Dynamical Systems with Applications using MAPLE

This book provides an introduction to the theory of dynamical systems with the aid of the Maple algebraic manipulation package. lt is written for both senior un­ dergraduates and first-year graduale students. The firsthalf of the book deals with continuous systems using ordinary differential equations (Chapters 1-12) and the second half is devoted to the study of discrete dynamical systems (Chapters 13-20). (The author has gone for breadth of coverage rather than fine detail and theorems with proof are kept at a minimum.) The material is not clouded by functional analytic and group theoretical definitions, and so is intelligible to readers with a general mathematical background. Some of the topics covered are scarcely cov­ ered elsewhere. Most of the material in Chapters 9-12, 16, 17, 19, and 20 is at postgraduale Ievel and has been influenced by the author's own research interests. It has been found that these chapters are especially useful as reference material for senior undergraduate project work. The book has a very hands-on approach and takes the reader from the basic theory right through to recently published research material. "The text treats a remarkable spectrum of topics and has a little for everyone. It can serve as an introduction to many of the topics of dynamical systems, and will help even the most jaded reader, such as this reviewer, enjoy some of the interactive aspects of studying dynamics using Maple." --UK Nonlinear News (Review of First Edition) "The book will be useful for all kinds of dynamical systems courses.... [It] shows the power of using a computer algebra program to study dynamical systems, and, by giving so many worked examples, provides ample opportunity for experiments. ... [It] is well written and a pleasure to read, which is helped by its attention to historical background." --Mathematical Reviews (Review of First Edition) Since the first edition of this book was published in 2001, MapleTM has evolved from Maple V into Maple 13. Accordingly, this new edition has been thoroughly updated and expanded to include more applications, examples, and exercises, all with solutions; two new chapters on neural networks and simulation have also been added. There are also new sections on perturbation methods, normal forms, Gröbner bases, and chaos synchronization. The work provides an introduction to the theory of dynamical systems with the aid of Maple. The author has emphasized breadth of coverage rather than fine detail, and theorems with proof are kept to a minimum. Some of the topics treated are scarcely covered elsewhere. Common themes such as bifurcation, bistability, chaos, instability, multistability, and periodicity run through several chapters. The book has a hands-on approach, using Maple as a pedagogical tool throughout. Maple worksheet files are listed at the end of each chapter, and along with commands, programs, and output may be viewed in color at the author's website. Additional applications and further links of interest may be found at Maplesoft's Application Center. Dynamical Systems with Applications using Maple is aimed at senior undergraduates, graduate students, and working scientists in various branches of applied mathematics, the natural sciences, and engineering. ISBN 978-0-8176-4389-8 § Also by the author: Dynamical Systems with Applications using MATLAB®, ISBN 978-0-8176-4321-8 Dynamical Systems with Applications using Mathematica®, ISBN 978-0-8176-4482-6 Front Matter....Pages i-xiii A Tutorial Introduction to Maple....Pages 1-11 Differential Equations....Pages 13-34 Linear Systems in the Plane....Pages 35-49 Nonlinear Systems in the Plane....Pages 51-64 Interacting Species....Pages 65-76 Limit Cycles....Pages 77-89 Hamiltonian Systems, Lyapunov Functions, and Stability....Pages 91-103 Bifurcation Theory....Pages 105-117 Three-Dimensional Autonomous Systems and Chaos....Pages 119-142 Poincaré Maps and Nonautonomous Systems in the Plane....Pages 143-167 Local and Global Bifurcations....Pages 169-180 The Second Part of David Hilbert’s Sixteenth Problem....Pages 181-192 Limit Cycles of Liénard Systems....Pages 193-204 Linear Discrete Dynamical Systems....Pages 205-222 Nonlinear Discrete Dynamical Systems....Pages 223-253 Complex Iterative Maps....Pages 255-265 Electromagnetic Waves and Optical Resonators....Pages 267-282 Analysis of Nonlinear Optical Resonators....Pages 283-294 Fractals....Pages 295-312 Multifractals....Pages 313-328 Controlling Chaos....Pages 329-345 Examination-Type Questions....Pages 347-351 Solutions to Exercises....Pages 353-373 Back Matter....Pages 375-399 This work covers material for an introductory course in the theory of dynamical systems. There is a short tutorial in MAPLE to facilitate the understanding of the theory. The text is divided into two continuous systems using differential equations and discrete dynamical systems. Differential equations are used to model examples taken from various topics such as mechanical systems, interacting species, electronic circuits, chemical reactions, and meterology. The second part of the text deals with real and complex dynamical systems. Examples are taken from population modelling, nonlinear optics, and materials science. Linear algebra and real and complex analysis are prerequisites. Since the first edition of this book was published in 2001, MapleTM has evolved from Maple V into Maple 13. Accordingly, this new edition has been thoroughly updated and expanded to include more applications, examples, and exercises, all with solutions; two new chapters on neural networks and simulation have also been added. The author has emphasized breadth of coverage rather than fine detail, and theorems with proof are kept to a minimum. This text is aimed at senior undergraduates, graduate students, and working scientists in various branches of applied mathematics, the natural sciences, and engineering. Suitable for many kinds of dynamical systems courses, this book shows the power of using a computer algebra program to study dynamical systems. It provides an introduction to the theory of dynamical systems with the aid of the Maple algebraic manipulation package.