe, Chaos and Fractals:

New Frontiers of Science (New York: Springer-Verlag, 1992).

The authors have compiled a veritable encyclopedia of Chaos. The text is very

readable, assumes little technical background, and explains fascinating connections among diverse Chaos applications. If you put only one Chaos book on your

shelf, this should be it.

System Dynamics Review, vol. 4, nos. 1–2, 1988.

This special issue assembles a fine collection of articles that discuss important issues of Chaos theory in great depth. The topics range from the very practical to the

philosophical. John D. Sterman, for instance, opens the issue with a well-written

introduction that surveys the basic concepts and results of Chaos theory; he also

contributes a strong paper on “Deterministic Chaos in Models of Human Behavior: Methodological Issues and Experimental Results.” This is another must-read

resource.

J.M.T. Thompson and H.B. Stewart, Nonlinear Dynamics and Chaos (New York:

John Wiley & Sons, 1986).

The authors aim this superb text at engineers and scientists, analysts and

experimentalists. They require as background only “a little familiarity with simple differential equations.” Step-by-step, they introduce Chaos, what to expect,

and how to interpret data sets with irregular behavior; they use numerous helpful

pictures and graphs. In addition, they present a healthy range of applications, focusing on the ways simple models can generate complicated dynamics in slender,

vibrating structures; resonances of off-shore oil production facilities; large-scale

atmospheric dynamics; particle accelerators; chemical kinetics; heartbeat and

nerve impulses; and animal population dynamics. They also include a fantastic

bibliography with more than four hundred entries. This is a great book from

which to learn Chaos theory.

John T. Dockery and A.E.R. Woodcock, The Military Landscape(Cambridge, England: Woodhead Publishing, 1993).

This book presents an exceptionally detailed analysis of several models and the

implications of their dynamics viewed through the lenses of catastrophe theory

and Chaos. New perspectives of combat dynamics and international competition

surface during the analysis of the models’ behaviors. The authors discuss extensive applications in strategy, posturing, and negotiation. In one of their many simulations, they uncover chaotic dynamics in the classic Lanchester equations for

force-on-force combat, with reinforcements. They demonstrate the use of many

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Chaos tools, and they take great pains to show relationships among the tools.

Overall, this book includes more analytical details than most recent reports, and it

is a thorough review of many models that exhibit chaotic dynamics.

John Argyris, Gunter Faust, and Maria Haase, An Exploration of Chaos, Texts on

Computational Mechanics, Vol. VII (New York: North-Holland, 1994).

Offered as an introductory text on Chaos theory, this book targets “aspiring physicists and engineers.” A good deal of general theory precedes a review of physical

and mechanical applications. The authors claim to assume no deep mathematical

background, but the reader really needs more than a casual familiarity with differential equations and vector calculus. The book has several strengths: a detailed

discussion of the logistic map; a nice compilation of classes of bifurcations; an interesting analysis of bone formation and regrowth. The applications are presented

in fine detail, making the results reproducible for interested readers. Most importantly, the authors outline a general process of theoretical and numerical investigation appropriate for technical applications of Chaos results. They conclude

with a spectacular bibliography of primary technical sources.

Richard A. Katz, ed., The Chaos Paradigm: Developments and Applications in Engineering and Science, American Institute of Physics (AIP) Conference Proceedings 296, Mystic, Conn. (New York: AIP Press, 1994).

This is a terrific survey of current research sponsored by the Office of Naval Research and the Naval Undersea Warfare Center. The list of participants is a useful

“Who’s Who” of many current research areas; the articles sample the diverse

fields where DOD engages in active research. Anywhere from two to four brief articles cover each of the following topics: Mathematical Foundations of Chaos,

Mechanical Sources of Chaos, Turbulence, Control of Chaos, Signal Modeling,

Noise Reduction, Signal Processing, and Propagation Modeling.

Todor Tagarev, Michael Dolgov, David Nicholls, Randal C. Franklin, and Peter

Axup, Chaos in War: Is It Present and What Does It Mean? Report to Air Command and Staff College, Maxwell AFB, Alabama, Academic Year 1994 Research Program, June 1994.

This is the best in-depth report examining historical data for evidence of Chaos.

The authors find chaotic dynamics in tactical, operational, and strategic levels of

military activity, examining data such as aircraft loss data for the entire Vietnam

War, Allied casualty data in their advance through western Europe in World War

II, and historical levels and trends in U.S. defense spending. The paper’s greatest

strength is the discussion of data collection and analysis, the obstacles the authors

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encountered, and details of their search process. This full report is much more

meaningful than the subsequent article they distilled for the Airpower Journal in

late 1994.123 Both the short article and the full essay contain some substantial

technical errors in the basics of Chaos, but the authors have clearly done their

homework.

T. Matsumoto et al., Bifurcations: Sights, Sounds and Mathematics (New York:

Springer-Verlag, 1993).

This textbook generally expects the reader to have an extensive mathematical background, but it starts with a section describing simple electronic circuits, which exhibit a vast array of chaotic dynamics. This is a great reference for those with access to

or interest in electronics applications. As its title implies, this book also includes a

thorough study of various classes of bifurcations common to many dynamical systems.

Edward Ott, Tim Sauer, and James A. Yorke, eds., Coping with Chaos: Analysis of

Chaotic Data and the Exploitation of Chaotic Systems (New York: John Wiley &

Sons, 1994).

Topic-wise, this book is the best end-to-end compilation of chapters and articles,

mostly published in other sources, which go from theoretical background to data

analysis and applications. The text includes more recent work on practical suggestions for calculating dimensions, Lyapunov exponents, time embeddings, and

control techniques. While the collection of articles is virtually all reprinted from

primary sources, it is a good collection and can save an interested reader many

hours of digging through periodical holdings. This book does require a solid

background in vector calculus and differential equations, but it is very practical.

The articles are generally at the level of papers from Physical Review and Physical

Review Letters. The bibliography is extraordinary.

G. Mayer-Kress, ed., Dimensions and Entropies in Chaotic Systems: Quantification of

Complex Behavior, Proceedings of an International Workshop at the Pecos

River Ranch, New Mexico, 11–16 September 1985 (New York:

Springer-Verlag, 1986).

This thin text offers the collection of papers contributed to the workshop cited. It

is an older reference describing some of the early results of Chaos calculations.

However, it presents a comprehensive review of techniques, modifications and

improvements, and explanations of how they are related. The papers cover the intense details of how to calculate, in both theory and experiment, fractal measures,

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cal work, not for the casual reader or weak of heart, and not a good place to first

learn about these measurements. However, it is necessary reading for serious analysts embarking on numerical explorations of dynamical systems.

Michael F. Barnsley and Lyman P. Hurd, Fractal Image Compression (Wellesley,

Mass.: AK Peters, 1993).

Perhaps more dense (i.e., slower) reading than Barnsley’s first text, Fractals Everywhere, this fine book focuses appropriately on only those details required to understand the fractal compression techniques patented by Iterated Systems, Inc. It

is a very thorough presentation, pleasant reading, and the text includes sample C

source code and many demonstrations of decompressed images.

Saul Krasner, ed.,The Ubiquity of Chaos(Washington, D.C.: American Association

for the Advancement of Science, 1990).

This is another nice review of Chaos applications in a wide variety of disciplines:

dynamical systems, biological systems, turbulence, quantized systems, global affairs, economics and the arms race, and celestial systems. Great bibliographies follow each individual article; most chapters have not been published elsewhere, as

is often the case in similar collections of contributions by many independent authors.

Naval Research Reviews, Office of Naval Research, vol. XLV, no. 3, 1993.

This special issue is devoted to ONR-sponsored research in engineering applications of Chaos. Nice overview articles cover the following topics: controlling

Chaos, noisy Chaos, communicating with Chaos, nonlinear resonance in

neurophysiological systems, and image compression.

99

Chaos Theory

VI

Further Questions to Research

I have assembled in this chapter a broad collection of research topics that deserve more careful study. For the benefit of students and prospective research

advisors, I have done my best to form the questions and issues into packages small

enough to address within a short research term during in-residence professional

military education.

Complexity: The Next Big Step. This report discusses how simple models can display complex behavior. However, once we develop a good intuition for Chaos,

other questions arise immediately. Here is a peek at one of the central issues,

only slightly oversimplified. Fact: fluids tend to move chaotically. The very nature of their dynamics makes them extremely sensitive to small disturbances.

Now, the mixture inside a chicken egg is a fluid; that mixture is surely subjected to

bumps and jostles during the formation of the baby chick inside. Question: if the

fluid is chaotic, and its motion and behavior is so unpredictable, how does the

creature inside always come out a chicken?

The answers to questions like these are the subject of the (even more recent)

science of Complexity. You may consider researching complexity and self-organization. When and why do complicated systems sometimes organize themselves to

behave “simply”? Which results of this theory are relevant for military decision

makers?

Exponents.Identify a few specific military systems, perhaps within the context of a

war game or through historical data, and calculate some Lyapunov exponents to

compare the systems’ relative sensitivity to perturbation. Prioritize the importance of various systems for protection or attack.
Additional Dynamics. Robert Axelrod’s aggregation model successfully predicts

the end states of two multi-party alliances, but there is still room to consider the

dynamics of these alliances. How long do the alignments take to form? How stable

are the end states? What sort of perturbations break the alliances? The analysis is

static only, so far, although it does discuss the presence of “basins of attraction” of

the end-state configurations.124

Feedback. Where are the feedback loops in current and future military systems?

Consider both friendly and hostile systems. Also investigate both mechanical and

social systems. Examine the strategic options for imposing feedback on these systems and protecting the systems from unwanted feedback. What behaviors and

system transitions should we expect?

Sensors. What sort of sensors can we identify as vulnerable to imposed feedback?

Where are they and how do they operate? What creative strategies can we devise to

exploit or reduce their sensitivity to disturbances?

War Games. Can we replace random variables in war games with simple chaotic

equations that produce comparable distributions? Can the underlying equations

lead to clues about which parameters are most important? How do our games behave now? Can any be driven into Chaos with the right combination of parameters? For a detailed discussion of the use of historical data for battlefield

predictions, see Colonel T.N. Dupuy’s Numbers, Predictions & War.

125 It thoroughly discusses the issues of data compilation, modeling, prediction, and tabulates exhaustive lists of relevant battlefield parameters.

The Nonlinear Battlefield. Sean B. MacFarland, at the Army School of Advanced

Military Studies (SAMS), defines “operational non-linearity” as the dispersed

state of a combat force characterized by a complex of interconnecting fire positions and carefully sighted long-range weapons.126 His paper highlights the difference between geometric nonlinearity and systemic (dynamical) nonlinearity.

If we think of a force’s physical disposition as its “state” in a combat system, old

ideas of “forward edge of the battle area” may be replaced by emerging perspectives of overlapping attractors.

J. Marc LeGare, also at SAMS, proposed operations on the nonlinear battlefield

organized in a “tactical cycle”: disperse, mass, fight, redisperse, and reconstitute.127Could we structure this cycle to protect our own dynamics and take advantage of enemy cycles to break down their systems? If our forces are limited, can we

exploit these cycles to apply our force efficiently and control the combat dynamThe Newport Papers

102
ics? What kind of small perturbations could we impose on such a combat system?

The answers to some of these questions may spring from other articles that consider the tactics of potential adversaries on the nonlinear battlefield.128

We should also note that the idea of dispersed, nonsequential operations is

not new. In 1967, J.C. Wylie contrasted two very different kinds of strategies.

One is sequential, a series of visible discrete steps that follow one another deliberately through time. The other iscumulative, “the less perceptible minute accumulation of little items piling one on top of the other until at some unknown point the

mass of accumulated actions may be large enough to be critical.” He observes that

in the Pacific from 1941 to 1945 “we were not about to predict the compounding

effect of the cumulative strategy (individual submarine attacks on Japanese tonnage) as it operated concurrently with and was enhanced by the sequential strategy [of the drive up the Pacific islands].”129 Strategies like these may lend

themselves to deeper analysis through Chaos theory.

Case Studies

For want of a nail the shoe is lost,

For want of a shoe the horse is lost,

For want of a horse the rider is lost,

For want of a rider the battle is lost,

For want of the battle the war is lost,

For want of the war the nation is lost,

All for the want of a horseshoe nail.

George Herbert (1593–1632)

We already noted one effort to examine theMayaguez crisis in the light of Chaos

results. This was, of course, only a rough beginning. Several historical case studies

(all entitled For Want of a Nail!) highlight the sensitivity of combat events to small

“disturbances.” The following references provide a list of candidate cases to consider for further Chaos analyses.

Robert Sobel composed a detailed counterfactual book of what would have

happened had British General John Burgoyne held Saratoga in the American

Revolution.130 Hugh R. Wilson studied the ineffective application of economic

sanctions against Italy in the winter of 1935–36 during the Italian military excursion intoEthiopia.131 Hawthorne Danielinvestigated the influence oflogistics on

war in several interesting case studies.

103

Chaos Theory
• American Revolution: New Jersey 1776; Lake Champlain and the Hudson

River 1777

• Peninsular War: Spain and Portugal, 1808 to 1814

• The Moscow campaign: Russia, 1812

• American Civil War: 1861 to 1865

• Sudan Campaign: The upper Nile, 1896 to 1898

• The Allied invasion of occupied Europe, World War II: 1944 and 1945.132

Bibliography. With the recent explosion in Chaos resources, the preparation of a

comprehensive bibliography would provide a great service to the general research

community. The reference lists in the texts noted above are a good place to start.

Many book reviews are also available to guide examinations of the most recent

texts.

Write More! Above all, this essay should be regarded as one voice in a continuing

conversation. Value always will be found through documenting other interesting

thoughts and research. Please continue the conversation. In particular, there is

plenty of room for open debate on issues this report has missed or overstated. It

would also be most helpful for other reports to be published on additional military

applications of which readers may be aware. I look forward to reading those reports.

The Newport Papers

104
VII

Conclusion

This report has focused on those issues of Chaos theory essential to military

decision makers. The new science of Chaos studies behavior that is characterized by erratic fluctuations, sensitivity to disturbances, and long-term

unpredictability. This paper has reviewed Chaos applications in military affairs

and, I hope, corrected some deficiencies in current publications on Chaos.

The study was centered in three areas. First, we reviewed the fundamentals of

chaotic dynamics to build some intuition for Chaos. Second, we surveyed the current military technologies that are prone to chaotic dynamics. Third, we saw how

the universal properties of chaotic systems point to practical suggestions for applying Chaos results to strategic thinking and decision making. The power of

Chaos comes from this universality: not just the vast number of chaotic systems

but the common types of behaviors and transitions that appear in completely unrelated systems. As a result, recent recognition of Chaos in social systems offers

new opportunities to apply these results to problems in decision making, strategic

planning, and policy formulation.

The evidence is clear: chaotic dynamics pervade the dynamics of military affairs. The implications of Chaos theory offer an extraordinary range of options unavailable only twenty years ago. Not only do current military systems naturally

exhibit chaotic dynamics, but many systems are vulnerable to new strategies that

exploit Chaos results. Because of the theory’s important potential, every military

leader needs to be familiar with the fundamentals of Chaos in order to expect chaotic dynamics in military systems, recognize Chaos when it occurs, and exploit

the vast array of tools for diagnosing and controlling those dynamics.

Appendix

What does it mean to be Random?

Our usual connotations of randomness carry images of erratic, completely unpredictable behavior. For a fair die on a craps table,randomness means thatsooner or

laterthat die will roll to a 6. It means there is no chance of that die rolling a string of

1’s forever. If that were the case, the die would be very predictable, and thus, not

random.

To be more precise, let us borrow an explanation from Batterman’s article, “Defining Chaos.”133 Start with an infinite string of perfectly alternating digits:

01010101 ...

How much information does it take to recognize, transmit, or repeat this string?

Suppose we had access to only a brief list of the first few elements of the sequence.

Could we draw any conclusions about the system’s behavior?

0 No.

0 1 Not yet.

0 1 0 Hmmm, we begin to see a pattern.

0101 Looks a little regular, but can’t tell yet.

0 1 0 1 0 We can start to guess some regularity....

After 20, or 50, or 1,000 new pieces of information (additional digits in the observed string) we think we have it: this string of data has period two; we need only

three pieces of information to repeat the string:

1. Print 0.

2. Print 1.

3. Repeat steps 1 and 2.

If we follow these steps, we’re confident we can completely replicate the series.

Now, if we do not know where or how the series was generated, we cannot be positive of its perfect periodicity. Nonetheless, as we get more and more information,

our confidence in our analysis improves.

So how would we characterize a random string of data? In terms of our data

string, it means we would need the ENTIRE infinite string—that is, an infinite list
of instructions—in order to accurately reproduce the original infinite data set. This

requirement for an unending set of instructions to communicate or reproduce the

data is sometimes offered as a formal definition of randomness.

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108
Notes

1. Michael F. Barnsley, Fractals Everywhere, 1st ed. (San Diego: Academic Press,

1988), p. 1.

2. James Gleick Chaos: Making a New Science (New York: Viking Penguin, 1987),

p. 304.

3. Todor Tagarev et al., “ChaosinWar:IsIt Present andWhat DoesIt Mean?”Report

to Air Command and Staff College, Maxwell Air Force Base, Alabama, Academic Year 1994

Research Program, June 1994, p. 52. See also Michael M. McCrea, “US Navy, Marine Corps,

and Air Force Fixed-wing Aircraft Losses and Damage in Southeast Asia (1962–1973)” (U),

Arlington, Va: Office of Naval Research, August 1976, Center for Naval Analyses report

CRC 305, downgraded to unclassified 8 September 1989.

4. Gleick, p. 69.

5. Department of Defense Critical Technologies Plan for the Committees on Armed

Services, U.S. Congress, 1 May 1991 [DTIC AD–A234 900], p. I–3.

6. David F. Andersen, “Foreword: Chaos in System Dynamics Models,” System Dynamics Review, vol. 5, nos. 1 and 2, 1988, pp. 6–7.

7. Gleick, p. 187.

8. Ibid., p. 43.

9. J.M.T. Thompson and H.B. Stewart, Nonlinear Dynamics and Chaos (New York:

John Wiley & Sons, 1986), pp. 291–310.

10. Gleick, p. 42.

11. “NOVA: The Strange New Science of Chaos,” PBS television broadcast NOVA

#1603, 31 January 1989, produced by WGBH, Boston, Mass.

12. Gleick, pp. 262–7.

13. John Argyris et al., An Exploration of Chaos, Texts on Computational Mechanics, vol.

VII (New York: North-Holland, 1994), p. ix.

14. Gleick, p. 266.

15. Argyris et al., p. 65.

16. Gleick, pp. 60–1.

17. Heinz-Otto Peitgen, Hartmut Jürgens, and Dietmar Saupe, Chaos and Fractals:

New Frontiers of Science (New York: Springer-Verlag, 1992), p. 697.

18. After ibid., p. 698.

19. After ibid., p. 514.

20. Gleick, p. 24.

21. After Barnsley, Fractals Everywhere, 1st ed., p. 103.

22. J.A. Dewar et al., “Non-Monotonicity, Chaos, and Combat Models,” RAND Library Collection (Santa Monica, Calif.: RAND, 1991).

23. After Barnsley, Fractals Everywhere, 1st ed., pp. 279–83.

24. T. Matsumoto, Bifurcations: Sights, Sounds and Mathematics (New York:

Springer-Verlag, 1993).

25. After Peitgen et al., p. 59.

26. Gleick, p. 48.

27. Gottfried Mayer-Kress, “Chaos and Crises in International Systems,” Lecture at

SHAPETechnologySymposiumonCrisisManagement,Mons,Belgium: 19–20March1992.
28. Paul W. Rapp et al., “Dynamical Characterization of Brain Electrical Activity,” in

Saul Krasner, ed., The Ubiquity of Chaos (Washington, D.C.: American Association for the

Advancement of Science (AAAS), 1990), p. 10.

29. After Peitgen et al., p. 526.

30. Tagarev et al., pp. 51–3.

31. David K. Campbell and Gottfried Mayer-Kress, “Nonlinearity and Chaos: The

Challenges and Limitations of Mathematical Modeling ofEnvironmental and Socio-PoliticalIssues,” in Proceedings of the Conference on Technology-Based Confidence Building: Energy

and Environment, Centerfor National Security Studies(CNSS) Papers, No. 22 (Los Alamos

National Laboratory, November 1989) [LA–11728–C], p. 84.

32. Gleick, p. 119.

33. Thompson and Stewart, pp. 291–310.

34. Solomon C.S. Yim, “Chaotic and Random Responses of Ocean Structural Systems,” Office of Naval Research Young Investigator Award, Research Project

N00014–88–K–0729, 1 March 1992 [DTIC AD–A247 561].

35. Zeki Okan Oral, “Hopf Bifurcations in Path Control of Marine Vehicles,” Thesis,

Naval Postgraduate School, Monterey, Calif.: June 1993 [DTIC AD–A268 818].

36. Michael F. Barnsley and Lyman P. Hurd, Fractal Image Compression (Wellesley,

Mass.: AK Peters, 1993).

37. Yuval Fisher et al., “Fractal Image Compression,” Quarterly Progress Report

8–12–91 (San Diego, Calif.: NETROLOGIC, Inc., July 1991) [DTIC AD–A240 636].

38. B.H. Tongue and K. Gu, “A Higher Order Method ofInterpolated Cell Mapping,”

Journal of Sound and Vibration, vol. 125, no. 169, 1988.

39. Max-Oliver Hongler, Chaotic and Stochastic Behaviourin AutomaticProductionLines

(Berlin: Springer-Verlag, 1994).

40. Department of Defense Critical Technologies Plan, forthe Committees on Armed

Services, U.S. Congress, 1 May 1991 [AD–A234900].

41. Kurt Wiesenfeld et al., “Observation of Antiphase States in a Multimode Laser,”

Physical Review Letters, vol. 65, no. 14, 1 October 1990, pp. 1749–52.

42. Tagarev et al., p. 32.

43. Frank Heppner and Ulf Grenander, “A Stochastic Nonlinear Model for Coordinated Bird Flocks,” in Krasner, The Ubiquity of Chaos, pp. 233–8.

44. Siegfried Grossman and Gottfried Mayer-Kress, “Chaos in the International

Arms Race,” Nature, 27 February 1989, pp. 701–4.

45. Ibid.

46. Margaret Wheatley, Leadership and the New Science (San Francisco: BerrettKoehler Publishers, 1992), p. 7.

47. Headquarters Department of the Army, Operations, FM 100–5 (Washington: 14

June 1993), p. 2–6.

48. Department of the Navy, Warfighting, FMFM–1 (Washington: 6 March 1989), pp.

66–7.

49. MichaelI. Handel, “German Strategic Alternatives,” Lecture,U.S.NavalWarCollege, Newport, R.I.: 26 September 1994.

50. Edward Teller, comments atthe Tenth Annual Space Symposium,Broadmoor Hotel, Colorado Springs, Colorado: February 1994.

The Newport Papers

110
51. J.P. Crutchfield, “Knowledge and Meaning: Chaos and Complexity,” in Lui Lam

and Vladimir Naroditsky, eds., Modeling Complex Phenomena (New York: Springer-Verlag,

1992), p. 68.

52. Andersen, pp. 3–13.

53. Interview with Dr. Stephen O. Fought, National Security Decision Making Department, Naval War College, Newport, R.I.: 1 February 1995.

54. Andreas S. Weigend and Neil A. Gershenfeld, eds., Time Series Prediction: Forecasting the Future and Understanding the Past, Proceedings of the NATO Advanced Research

Workshop on Comparative Time Series Analysis held in Santa Fe, New Mexico, 14-17 May

1992, Proceedings Vol. XV, Santa Fe Institute Studiesin the Sciences of Complexity (Reading, Mass.: Addison-Wesley, 1994), pp. 175–3.

55. Ibid., p. 191.

56. Ibid.

57. M. Casdagli, “Nonlinear Forecasting, Chaos and Statistics,” in Lam and

Naroditsky, Modeling Complex Phenomena, pp. 131–52.

58. Edward Ott et al., eds., Coping with Chaos: Analysis of Chaotic Data and the Exploitation of Chaotic Systems (New York: John Wiley & Sons, 1994).

59. J.D. Farmer and John J. Sidorowich, “Exploiting Chaos to Predict the Future and

ReduceNoise,” in Y.C. Lee, ed.,Evolution,Learning and Cognition (Teaneck,N.J.:World Scientific Publishing, 1988), pp. 277–330.

60. William W. Taylor, “Chaotic Evolution and Nonlinear Prediction in Signal Separation Applications,” RAND Report P–7769 (Santa Monica, Calif.: RAND, April 1994).

61. Interview with Dr. John Hanley,Program Director of U.S.NavalWarCollege Strategic Studies Group, and Bill Millward (Commander, USN), Naval War College, Newport,

R.I.: 16 December 1994.

62. Thompson and Stewart, p. xii.

63. Patrick Kelly III (Major, USA), “Modern Scientific Metaphors of Warfare: Updating the Doctrinal Paradigm,” Monograph, School of Advanced Military Studies, U.S.

Army Command and General Staff College, Fort Leavenworth, Kans., Second Term Academic Year 1992–1993 [DTIC AD–A274 366), p. 38.

64. Edward Ott et al., “Controlling Chaos,” Physical Review Letters, vol. 64, no. 11, 12

March 1990, pp. 1196–9.

65. Ibid.

66. Troy Shinbrot,Celso Grebogi,Edward Ott, and James A. Yorke, “Using Small Perturbations to Control Chaos,” Nature, vol. 363, 3 June 1993, p. 415.

67. Ott et al., Coping with Chaos.

68. Elizabeth Bradley, “Control Algorithms for Chaotic Systems,” MIT Artificial Intelligence Laboratory, AI Memo No. 1278 (Cambridge, Mass.: MIT, March 1991) [DTIC

AD–260 067].

69. Shinbrot et al., p. 415.

70. W.L. Ditto et al., “ExperimentalControl ofChaos,”Physical Review Letters, vol. 65,

no. 35, 24 December 1990, pp. 3211–4.

71. Henry D.I. Abarbanel, “Nonlinearity and Chaos at Work,” Nature, vol. 364, 19 August 1993, pp. 672–3.

111

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72. Martinus M. Sarigul-Klijn, “Application of Chaos Methods to Helicopter VibrationReductionUsing HarmonicControl,”Ph.D. Dissertation,NavalPostgraduate School,

Monterey, Calif., March 1990 [DTIC AD–A226 736].

73. Ed Paisley, “Out of Chaos, Profits,” Far Eastern Economic Review, 7 October 1993,

p. 84.

74. Raj Roy et al., “Dynamical Control of a Chaotic Laser: Experimental Stabilization of a Globally Coupled System,” Physical Review Letters, vol. 68, 2 March 1992, pp.

1259–62.

75. Jeffrey J. Leader, “Chaotic Keystream Generators for Additive Stream Ciphers,”

Technical Report NPS–MA–93–015 (Monterey, Calif.: Naval Postgraduate School, January 1993–April 1993) [DTIC AD–A265 824].

76. Joseph Neff and Thomas L. Carroll, “Circuits That Get Chaos in Sync,” Scientific

American, August 1993, pp. 120–2.

77. Elizabeth Bradley, “Taming Chaotic Circuits,” MIT Artificial Intelligence Laboratory, AI–TR 1388 (Cambridge, Mass.: MIT, September 1992) [DTIC AD–A260 067].

78. Ott et al., “Controlling Chaos.”

79. Price T. Bingham (Lieutenant Colonel, USAF, Retired), “The Air Force’s New

Doctrine,” Military Review (Fort Leavenworth, Kans.: U.S. Army Command and General

Staff College, November 1992), pp. 13–4.

80. Gleick, p. 278.

81. Thompson and Stewart, p. xi.

82. Wheatley, p. 126.

83. Jim Yorke, “Calculated Trajectories in Chaotic Systems,” Lecture at School of

Mathematics, Georgia Institute of Technology, Atlanta, Ga.: Fall 1989.

84. Argyris et al., p. 232.

85. Ibid., p. 72.

86. Ralph Abraham et al., “Computational Unfolding of Double-Cusp Models of

Opinion Formation,” International Journal of Bifurcation and Chaos, vol. 1, no. 2, 1991, pp.

417–30.

87. Y.Y. Azmy and V. Protopopescu, “Two Dimensional Maps Generated by Competitive Systems,” Oak Ridge National Laboratory (ORNL) TM–11026 (Oak Ridge, Tenn.:

ORNL, February 1989).

88. See Dewar et al.

89. Ibid., p. 42.

90. John T. Dockery and A.E.R. Woodcock, The Military Landscape (Cambridge, England: Woodhead Publishing, 1993).

91. Ibid., p. 138.

92. Glenn E. James et al., “Eliminations of Chaosin an Intracavity-Doubled Nd:YAG

Laser,” Optics Letters, vol. 15, no. 20, 15 October 1990, pp. 1141–3.

93. Alvin M. Saperstein, “Chaos and the Making of International Security Policy,” in

Krasner, The Ubiquity of Chaos, pp. 167–80.

94. David Nicholls (Major, USAF) and Todor D. Tagarev (Major, Bulgarian Air

Force), “What Does Chaos Theory Mean for Warfare?” Airpower Journal, Fall 1994, pp.

48–57.

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112
95. JamesR. Stright(Captain,USAF), “EmbeddedChaotic Time Series: Applications

in Prediction and Spatio-Temporal Classification” (Ph.D. Dissertation, Air Force Institute

of Technology,Wright-PattersonAirForceBase,Ohio, June 1994)[DTICAD–A280 690].

96. “Project 2025 Briefing for the Secretary of Defense and the Service Chiefs,” Institute for National Strategic Studies, National Defense University, 1994.

97. Saperstein, pp. 167–80.

98. Gleick, pp. 107–8.

99. Telephone conversations with Gerald Diaz (Lieutenant Colonel, USAF), Defense

Airborne Reconnaissance Office, Washington, D.C.: November 1994.

100. Patrick Seitz, “C-Band Shortage Forces Scramble for Alternatives,” Space News,

vol. 5, no. 43, 7–13 November 1994, p. 1.

101. Michael F. Barnsley, Fractals Everywhere, 2d ed. (Cambridge, Mass.: Academic

Press Professional, 1993), pp. 94–5.

102. Telephone interview with Rick Darby, Iterated Systems, Inc., Norcross, Ga.: 28

November 1994.

103. Information based on published descriptions of off-the-shelf products available

from Iterated Systems, Inc., 5550–A Peachtree Parkway, Suite 650, Norcross, Ga., 30092.

104. Gleick, p. 262.

105. Theodore H. Mueller(Lieutenant Colonel, USA), “Chaos Theory: The Mayaguez

Crisis,”U.S. ArmyWarCollege,Military StudiesProgram,CarlisleBarracks,Pa., 15March

1990.

106. JoshuaM.Epstein, “On theMathematicalBiology of ArmsRaces,Wars, andRevolutions,” in Lynn Nadel and Daniel L. Stein, eds., “1992 Lectures in Complex Systems,”

Lecture Vol. V, Santa Fe Institute Studies in the Sciences of Complexity (Reading, Mass.:

Addison-Wesley, 1993), pp. 425–36.

107. John D. Sterman, “Deterministic Chaos in Models of Human Behavior: Methodological Issues and Experimental Results,” System Dynamics Review, vol. 4, nos. 1–2, 1988,

pp. 148–78.

108. Gregoire Nicolis and Ilya Prigogine, Exploring Complexity (New York: W.H. Freeman, 1989), p. 232.

109. Abraham et al., pp. 417–30.

110. Robert Axelrod and D. Scott Bennett, “A Landscape Theory of Aggression,” British Journal of Political Science, part 2, vol. 23, April 1993, pp. 211–33.

111. Diana Richards, “Is Strategic Decision Making Chaotic?” Behavior Science, vol.

35, 1990, pp. 219–32.

112. Tagarev et al.

113. Sterman, pp. 148–78.

114. Ibid., p. 155.

115. William J.Ray, “EEGandChaos: DescriptionofUnderlying Dynamics and ItsRelation to Dissociative States,” Final Progress Report: NASA Grant #NAG-1-1441, February 1994 [NASA–CR–195171].

116. Hal Gregersen and Lee Sailer, “Chaos Theory and Its Implications for Social Science Research,” Human Relations, vol. 46, no. 7, 1993, pp. 777–803.

117. Tagarev et al.

118. Sterman, p. 172.

119. Grossman and Mayer-Kress, pp. 701–4.

113

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120. Dick Axelrod, “Getting Everyone Involved: How One Organization Involved Its

Employees, Supervisors and Managers in Redesigning the Organization,” The Journal of

Applied Behavioral Science, vol. 28, no. 4, December 1992, pp. 499–509.

121. Sterman, pp. 148–78.

122. David F. Andersen and Jeppe Sturis, “Chaotic Structuresin Generic Management

Models: Pedagogical Principles and Examples,” System Dynamics Review, vol. 4, nos. 1-2,

1988, pp. 218–45.

123. Nicholls and Tagarev.

124. Axelrod and Bennett.

125. T.N. Dupuy, Numbers, Predictions & War: The Use of History to Evaluate and Predict

the Outcome of Armed Conflict, 2d ed. (Fairfax, Va.: HERO Books, 1985).

126. Sean B. MacFarland (Major, USA), “Non-Linear Operations: A New Doctrine for

a New Era,” School of Advanced Military Studies, U.S. Army Command and General Staff

College, Fort Leavenworth, Kans., Second Term Academic Year 1993–94, 20 April 1994

[DTIC AD–A284 137], p. 15.

127. J. Marc LeGare (Major, USA), “Paradigm Found—The Nuclear and Nonlinear

Battlefields,” School of Advanced Military Studies, U.S. Army Command and General

StaffCollege,Fort Leavenworth, Kans.,First Term Academic Year 1992–93,February 1993

[DTIC AD–A264 451], p. 23.

128. Lester W. Grau (Lieutenant Colonel, USA), “Soviet Non-Linear Combat: The

Challenge ofthe 90s.” Soviet Army StudiesOffice,U.S. ArmyCombined ArmsCenter,Fort

Leavenworth, Kans., September 1990.

129. J.C. Wylie (Rear Admiral, USN), “Military Strategy: A General Theory of Power

Control,” in George Edward Thibault, ed., The Art and Practice of Military Strategy (New

York: Rutgers, The State University, 1967), pp. 200–1.

130. Robert Sobel, For Want of a Nail....: If Burgoyne Had Won at Saratoga (New York:

Macmillan, 1973).

131. Hugh R. Wilson, Jr., For Want of a Nail: The Failure of the League of Nations in Ethiopia (New York: Vantage Press, 1959).

132. Hawthorne Daniel, For Want of a Nail: The Influence of Logistics on War (New York:

McGraw-Hill, 1948).

133. RobertW.Batterman, “DefiningChaos,”Philosophy ofScience, vol. 60, 1993, p. 56.

The Newport Papers

114
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About the Author

Glenn James entered the Service in 1982 as the Air Force Academy’s top academic graduate. His four years as a cadet included a one-semester exchange at

Ecole de l’Air, the French Air Force Academy. Proceeding directly to graduate

school at the Georgia Institute of Technology, he earned a master’s degree in

mathematics, after which he took his first research assignment at the Air Force

Weapons Laboratory in Albuquerque, New Mexico.

In 1987, the Air Force Academy Department of Mathematical Sciences sponsored his return to Georgia Tech for a Ph.D., with a follow-on assignment as a

mathematics instructor. In the Department of Mathematical Sciences, Captain

James earned several teaching honors: the Tony M. Johnson Award for Excellence

in Teaching Mathematics, the department’s Outstanding Military Educator, and

the academic rank of associate professor.

An early promotion to major led to his selection to attend the College of Naval

Command and Staff at the Naval War College in Newport, Rhode Island. After

graduating “with distinction,” he was assigned to be Deputy Director of the Propulsion Sciences Division, the basic research arm of the Phillips Laboratory’s

Propulsion Directorate. He lives there now, at Edwards Air Force Base, California, with his wife Patricia and his children, Christine and Phillip.

THE NEWPORT PAPERS

Index

1991-1995

1. “‘Are We Beasts?’ Churchill and the Moral Question of World War II ‘Area

Bombing,’” by Christopher C. Harmon (December 1991).

2. “Toward a Pax Universalis: A Historical Critique of the National Military

Strategy,” by Lieutenant Colonel Gary W. Anderson, U.S. Marine Corps (April

1992).

3. “The ‘New’ Law of the Sea and the Law of Armed Conflict at Sea,” by Horace B. Robertson, Jr. (October 1992).

4. “Global War Game: The First Five Years,” by Bud Hay and Bob Gile (June

1993).

5. “Beyond Mahan: A Proposal for a U.S. Naval Strategy in the Twenty-First

Century,” by Colonel Gary W. Anderson, U.S. Marine Corps (August 1993).

6. “The Burden of Trafalgar: Decisive Battle and Naval Strategic Expectations

on the Eve of the First World War,” by Jan S. Breemer (October 1993).

7. “Mission in the East: The Building of an Army in a Democracy in the New

German States,” by Colonel Mark E. Victorson, U.S. Army (June 1994).

8. “Physics and Metaphysics of Deterrence: The British Approach,” by

Myron A. Greenberg (December 1994).

9. “A Doctrine Reader: The Navies of the United States, Great Britain, France,

Italy, and Spain,” by James J. Tritten and Vice Admiral Luigi Donolo, Italian

Navy (Retired), (December 1995).