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CISM COURSES AND LECTURES
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The series presents lecture notes, monographs, edited works and
proceedings in the field of Mechanics, Engineering, Computer Science
and Applied Mathematics.
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and technical community results obtained in some of the activities
organized by CISM, the International Centre for Mechanical Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECTURES - No. 476
NONLINEAR DYNAMICAL SYSTEMS
IN ECONOMICS
EDITED BY
MARJI LINES
UNIVERSITY OF UDINE, ITALY
SpringerWien NewYork
This volume contains 90 illustrations
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© 2005 by CISM, Udine
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ISBN-10 3-211-26177-X SpringerWienNewYork
ISBN-13 978-3-211-26177-4 SpringerWienNewYork
PREFACE
Many problems in theoretical economics are mathematically formalized as dynam
ical systems of difference and differential equations. In recent years a truly open
approach to studying the dynamical behavior of these models has begun to make
its way into the mainstream. That is, economists formulate their hypotheses and
study the dynamics of the resulting models rather than formulating the dynamics
and studying hypotheses that could lead to models with such dynamics. This is a
great progress over using linear models, or using nonlinear models with a linear
approach, or even squeezing economic models into well-studied nonlinear systems
from other fields.
There are today a number of economic journals open to publishing this type
of work and some of these have become important. There are several societies
which have annual meetings on the subject and participation at these has been
growing at a good rate. And of course there are methods and techniques avail
able to a more general audience, as well as a greater availability of software for
numerical and graphical analysis that makes this type of research even more excit
ing. The lecturers for the Advanced School on Nonlinear Dynamical Systems in
Economics, who represent a wide selection of the research areas to which the the
ory has been applied, agree on the importance of simulations and computer-based
analysis. The School emphasized computer applications of models and methods,
and all contributors ran computer lab sessions. The exigencies of space left us
no room to include the related exercises and software, but you can get a taste of
those (and access to a wealth of other useful material) by referring to contributors
home pages.
The volume is structured as follows. The first three chapters are introductory
(though not necessarily elementary). The first provides a quick introduction to
nonlinear analysis: a short review of what is useful from linear systems theory in
the analysis of nonlinear systems through first-order approximations; the essential
theorems useful for local analysis; definitions and terminology for stability analy-
sis; a discussion of limit sets and local bifurcation theory. The second chapter is
a discussion of chaos and complexity at an intermediate level of difficulty. Typi
cal examples of systems with chaotic trajectories are provided in order to discuss
deeper issues including chaotic attractors as a form of global stability, random
versus deterministic chaotic series, predictability of chaotic systems, statistical
predictability of chaotic systems and financial and economic implications of de
terministic chaos. The section on complexity focuses on the cellular automata
approach, considering complexity classes, predictability and agent-based modeling
in economics. The third chapter is an introduction to a relatively new line of
research in economics, the ergodic approach, which investigates the probabilistic
properties of dynamical systems. The basic concepts of elementary measure the
ory are used to understand the dynamics of nonlinear models. Concepts such as
invariant, ergodic, absolutely continuous and natural measures are explained with
simple examples. The issue of deterministic chaos and randomness is discussed
from the point of view of predictability, by means of the notion of metric entropy.
These three chapters, coming from, quite different approaches, give a broad intro
duction to definitions, concepts and methods that are useful for the more applied
chapters that follow.
The final four chapters are applications of local and global bifurcation theory
to models coming from different approaches and fields in economics. In Chapter
4 the local approximation is used to understand the dynamics in two versions of
one of the models currently dominating macroeconomics, the Overlapping Genera
tions (OLG) model. From the basic 1-dimensional Diamond model, with standard
choices of functions, hypotheses are altered to develop two other models, each il
lustrating problems that arise once the basic hypotheses are abandoned. A second
type of OLG model is studied, in 2 and 3 dimensions, for which the Neimark-
Sacker bifurcation is typical and invariant curves are common limit sets. Chapter
5 focuses on some very interesting work in modeling the dynamics of financial
markets with heterogeneous agents, an approach receiving much attention in the
field. The first part employs the cobweb model with rational versus naive agents
to demonstrate a rational route to randomness as well as the existence of a ho-
moclinic orbit. The second part develops an asset pricing model in which agents
switch between different forecasting or trading strategies, using an evolutionary
fitness measure. Prices and beliefs co-evolve over time, leading to instability and
complicated price fluctuations. Chapter 6 is dedicated to complex dynamics in
models from oligopoly theory, which is one of the fields that pioneered in the ap
plication of nonlinear dynamics. A model of duopolists with nonlinear reaction
curves gives rise to a period-doubling scenario to chaos and the coexistence of
cycles and complicated basins of attraction. The model is extended to include
adaptive expectations, and stability is lost through the Neimark-Sacker bifurca
tion giving rise to a very complicated structure of periodic ArnoVd tongues ob
servable in the two-parameter bifurcation diagram. The critical line approach is
used to define the absorbing areas. In Chapter 7 an excellent review of defini
tions and properties concerning noninvertible maps is followed by a description
of the method of critical lines and curves in determining the trapping region, with
examples of global bifurcations causing nonconnected basins of attraction. These
methods are applied to a Cournot duopoly game with best reply, naive expectations
and adaptive behavior, and to a duopoly game with gradient dynamics. Finally^
the related phenomena of chaos synchronization and riddled basins are studied in
a dynamic brand competition model with market shares and marketing effort.
The organization of a learning experience such as the Advanced School is
amazingly complex in itself and I wish to thank Prof Manuel Velarde, current
Rector of CISM, for his support at all stages, the CISM staff for their patience
and competence, my fellow lecturers for their efforts in preparing presentations,
lab sessions, and the chapters that follow, and of course, the students, who gave
themselves up for an entire week to the joys of nonlinearity.
Marji Lines
CONTENTS
Introductory notes on the dynamics of linear and linearized systems
by M. Lines and A. Medio 1
Complex and chaotic dynamics in economics
by D. Foley 27
Ergodic approach to nonlinear dynamics
by A, Medio 67
Local bifurcation theory apphed to OLG models
by M. Lines 103
Heterogeneous agent models: two simple examples
by C. Hommes 131
Complex oligopoly dynamics
by T. Puu 165
Coexisting attractors and complex basins in discrete-time economic
models
by G. Bischi and F. Lamantia 187
Introductory Notes on the Dynamics of Linear and
Linearized Systems
Marji Lines and Alfredo Medio
Department of Statistics, University of Udine, Udine, Italy
Abstract In the following we provide terminology and concepts which are central
to understanding the dynamical behavior of nonlinear systems. The first four sec
tions are a necessarily very brief introduction to the dynamics of linear systems,
in which we concentrate on those aspects most useful for acquiring a sense of the
basic behaviours characterising systems of differential and difference equations. The
last four sections introduce basic notions of stability, the linear approximation and
the Hartman-Grobman Theorem, the use of the Centre Manifold Theorem, local
bifurcation theory.*
1 Linear systems in continuous time
In this section we discuss the form of the solutions to the general system of linear differ
ential equations
x = Ax X G E^ (1.1)
where A is a m x m matrix of constants, also called the coefficient matrix.
An obvious solution to equation (1.1) is x{t) = 0, called the equilibrium solution
because if x = 0, i: = 0 as well. That is, a system starting at equilibrium stays there
forever. Notice that if A is nonsingular, x = 0 is the only equilibrium for linear systems
like (1.1). Nontrivial solutions will be of the form
x{t) = e^'u (1.2)
where ix is a vector of real or complex constants and A real or complex constants. Differ
entiating (1.2) with respect to time, and substituting into (1.1), we obtain Xe^^u = Ae^^u
which, for e^^ ^0, implies
(A-A/)ix-0 (1.3)
where 0 is an m-dimensional null vector. A nonzero vector u satisfying (1.3) is called an
eigenvector of matrix A associated with the eigenvalue A. Equation (1.3) has a nontrivial
solution w 7^ 0 if and only if
det{A - A7) - 0 (1.4)
*For suggestions on further reading and an extended bibliography please see Medio and Lines,
Nonlinear Dynamics: A Primer^ Cambridge: Cambridge University Press, 2001.
