Friday, October 17, 2008

Nonlinear Dynamics

Nonlinear dynamics is discussed in several chapters in the 4th Edition of Intermediate Physics for Medicine and Biology, and particularly in chapter 10. Where can you go to get a more information about this topic? Several fine books are cited in our references.

One book that Russ Hobbie cited in earlier editions of our text is
Nonlinear Dynamics and Chaos, by Steven Strogatz. Even though we still cite the book in the 4th edition, I was not really familiar with it until last year, when my daughter Stephanie used it in a class on nonlinear dynamics at the University of Michigan. She loved the book, and recommended it to me. In the preface, Strogatz writes
"This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. It is based on a one-semester course I've taught for the past several years at MIT. My goal is to explain the mathematics as clearly as possible, and to show how it can be used to understand some of the wonders of the nonlinear world....A unique feature of the book is its emphasis on applications. These include mechanical vibrations, lasers, biological rhythms, superconducting circuits, insect outbreaks, chemical oscillators, genetic control systems, chaotic waterwheels, and even a technique for using chaos to send secret messages. In each case, the scientific background is explained at an elementary level and closely integrated with the mathematical theory."
For anyone who reads Intermediate Physics for Medicine and Biology and wants to learn more about nonlinear dynamics, I recommend this delightful text. Strogatz, the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University, was a collaborator and student of one of my heroes, the late Art Winfree. Art's book When Time Breaks Down had a tremendous influence on my early career. I have not yet read Strogatz's newest book Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life, but reading it is on my list of things to do.

A book that came out when I was working at the National Institutes of Health is Leon Glass and Michael Mackey's
From Clocks to Chaos. While this book may not present the math as elegantly as Strogatz's book, it does focus specifically on biological and medical problems. In particular, it introduces the idea of a "dynamical diseases": diseases characterized by abnormal temporal organization.

When explained well, nonlinear dynamics is a very visual subject. An older book that makes much use of pictures is Dynamics: The Geometry of Behavior by Abraham and Shaw. In fact it could be called a "picture book," but don't let that description fool you. The math is presented at a high level, but in a primarily visual way. It was a great help to me when I was first learning the subject.

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