Dynamical Systems with Applications Using Mathematica

By Ulrich Hoensch
Print

Dynamical Systems with Applications Using Mathematica

Stephen Lynch
Birkhäuser (2007), 484 pp., Price: US$59.95 (softcover)
ISBN 978-0-8176-4482-6.
Reviewer: Ulrich Hoensch
Rocky Mountain College
Billings, Montana, U.S.A.

The book reviewed here is one of three different versions, all authored by Stephen Lynch. It includes code for the computer algebra system (CAS) Mathematica, whereas the other two versions use Matlab and Maple code. These have not been reviewed, but it is to be expected that they are very similar to the Mathematica version. Lynch's book gives a broad introduction to various topics in the area of dynamical systems. Some are standard methods and classical results as they may be found in undergraduate or graduate textbooks. Others are subjects of current research and may be of interest to researchers both within and outside the area of dynamical systems. The variety of subjects covered makes this book excellent reading not only for people new to the field; it is also a great way for the specialist to inform herself or himself about other areas and applications of dynamical systems. In the opinion of this reviewer, it is the applied aspects of dynamical systems that make this book an exciting and worthwhile read.

In each chapter, basic definitions and results related to the topic under discussion are presented. The presentation is very clear and focused, and, as far as each subject allows, non-technical. Formal proofs are given sparingly, and only for results that admit short and straight-forward arguments. The unifying theme of this book is to give the reader an idea of each subject by discussing well-chosen examples. Mostly, this discussion involves using a CAS to check conditions of theorems, perform numerical or symbolic calculations, or utilize the CAS's graphing capabilities. Readers are invited to download the CAS code from a web site. Practice problems at the end of each chapter give the reader the opportunity to use and adapt this code. In general, these practice problems were found to be rather straightforward, and many of them are suitable for undergraduate students. At the end of each chapter a list of relevant references, comprising both standard reference texts and research articles, is given. These references offer a great inroads for readers who desire to learn more about the subject discussed in the chapter. In the following, we give a brief chapter-by-chapter overview of the book.

Chapter 0 is meant as a quick introduction to Mathematica and briefly touches upon commonly used features of the software. This introduction is useful for readers who generally are familiar with another CAS or who need a quick refresher in using Mathematica. Readers who wish to gain a thorough understanding of Mathematica are advised to turn to some of the references given at the end of this chapter.

Chapter 1 presents basic solution methods for first-order differential equations. Two important types of applications that appear in later chapters of the book are introduced: chemical reaction equations and electrical circuits. Chapter 2 deals with qualitative and geometric methods of analyzing two-dimensional systems of autonomous differential equations. Chapter 3 presents applications to population dynamics. Models of competition and coexistence of two species and predator-prey models including the Lotka-Volterra model and the Holling-Tanner model are discussed.

Chapter 4 marks a departure from material customarily found in differential equations textbooks. It focuses on criteria for existence, non-existence or uniqueness of limit cycles. In particular, the Poincaré-Bendixson Theorem, Dulac's Criterion, and Bendixson's Criterion are stated. Examples include Liénard systems, and perturbation methods for obtaining general solutions to Duffings's and van der Pol's equation are presented. Here, the advantages of using a CAS instead of a numerical solver come to bear. Chapter 5 gives a brief introduction to two-dimensional Hamiltonian systems and introduces Lyapunov functions. Chapter 6 deals with bifurcations of planar systems. Generic bifurcations are presented, examples of bi-stable systems and a system with a large-amplitude limit cycle bifurcation are given.

Chapter 7 deals with three-dimensional systems of autonomous differential equations. Examples of chaotic systems include the Rössler system, the Lorenz system, Chua's circuit and the Belousov-Zhabotinski reaction. In Chapter 8, Poincaré maps are used to analyze higher-dimensional systems such as Hamiltonian systems with two degrees of freedom and the Duffing equation as an example of a non-autonomous system in the plane. Chapter 9 deals with the process of bifurcating limit cycles from a center. Gröbner bases and Melnikov integrals are introduced. This chapter is rather technical and relies heavily on the use of CAS. Chapter 10 opens the door to a subject of intense current research: the second part of Hilbert's Sixteenth Problem. Some results related to this problem are stated. The remainder of the chapter is devoted to introducing Poincaré compactification, and presenting global and local results for Liénard systems.

A return of more basic material occurs in Chapter 11, where linear discrete dynamical systems are discussed. The Leslie model serves as a nice example of a linear but non-trivial population model. Chapter 12 deals with nonlinear discrete dynamical systems. The focus is on bifurcations and chaotic systems, examples of which are furnished in the form of the tent map, the logistic map, the Gaussian map, and the Hénon map. As usual, graphical and numerical explorations dominate the discussion. Chapter 13 gives a short overview of the basics of discrete complex dynamical systems. The complex quadratic map, corresponding Julia sets and the Mandelbrot set are investigated.

Chapter 14 serves as an introduction to the theory of non-linear optics. A short introduction to Maxwell's equations and a historical overview of the field precede a discussion of optical resonators. Chapter 15 deals with fractals and multifractals. Some classical fractals are presented and their fractal dimension is computed. Box-counting dimension is introduced and multifractals are discussed. Chapter 16 touches upon the subjects of chaos control and synchronization. Again, a historical overview gives the reader an overview of the field, and the examples that follow serve as illustrations of the problems encountered in these areas. Chapter 17 is, in the mind of this reviewer, one of the highlights of this book. Here, the theory of neural networks is introduced, and concrete examples of data analysis with neural networks are given. An example of a discrete Hopfield network that uses attractors as fundamental memories is a delightful example of applied dynamical systems.

Chapter 18 is a collection of exam questions that apparently were used by the author when teaching a course based on the material in the book. Chapter 19 gives solutions or hints to most of the practice problems. Finally, a long set of references marks the end of the book.

Stephen Lynch's book offers a comprehensive introduction to the theory and application of differential equations and dynamical systems methods. Its focus on applications and avoidance of overly technical arguments makes it a an equally good choice for teaching an undergraduate course in dynamical systems, as self-study for graduate students interested in dynamical systems, or as an introductory text for researchers seeking an overview of some current developments in applied dynamical systems. Most importantly, its content and presentation style convey the excitement that has drawn many students and researchers to dynamical systems in the first place.

Categories: Magazine, Book Reviews
Tags:

Please login or register to post comments.

Name:
Email:
Subject:
Message:
x