Differential Equations: A Dynamical Systems Approach to Theory and Practice

This book offers an attractive introduction to the modern theory of ordinary differential equations and dynamical systems at the graduate level. In many respects, the text creates a development of the subject in accordance with Poincaré’s vision of more than a century ago. Of course, the field has evolved extensively since his time but the emphasis on using a combination of qualitative analysis and numerical calculation of solutions remains very relevant. The authors succeed in coordinating both these elements to tell a pleasingly coherent and logical story.

 

That story begins with foundations: existence, uniqueness, and the robustness of solutions under parameter changes. Following that the reader is introduced to the basic tools. This begins - contrary to the usual order – with numerical methods for solving differential equations. It then moves on to the qualitative theory and its theoretical formalism: flows, the Poincaré map, conjugacy and equivalence of flows. Here we first see some of the strength of the qualitative theory with the Poincaré recurrence theorem.

 

Linear differential equations – first autonomous and then nonautonomous are treated next. After the basics, this introduces the readers to topics like the topological classification of hyperbolic flows and Floquet’s theorem for homogeneous linear equations with periodic coefficients. The discussion of Lyapunov stability continues to reinforce the benefits of qualitative analysis. This approach is one that was developed in Poincaré’s time and is very much consistent with his program.

 

After this, there are two chapters devoted to big theorems of the local theory of dynamical systems: the Hartman-Grobman theorem and Stable Manifold theorem. In many respects, these represent the culmination of the basic theory of dynamical systems. Something of the global theory is presented at the end of the book with Poincaré -Bendixson and Poincaré - Hopf theorem. 

 

Unlike some comparable texts, this one has very little to say about chaotic dynamics. Instead, the authors limit their discussion to the Lorenz attractor in the context of a numerical experiment. Another unusual feature of the book is that every chapter proposes a “numerical experiment” for students to analyze with computational methods. It is presented in a way that encourages exploration (e.g., “Find interesting solutions …”).

 

Not many texts are available with an approach and content like this one. The differential equation books by Coddington and Levinson, Arnold, Hale, and Hartman incorporate many of the standard results, but none examine numerical questions in the same context. Furthermore, none of these give the same attention to the historical development of the subject.

 

The book is aimed at beginning graduate students and was originally developed for a Masters degree program. It has more material than could be covered in a single course. The authors expect readers to have a basic background at the undergraduate level in general topology, linear algebra and analysis. The last two chapters on the global theory require a bit of the theory of differential manifolds. Some background for this is provided in an appendix.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.