Synopses & Reviews
The book integrates both classical and modern treatments of difference equations. It contains the most updated and comprehensive material, yet the presentation is simple enough for the book to be used by advanced undergraduate and beginning graduate students. This third edition includes more proofs, more graphs, and more applications. The author has also updated the contents by adding a new chapter on Higher Order Scalar Difference Equations, along with recent results on local and global stability of one-dimensional maps, a new section on the various notions of asymptoticity of solutions, a detailed proof of Levin-May Theorem, and the latest results on the LPA flour-beetle model. Saber Elaydi is Professor of Mathematics at Trinity University. He is also the author of Discrete Chaos (1999), and the Editor-In-Chief of the Journal of Difference Equations and Applications. About the Second Edition: The book is a valuable reference for anyone who models discrete systems. Dynamicists have the long-awaited discrete counterpart to standard textbooks such as Hirsch and Smale ('Differential Equations, Dynamical Systems, and Linear Algebra'). It is so well written and well designed, and the contents are so interesting to me, that I had a difficult time putting it down. - Shandelle Henson, Journal of Difference Equations and Applications Among the few introductory texts to difference equations this book is one of the very best ones. It has many features that the other texts don't have, e.g., stability theory, the Z-transform method (including a study of Volterra systems), and asymptotic behavior of solutions of difference equations (including Levinson's lemma) are studied extensively. It also contains very nice examples that primarily arise in applications in a variety of disciplines, including neural networks, feedback control, biology, Markov chains, economics, and heat transfer... -Martin Bohner, University of Missouri, Rolla
Synopsis
This book integrates both classical and modern treatments of difference equations. It contains the most updated and comprehensive material, yet the presentation is simple enough for the book to be used by advanced undergraduate and beginning graduate students. This third edition includes more proofs, more graphs, and more applications. The author has also updated the contents by adding a new chapter on Higher Order Scalar Difference Equations, and also recent results on local and global stability of one-dimensional maps, a new section on the various notions of asymptoticity of solutions, a detailed proof of Levin-May Theorem, and the latest results on the LPA flour-beetle model.
Synopsis
In contemplating the third edition, I have had multiple objectives to achieve. The ?rst and foremost important objective is to maintain the - cessibility and readability of the book to a broad readership with varying mathematical backgrounds and sophistication. More proofs, more graphs, more explanations, and more applications are provided in this edition. The second objective is to update the contents of the book so that the reader stays abreast of new developments in this vital area of mathematics. Recent results on local and global stability of one-dimensional maps are included in Chapters 1, 4, and Appendices A and C. An extension of the Hartman Grobman Theorem to noninvertible maps is stated in Appendix D. A whole new section on various notions of the asymptoticity of solutions and a recent extension of Perron s Second Theorem are added to Chapter 8. In Appendix E a detailed proof of the Levin May Theorem is presented. In Chapters 4 and 5, the reader will ?nd the latest results on the larval pupal adult ?our beetle model. The third and ?nal objective is to better serve the broad readership of this book by including most, but certainly not all, of the research areas in di?erence equations. As more work is being published in the Journal of Di?erence Equations and Applications and elsewhere, it became apparent that a whole chapter needed to be dedicated to this enterprise. With the prodding and encouragement of Gerry Ladas, the new Chapter 5 was born."
Synopsis
A must-read for mathematicians, scientists and engineers who want to understand difference equations and discrete dynamics Contains the most complete and comprehenive analysis of the stability of one-dimensional maps or first order difference equations. Has an extensive number of applications in a variety of fields from neural network to host-parasitoid systems. Includes chapters on continued fractions, orthogonal polynomials and asymptotics. Lucid and transparent writing style
Table of Contents
* Preface * List of Symbols * Dynamics of First-Order Difference Equations * Linear Difference Equations of Higher Order * Systems of Linear Difference Equations * Stability Theory * Higher Order Scalar Difference Equations * The Z-Transform Method and Volterra Difference Equations * Oscillation Theory * Asymptotic Behavior of Difference Equations * Applications to Continued Fractions and Orthogonal Polynomials * Control Theory * Answers and Hints to Selected Problems * Appendix A: Stability of Nonhyperbolic Fixed Points of Maps on the Real Line * Vandermonde Matrix * Stability of Nondifferentiable Maps * Stable Manifold and Hartman-Grobman-Cushing Theorems * Levin-May Theorem * Classical Orthogonal Polynomials * Identities and Formulas * References * Index