Synopses & Reviews
The authors stress a more balanced approach, one that includes analytic, numeric, and graphical techniques. The book emphasizes modeling and qualitative theory throughout the course. It employs technology significantly and consistently, presents linear and nonlinear systems in parallel, and includes an introduction to discrete dynamical systems. This text grew out of the Boston University Differential Equations Project, funded in part by the National Science Foundation.
About the Author
Paul Blanchard is Associate Professor of Mathematics at Boston University. Paul grew up in Sutton, Massachusetts, spent his undergraduate years at Brown University, and received his Ph.D. from Yale University. He has taught college mathematics for twenty-five years, mostly at Boston University. In 2001, he won the Northeast Section of the Mathematical Association of America's Award for Distinguished Teaching in Mathematics. He has coauthored or contributed chapters to four different textbooks. His main area of mathematical research is complex analytic dynamical systems and the related point setsJulia sets and the Mandelbrot set. Most recently his efforts have focused on reforming the traditional differential equations course, and he is currently heading the Boston University Differential Equations Project and leading workshops in this innovative approach to teaching differential equations. When he becomes exhausted fixing the errors made by his two coauthors, he usually closes up his CD store and heads to the golf course with his caddy, Glen Hall.Robert L. Devaney is Professor of Mathematics at Boston University. Robert was raised in Methuen, Massachusetts. He received his undergraduate degree from Holy Cross College and his Ph.D. from the University of California, Berkeley. He has taught at Boston University since 1980. His main area of research is complex dynamical systems, and he has lectured extensively throughout the world on this topic. In 1996 he received the National Excellence in Teaching Award from the Mathematical Association of America. When he gets sick of arguing with his coauthors over which topics to include in the differential equations course, he either turns up the volume of his opera CDs, or heads for waters off New England for a long distance sail.Glen R. Hall is Associate Professor of Mathematics at Boston University. Glen spent most of his youth in Denver, Colorado. His undergraduate degree comes from Carleton College and his Ph.D. comes from the University of Minnesota. His research interests are mainly in low-dimensional dynamics and celestial mechanics. He has published numerous articles on the dynamics of circle and annulus maps. For his research he has been awarded both NSF Postdoctoral and Sloan Foundation Fellowships. He has no plans to open a CD store since he is busy raising his two young sons. He is an untalented, but earnest, trumpet player and golfer. He once bicycled 148 miles in a single day.
Table of Contents
1. First-Order Differential Equations. Modeling via Differential Equations. Analytic Technique: Separation of Variables. Qualitative Technique: Slope Fields. Numerical Technique: Eulers Method. Existence and Uniqueness of Solutions. Equilibria and the Phase Line. Bifurcations. Linear Differential Equations. Changing Variables. Labs for Chapter 1. 2. First-Order Systems. Modeling Via Systems. The Geometry of Systems. Analytic Methods for Special Systems. Eulers Method for Systems. Qualitative Analysis. The Lorenz Equations. Labs for Chapter 2. 3. Linear Systems. Properties of Linear Systems and the Linearity Principle. Straight-Line Solutions. Phase Planes for Linear Systems with Real Eigenvalues. Complex Eigenvalues. The Special Cases: Repeated and Zero Eigenvalues. Second-Order Linear Equations. The Trace-Determinant Plane. Linear Systems in Three Dimensions. Labs for Chapter 3. 4. Forcing and Resonance. Forced Harmonic Oscillators. Sinusoidal Forcing and Resonance. Undamped Forcing and Resonance. Quantitative Analysis of the Forced Harmonic Analysis. The Tacoma Narrows Bridge. Labs for Chapter 4. 5. Nonlinear Systems. Equilibrium Point Analysis. Qualitative Analysis. Hamiltonian Systems. Dissipative Systems. Nonlinear Systems in Three Dimensions. Labs for Chapter 5. 6. Laplace Transforms. Laplace Transforms. Discontinuous Functions. Second-Order Equations. Delta Functions and Impulse Forcing. Convolutions. The Qualitative Theory of Laplace Transforms. Labs for Chapter 6. 7. Numerical Methods. Numerical Error in Eulers Method. Improving Eulers Method. The Runge-Kutta Method. The Effects of Finite Arithmetic. Labs for Chapter 7. 8. Discrete Dynamical Systems. The Discrete Logistic Equation. Fixed and Periodic Points. Bifurcations. Chaos. Chaos in the Lorenz System. Labs for Chapter 8. Hints and Answers. Index.