Nonlinear Difference Equations

MATHEMATICAL MODELLING: Theory and Applications

VOLUME 15

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Editorial Board: I.-P. Aubin (Universite de Paris IX, France) E. Jouini (University of Paris 1 and ENSAE, France) GJ. Klir (New York, U.S.A.) P.G. Mezey (Saskatchewan, Canada) F. Pfeiffer (Miinchen, Germany) A. Stevens (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany) H.-J. Zimmerman (Aachen, Germany)

Nonlinear Difference Equations Theory with Applications to Social Science Models

by

Hassan Sedaghat Department of Mathematics, Virginia Commonwealth University, Richmond, Virginia, U.S.A.

• " SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-6215-4 ISBN 978-94-017-0417-5 (eBook) DOI 10.1007/978-94-017-0417-5

Printed on acid-free paper

AII Rights Reserved © 2003 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2003 Softcover reprint of the hardcover 1 st edition 2003 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Dedicated, with love and gratitude, to my

father Ali and my mother Zahra

Contents

Preface Xl

Acknowledgments xv

Part I THEORY

l. PRELIMINARIES 3

2. DYNAMICS ON THE REAL LINE 13 1 Equilibria and their Stability 13

A. An inverse map characterization 13 B. Asymptotic Stability 21 C. Nonlinear local stability 26 D. Further results 31 E. Notes 33

2 Cycles and Limit Cycles 33 A. Coexistence of cycles 33 B. Limit cycles 42 C. Notes 50

3 Elementary Bifurcations 51 A. The tangent bifurcation 53 B. The period-doubling bifurcation 58 C. Other one-parameter bifurcations 64 D. Notes 67

3. VECTOR DIFFERENCE EQUATIONS 71

1 Stability 71 A. The invariance principle 71 B. Boundedness and stability 74

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C. Notes 78

2 Semiconjugates of maps of the line 79 A. Definitions and examples 80 B. The basic dynamics 87 C. Fiber bifurcations 94 D. Invariants 105 E. Semiconjugate links as Liapunov functions 107 F. Notes 111

3 Chaotic Maps 112 A. Chaos and period 3 113 B. Snap-back repellers: Beyond period 3 116 C. R-semiconjugate maps and chaos 124 D. Notes 139

4 Polymodal systems and thresholds 140 A. Basic concepts and examples 142 B. Ejector cycles: Continuity revisited 149 C. Mode structures 155 D. Notes 163

4. HIGHER ORDER SCALAR DIFFERENCE EQUATIONS 165

1 Boundedness and persistent oscillations 165 A. Persistent oscillations 165 B. Absorbing intervals 171 C. The equation xn+1 = CXn + g(xn - Xn-l) 172 D. Notes 182

2 Permanence 185 A. Semi-permanence 185 B. The equation Xn = xn-lg(Xn-l, ... 'xn- m) 190 C. Additional equations 195 D. Notes 200

3 Global attractivity and related results 200 A. Weak contractions and stability 201 B. Weak expansions and instability 210 C. Coordinate-wise monotonicity and global attractivity 214 D. The equation xn+1 = 2.::0 aiXn-i + 9 (2.::0 bixn-i) 224 E. The equation Xn = ~-k [1 + 9 (2.::1 !i(Xn-i))] 227 F. Notes 236

Contents ix

Part II APPLICATIONS TO SOCIAL SCIENCE MODELS

5. CHAOS AND STABILITY IN SOME MODELS 243

1 The Accelerator-Multiplier Business Cycle Models 243 A. The Goodwin-Hicks model 244 B. Full consumption of savings: Puu's model 250 C. The classical model of Hicks 261 D. Notes 268

2 A productivity growth model 269 A. The model 270 B. Stability and oscillations 271 C. Notes 273

3 Chaos and competition in a model of consumer behavior 274 A. The model 274 B. Snap-back repellers and chaotic behavior 276 C. Chaotic competition and exclusion 281 D. Notes 285

4 An overlapping generations consumption-loan model 287 A. The model 287 B. Chaotic behavior 289 C. Notes 291

5 A dynamical model of consumer demand 291 A. The model 291 B. General asymptotics and stability 294 c. Demand functions and their properties 298 D. Notes 309

6 A bimodal model of combat 310 A. The model 311 B. General asymptotics 314 C. Transient behavior 319 D. The invariant plane z = 1 322 E. Notes 334

6. ADDITIONAL MODELS 339

1 Addiction and habit formation 339

2 Budgetary competition 341

3 Cournot duopoly 343

4 Chaos in real exchange rates 345

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5 Real wages and mode switching 346 6 Chaos in a dynamic equilibrium model 349 7 Oscillatory behavior in an OLG model 352 8 Attractor basins and critical curves in two models 353 9 Reducing inflation: Gradual vs. shock treatments 355 10 Walrasian tatonnement with adaptive expectations 358 11 Socia-spatial dynamics 361 12 Models of arms race 363

Bibliography 367

Index 385

Preface

It is generally acknowledged that deterministic formulations of dy­namical phenomena in the social sciences need to be treated differently from similar formulations in the natural sciences. Social science phe­nomena typically defy precise measurements or data collection that are comparable in accuracy and detail to those in the natural sciences. Con­sequently, a deterministic model is rarely expected to yield a precise description of the actual phenomenon being modelled. Nevertheless, as may be inferred from a study of the models discussed in this book, the qualitative analysis of deterministic models has an important role to play in understanding the fundamental mechanisms behind social sci­ence phenomena. The reach of such analysis extends far beyond tech­nical clarifications of classical theories that were generally expressed in imprecise literary prose.

The inherent lack of precise knowledge in the social sciences is a fun­damental trait that must be distinguished from "uncertainty." For in­stance, in mathematically modelling the stock market, uncertainty is a prime and indispensable component of a model. Indeed, in the stock market, the rules are specifically designed to make prediction impossible or at least very difficult. On the other hand, understanding concepts such as the "business cycle" involves economic and social mechanisms that are very different from the rules of the stock market. Here, far from seeking unpredictability, the intention of the modeller is a scientific one, i.e., to clarify and explain the phenomenon in an objective way that will make it possible to apply what is learned (e.g., to moderate down-turns or busts, and thus help lessen human suffering).

Although undesirable in a scientific study of the business cycle and similar concepts, uncertainty is impossible to avoid completely in the social sciences. For this reason, there is considerable room for stochas­tic formulations in mathematical models, as long as one is careful to

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not attribute complex, but deterministic phenomena to random effects. For instance, it is known that the mathematical equations that result in persistent oscillatory behavior that is asymptotically and structurally stable cannot be linear (see Section 4.1 below). Thus, in part to as­sure that a system of linear equations generated persistent oscillations of the type seen in the economic data, it was common in the 1960's and '70's to add stochastic terms (called "random shocks") to the lin­ear equations. But this was tantamount to assuming that the economic mechanisms behind the business cycle were incapable of generating sus­tained oscillations. Such an assumption never had any support, either philosophically or on economic grounds. Moreover, its tacit acceptance provided no credible information about the economic reality behind the business cycle - almost anything could be attributed to random effects when linear oscillations dissipated. A similar comment applies to other social science models, and the "knowledge" gained about them through statistical analyses based on linear equations. A more realistic, and in­formative, approach would be to propose a deterministic model whose properties can be analyzed qualitatively, and then add stochastic terms to the deterministic formalism for the sake of better fitting the existing data, if any.

The lack of precision in the social sciences that was noted above has important consequences for both the modelling, and the associated mathematical theory. In particular, if a model is specified by a mapping F of the Euclidean space ]Rm, then typically F is to be given by means of the various properties that it is supposed to have, rather than by specific analytical expressions for which no satisfactory empirical justi­fication can be found. The properties of F are generally deduced from the semantic context of the model through a process of abstraction that has matured considerably in recent times. Most (though not all) of the mathematical models encountered in this book involve partially speci­fied mappings that reflect the characteristic coarseness of models. The mathematical results about such mappings typically involve rigorously establishing such qualitative properties as permanence and boundedness, persistent oscillations (periodic or aperiodic), sensitivity to initial con­ditions, and stability, instability, and the global attractivity of equilibria and cycles.

This monograph is split into two parts: The first gives a rigorous yet general mathematical treatment of maps and equations (Chapters 2-4) containing both some of the best known results in the literature, and many results that are quite recent (including a few hitherto unpublished ones). Several new concepts such as semiconjugates, polymodal sys­tems and ejector cycles, persistent oscillations and absorbing intervals,

PREFACE xiii

as well as recently developed analytical techniques are presented in this book, many for the first time. The choice of topics in Chapters 2-4 is motivated to a large extent by the preceding observations about social science models and the formalism that seems appropriate to them. The mathematical treatment is rigorous, and the vast majority of theorems come with complete proofs. For readers who do not wish to study the sometimes long and technical proofs, a liberal supply of corollaries, re­marks and examples provide a good sense of the boundaries of the main results, i.e., their applications and their limitations.

In Chapter 2, the theory on the real line is studied. Major topics include necessary and sufficient conditions for the asymptotic stability and instability of fixed points, coexistence and the Sharkovski ordering of cycles, the Singer-Allwright theory of limit cycles and the modern the­ory of one-parameter bifurcations. Chapter 3 begins with a presentation of LaSalle's approach to Liapunov stability for maps, and then proceeds to a discussion of mappings of Euclidean spaces that are semiconjugate to maps of the real line. Semiconjugacy as presented here, extends the notion of invariants to more general mappings and is intimately related to Liapunov functions. Further, it permits the extension of certain top­ics from the real line to the higher dimensional context. For example, it is possible to obtain extensions of the Li-Yorke Theorem that are very different from known analogs such as Marotto's Theorem. The last sec­tion of Chapter 3 is concerned with a light-hearted, though systematic study of complex threshold systems as special types of piecewise linear or polymodal systems. Certain constructs, such as ejector cycles, are seen to allow an application of the continuous theory to obtaining results about the global behavior of trajectories. Chapter 4 is concerned with the relatively better developed topic of higher order, scalar difference equations and here the level of rigor is proportionately higher.

The second half of the book (Chapters 5,6) presents several models from economics and other social sciences. In Chapter 5, a few of these models have been rigorously (though often not exhaustively) analyzed, following a brief description and derivation for each model. Readers who are interested in semantic aspects beyond what is presented here will find it more useful to learn them first hand from the original authors who introduced the models. Chapter 6 contains more models, each of which is presented in a brief format. In some cases, the mathematical analysis is similar to what is found in Chapter 5, and in these cases, it is left to the reader to complete the tasks. In the remaining cases, the analysis either requires material that goes beyond the scope of this monograph, or there is no known analytical work beyond what is provided by their authors' (usually in the form of numerical simulations). In a few of these

XIV NONLINEAR DIFFERENCE EQUATIONS

latter models, the fundamental equations may need to be modified or restricted so as to permit the existence of an adequatE. number (e.g., an open set) of bounded, semantically viable trajectories within the positive cone of ]Rffi.

This book has been written in such a way that it can be read and the gist of it understood not only by professional mathematicians, but also by readers with limited expertise (or interest) in rigorous proofs. Such readers may wish to study the statements of theorems, and then proceed to study their consequences in various examples and models. The flavor of the book is mathematical however; it is assumed that the typical reader has been exposed to college undergraduate-level math­ematics, and that he or she has gained some appreciation for precise mathematical analysis as a result. The rigor and universal validity of the mathematical language permits an objective discussion of scientific concepts and theories, and as such it is an invaluable tool for developing a scientific understanding of the complex and surprisingly non-random area covered by the social science models.

HASSAN SEDAGHAT

Acknowledgments

The general idea for this book took shape in my mind after a confer­ence in San Antonio in 1999, where a conversation with professor Saber Elaydi convinced me that the time had come for a book that contained a rigorous mathematical presentation of nonlinear models from the social sciences. Thereafter, the early support and encouragement of Dr. Lies­beth Mol, my editor at Kluwer was essential in getting this project off the ground. Discussions with professor Gerry Ladas and other faculty and graduate students of the University of Rhode Island were influential in developing certain parts of this book such as Sections 4.2 and 4.3. I would like to further acknowledge inputs by the many colleagues, stu­dents and friends who read various parts of this monograph and provided suggestions and corrections, and to the staff of Kluwer Academic Pub­lishers for their help in the preparation of the manuscript. Any mistakes or misunderstandings still remaining are my oversights only.

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