Preface xv original theorem number. The material on competitive populations and predator– prey systems is contained in one of the beginning sections of the chapters in which these topics are covered, rather than in the applications at the end of the chapters, because these topics serve to develop the main techniques presented. Also, some proofs are contained in the main sections when they are more computational and serve to make the concepts clearer. Longer and more technical proofs and further theoretical discussion are presented separately at the end of the chapter. A course that covers the material from the primary sections, without covering the sections at the end of the chapter on applications and more theoretical material, results in a course on the concepts of dynamical systems with some motivation from applications. The applications provide motivation and illustrate the usefulness of the con- cepts. None of the material from the sections on applications is necessary for treating the main sections of later chapters. Treating more of this material would result in a more applied emphasis. Separating the harder proofs allows the instructor to determine the level of theory of the course taught using this book as the text. A more theoretic course could consider most of the proofs at the end of the chapters. Computer Programs This book does not explicitly cover aspects of computer programming. How- ever, a few selected problems require computer simulations to produce phase por- traits of differential equations or to iterate functions. Sample Maple worksheets, which the students can modify to help with some of the more computational prob- lems, will be available on the webpage: http://www.math.northwestern.edu/∼clark/dyn-sys. (Other material on corrections and updates of the book will also be available at this website.) There are several books available that treat dynamical systems in the context of Maple or Mathematica: two such books are [Kul02] by M. Kulen- ovi´ c and [Lyn01] by S. Lynch. The book [Pol04] by J. Polking and D. Arnold discusses using Matlab to solve differential equations using packages available at http://math.rice.edu/∼dfield. The book [Nus98] by H. Nusse and J. Yorke comes with its own specialized dynamical systems package. Acknowledgments I would like to acknowledge some of the other books I have used to teach this material, since they have influenced my understanding of the material, especially with regard to effective ways to present material. I will not attempt to list more advanced books which have also affected my understanding. For the material on differential equations, I have used the following books: F. Brauer and J. Nohel [Bra69], M. Hirsch and S. Smale [Hir74], M. Braun [Bra73], I. Percival and D. Richards [Per82], D.W. Jordan and P. Smith [Jor87], J. Hale and H. Ko¸cak [Hal91], and S. Strogatz [Str94]. For the material on iteration of functions, I have used the following books: the two books by R. Devaney [Dev89] and [Dev92], D. Gulick [Gul92], and K. Alligood, T. Sauer, and J. Yorke [All97].

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2012 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.