Devaney Robert L. - Differential Equations, Dynamical Systems, and an Introduction to Chaos

In the thirty years since the publication of the first edition of this book, much has changed in the field of mathematics known as dynamical systems. In the early 1970s, we had very little access to high-speed computers and computer graphics. The word chaos had never been used in a mathematical setting. Most of the interest in the theory of differential equations and dynamical systems was confined to a relatively small group of mathematicians.

Things have changed dramatically in the ensuing three decades. Computers are everywhere, and software packages that can be used to approximate solutions of differential equations and view the results graphically are widely available. As a consequence, the analysis of nonlinear systems of differential equations is much more accessible than it once was. The discovery of complicated dynamical systems, such as the horseshoe map, homoclinic tangles, the Lorenz system, and their mathematical analysis, convinced scientists that simple stable motions such as equilibria or periodic solutions were not always the most important behavior of solutions of differential equations. The beauty and relative accessibility of these chaotic phenomena motivated scientists and engineers in many disciplines to look more carefully at the important differential equations in their own fields. In many cases, they found chaotic behavior in these systems as well.

Now dynamical systems phenomena appear in virtually every area of science, from the oscillating BelousovZhabotinsky reaction in chemistry to the chaotic Chua circuit in electrical engineering, from complicated motions in celestial mechanics to the bifurcations arising in ecological systems.

As a consequence, the audience for a text on differential equations and dynamical systems is considerably larger and more diverse than it was in the 1970s. We have accordingly made several major structural changes to this book, including:

1. The treatment of linear algebra has been scaled back. We have dispensed with the generalities involved with abstract vector spaces and normed linear spaces. We no longer include a complete proof of the reduction of all nn matrices to canonical form. Rather, we deal primarily with matrices no larger than 4 4.

2. We have included a detailed discussion of the chaotic behavior in the Lorenz attractor, the Shilnikov system, and the double-scroll attractor.

3. Many new applications are included; previous applications have been updated.

4. There are now several chapters dealing with discrete dynamical systems.

5. We deal primarily with systems that are C, thereby simplifying many of the hypotheses of theorems.

This book consists of three main parts. The first deals with linear systems of differential equations together with some first-order nonlinear equations. The second is the main part of the text: here we concentrate on nonlinear systems, primarily two-dimensional, as well as applications of these systems in a wide variety of fields. Part three deals with higher dimensional systems. Here we emphasize the types of chaotic behavior that do not occur in planar systems, as well as the principal means of studying such behaviorthe reduction to a discrete dynamical system.

Writing a book for a diverse audience whose backgrounds vary greatly poses a significant challenge. We view this one as a text for a second course in differential equations that is aimed not only at mathematicians, but also at scientists and engineers who are seeking to develop sufficient mathematical skills to analyze the types of differential equations that arise in their disciplines.