NONLINEAR DIFFERENCE EQUATIONS:
THEORY WITH APPLICATIONS TO SOCIAL SCIENCE MODELS

A monograph by Hassan Sedaghat, Mathematics Professor
Virginia Commonwealth University, Richmond, Virginia, USA

This book provides mathematical treatments of models from various social science disciplines. It also contains a detailed exposition of some the latest theory behind the mathematical analysis. Many theoretical results are new and some are presented for the first time in this book. These results, accompanied by various older and better known results that are also presented in detail, help create a comprehensive theoretical framework for the study of nonlinear models in discrete time (many of which have no continuous-time analogs in terms of differential equations).
Each section in every chapter ends in a Notes segment which gives additional or related information about the material covered in that section plus some history and complete references. In addition to technical results, the book contains more than 80 diagrams, lots of examples and counter-examples for motivation and clarification, an extensive bibliography and a detailed index. The book has material of interest to mathematicians and theortical researchers in social sciences and graduate students in applied mathematics as well as graduate students working on mathematical social science models.

Excerpts from reviews by experts in the field:

The Mathematical Reviews: "The level of rigor is substantial ... especially the first part where theorems are stated formally and proofs are given for nearly all of them... [The] final two chapters provide a rich collection of interesting applications from the social sciences, especially economics...These chapters constitute a valuable resource of important applications of discrete dynamical systems." Reviewer: F.R. Marotto, Rev#2004f:39001

The Journal of Difference Equations and Applications: "One of the major strengths of the book is the way Sedaghat has condensed and simplified many of the complex and technical properties of different mathematical social models...Beginning graduate students in mathematics, scientists in the natural or social sciences or mathematicians who want to enter the field of mathematical economics and mathematical [modeling] in the social sciences will find this book useful." Reviewer: A-A. Yakubu, Vol.10, No.10, 2004

TABLE OF CONTENTS (brief version - download a detailed version below)

Preface (click to download as a pdf file)

PART I: THEORY
Chapter 1: Preliminaries
Chapter 2: Dynamics on the Real Line
2.1 Equilibria and their stability
2.2 Cycles and limit cycles
2.3 Elementary bifurcations
Chapter 3: Vector Difference Equations
3.1 Stability
3.2 Semiconjugates of maps of the line
3.3 Chaotic maps
3.4 Polymodal systems and thresholds
Chapter 4: Higher Order Scalar Difference Equations
4.1 Boundedness and persistent oscillations
4.2 Permanence
4.3 Global attractivity and related results

PART II: APPLICATIONS TO SOCIAL SCIENCE MODELS
Chapter 5: Chaos and Stability in some Models
5.1 The accelerator-multiplier business cycle models
5.2 A productivity growth model
5.3 Chaos and competition in a model of consumer demand
5.4 An overlapping generations consumption-loan model
5.5 A dynamical model of consumer demand
5.6 A bimodal model of combat
Chapter 6: Additional Models
6.1 Addiction and habit formation
6.2 Budgetary competition
6.3 Cournot duopoly
6.4 Chaos in real exchange rates
6.5 Real wages and mode-switching
6.6 Chaos in a dynamic equilibrium model
6.7 Oscillatory behavior in an OLG model
6.8 Attractor basins and critical curves in two models
6.9 Reducing inflation: Gradual vs. shock treatments
6.10 Walrassian tatonnement with adaptive expectations
6.11 Socio-spatial dynamics
6.12 Models of arms race

SAMPLES FROM THE BOOK (click each topic to download it as a pdf file)

Table of Contents (detailed version)

From Section 2.1B
From Section 2.2C
From Section 3.3C
From Section 4.3A
From Section 5.3C
From Section 5.6A

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