xiv Preface course on differential equations. From the multivariable calculus, the material on partial derivatives is used extensively, and in a few places multiple integrals and sur- face integrals are used. (See Appendix A.1.) Eigenvalues and eigenvectors are the main concepts used from linear algebra, but further topics are listed in Appendix A.3. The material from the standard introductory course on differential equations is used only in Part 1 we assume that students can solve first-order equations by separation of variables, and that they know the form of solutions from second-order scalar equations. Students who have taken an introductory course on differential equations are usually familiar with linear systems with constant coeﬃcients (at least the real-eigenvalue case), but this material is repeated in Chapter 2, where we also introduce the reader to the phase portrait. At Northwestern, some students have taken the course covering part one on differential equations without this in- troductory course on differential equations they have been able to understand the new material when they have been willing to do the extra work in a few areas that is required to fill in the missing background. Finally, we have not assumed that the student has had a course on real analysis or advanced calculus. However, it is convenient to use some of the terminology from such a course, so we include an appendix with terminology on continuity and topology. (See Appendix A.) Organization This book presents an introduction to the concepts of dynamical systems. It is divided into two parts, which can be treated in either order: The first part treats various aspects of systems of nonlinear ordinary differential equations, and the second part treats those aspects dealing with iteration of a function. Each separate part can be used for a one-quarter course, a one-semester course, a two- quarter course, or possibly even a year course. At Northwestern University, we have courses that spend one quarter on the first part and two quarters on the second part. In a one-quarter course on differential equations, it is diﬃcult to cover the material on chaotic attractors, even skipping many of the applications and proofs at the end of the chapters. A one-semester course on differential equations could also cover selected topics on iteration of functions from Chapters 9 –11. In the course on discrete dynamical systems using Part 2, we cover most of the material on iteration of one-dimensional functions (Chapters 9 –11) in one quarter. The material on iteration of functions in higher dimensions (Chapters 12 –13) certainly depends on the one-dimensional material, but a one-semester course could mix in some of the higher dimensional examples with the treatment of Chapters 9 – 11. Finally, Chapter 14 on fractals could be treated after Chapter 12. Fractal dimensions could be integrated into the material on chaotic attractors at the end of a course on differential equations. The material on fractal dimensions or iterative function systems could be treated with a course on iteration of one-dimensional functions. The main concepts are presented in the first sections of each chapter. These sections are followed by a section that presents some applications and then by a section that contains proofs of the more diﬃcult results and more theoretical material. The division of material between these types of sections is somewhat arbitrary. The theorems proved at the end of the chapter are restated with their

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