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Discovering Dynamical Systems Through Experiment and Inquiry

Thomas LoFaro and Jeff Ford
Chapman and Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Pierre-Olivier Parise
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Computer simulations give us a powerful new tool for approaching the teaching and learning the fundamentals of the theory of dynamical systems. The authors of the book under review take full advantage of that tool in this innovative book.
The book is quite different from classical books on dynamical systems like Devaney’s A First Course in Chaotic Dynamical Systems: Theory and Experiment. Indeed, the authors have written the book with a different perspective in mind: the inquiry-based learning-teaching approach. This teaching approach gives all the power to the students in their learning process by asking questions first, using software to experiment and verify conjectures, and then creating new theorems as output. The authors concretely implemented the learning-teaching approach by creating an innovative method for their book entitled ECAP, or Explore, Conjecture, Apply, and Prove.
To illustrate ECAP, take section 3.1.1 from chapter 3 on fixed points of linear systems. To execute the first step, Explore, the authors ask the reader to use software on a website that accompanies the book to analyze typical aspects of linear systems. For example, the reader is invited to explore the dynamical system generated by the function \( f(x) = mx \) by varying the parameter m. The second step, Conjecture, invites the reader to draw his or her own conclusions based on the experimentation just performed. Generally, the authors present a statement that the reader has to complete. The third step, Apply, then invites the reader to use what s/he discovered in another applied or pure math problem. For example, the reader is asked to model and to analyze the population of walleye in Teelo Lake as a linear system.  The final step, Prove, is to construct a mathematical argument supporting the conjectures just made. The authors usually provide hints to help the reader build the proof. Most of the material presented in this book uses the ECAP method, although the steps are not always in the same order that I presented here.
Even though this book presents a more avant-garde approach, definitions of concepts and fundamental theorems are no less thorough. The material covered is equivalent to a standard course in dynamical systems. This being said, I would highly recommend this book for instructors teaching dynamical systems. It’s an excellent companion to explore this field of mathematics with your students. However, I would also recommend having a copy of Devaney’s books An Introduction to Chaotic Dynamical Systems and A First Course in Chaotic Dynamical systems: Theory and Experiment at hand. It will help complete the proofs left for the reader. In my opinion, the book under review combined with one of Devaney’s books is a perfect match.

Pierre-Olivier Parisé got his PhD from Laval University under the supervision of Thomas J. Ransford.  He was recently hired as a temporary assistant professor at the University of Hawaii at Manoa (UHM) where he works alongside Malik Younsi. His main research interests are in complex analysis, operator theory and its relationship to complex analysis, summability theory and its application in polynomial approximation theory, multicomplex analysis, and dynamical systems.  He recently got interested in geometric function theory, especially conformal welding.