Discrete Dynamical Systems
This book provides an introduction to discrete dynamical systems -- a framework of analysis commonly used in the fields of biology, demography, ecology, economics, engineering, finance, and physics. The book characterizes the fundamental factors that gove
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Oded Galor
Discrete Dynamical Systems
Prof. Oded Galor Brown University Department of Economics 64 Waterman St. Providence, RI 02912 USA oded [email protected]
Library of Congress Control Number: 2006934984 ISBN-10 3-540-36775-6 Springer Berlin Heidelberg New York ISBN-13 978-3-540-36775-8 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Integra Software Services Pvt. Ltd., Pondicherry, India Cover design: WMX Design GmbH Printed on acid-free paper
SPIN: 11791140
42/3100/Integra
5 4 3 2 1 0
To my sons: Alon, Yuval, and Omri
Acknowledgements
This book has evolved over the past decade while I have been teaching master and doctoral courses at Brown University and the Hebrew University of Jerusalem on the subjects of discrete dynamical systems and their applications for intertemporal macroeconomics and economic growth. I benefited from the interaction with several cohorts of students who have read the manuscript in its various stages and provided constructive comments on its different segments. I am especially grateful for Benny Berdugo, Ruben Durante, Leo Feler, Phillip Garner, Takuma Kunieda, Yishay Maoz, Petros Milionis and Carmina Vargas, who have carefully read and commented on the final draft of the manuscript.
Preface
This book provides an introduction to discrete dynamical systems – a framework of analysis that is commonly used in the fields of biology, demography, ecology, economics, engineering, finance, and physics. The book characterizes the fundamental factors that govern the quantitative and qualitative trajectories of a variety of deterministic, discrete dynamical systems, providing solution methods for systems that can be solved analytically and methods of qualitative analysis for those systems that do not permit or necessitate an explicit solution. The analysis focuses initially on the characterization of the factors that govern the evolution of state variables in the elementary context of one-dimensional, first-order, linear, autonomous systems. The fundamental insights about the forces that affect the evolution of these elementary systems are subseq
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