Introduction to Dynamical Systems Continuous and Discrete

This book is intended for an advanced undergraduate course in dynamical systems or nonlinear ordinary differential equations. There are portions that could be beneficially used for introductory master level courses. The goal is a treatment that gives examples and methods of calculation, at the same time introducing the mathematical concepts involved. Depending on the selection of material covered, an instructor could teach a course from this book that is either strictly an introduction into the concepts, that covers both the concepts on applications, or that is a more theoretically mathematical introduction to dynamical systems. Further elaboration of the variety of uses is presented in the subsequent discussion of the organization of the book. The assumption is that the student has taken courses on calculus covering both single variable and multivariables, a course on linear algebra, and an introductory course on differential equations. From the multivariable calculus, the material on partial derivatives is used extensively, and in a few places multiple integrals and surface integrals are used. (See Appendix A.) Eigenvalues and eigenvectors are the main concepts used from linear algebra, but further topics are listed in Appendix C. The material from the standard introductory course on differential equations is used only in part one; 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 a 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. Students have taken the course covering part one on differential equations without this introductory course on differential equations; they have been able to understand the new material where 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 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. 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 semester course on differential equations could also cover selected topics on iteration of functions from chapters nine through eleven. In the course on discrete dynamical systems from part two, we cover most of the material on iteration of one dimensional functions (chapters nine through eleven) in one quarter. The material on iteration of higher dimensional functions (chapters twelve through thirteen) 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 nine through eleven. Finally, chapter fourteen on fractals can be treated after a number of the earlier chapters. 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 iteratiRobinson, R. Clark is the author of 'Introduction to Dynamical Systems Continuous and Discrete', published 2004 under ISBN 9780131431409 and ISBN 0131431404.