Instabilities, Chaos and Turbulence
is an exploration of instability and turbulence in fluid dynamics. This text is not a litany of techniques but rather introduces non-linear dynamics and chaos in complex systems by telling the story of a fluid's transition to turbulence in various settings.
After an introductory chapter and a review of two-dimensional systems of linear differential equations, Manneville introduces some classical examples of two-dimensional non-linear systems (the Duffing oscillator, the van der Pol oscillator) and discusses several techniques for solving these systems. The remainder of the text is devoted to modeling instability in fluids.
A fundamental example is: apply heat to the bottom of pot containing water and model how the water transitions from conduction to convection to turbulence. Manneville describes how the parameters of the system (the depth of the water, the viscosity of the water, the dimensions of the pot) establish thresholds (bifurcation points) which determine these transition points. He then turns his attention to the appearance of instability and turbulence in open flows such as occurs when an obstacle is placed in a flowing river. The models presented are meant to be practical representations of real-world behavior. As such, Manneville makes frequent references to experimental results, comparing the theoretical results to the empirical data and discussing hypotheses about why discrepancies exist.
The final chapter of the text is a generic discussion of why it is appropriate to use non-linear dynamics to model the Earth's climate; it does not introduce any additional mathematics or present any specific models of the Earth's climate (except for a very simple one as an exercise). The first appendix reviews the linear algebra used in the text. (It contains much more than is typically taught in a first course in linear algebra.) In a second appendix, the author provides a survey of numerical techniques for approximating solutions to initial value problems. Each of the eight chapters ends with some quite interesting exercises. They are often very involved, sometimes running for 2 or 3 pages each.
I like Manneville's decision to tell a story about instabilities in fluid dynamics. Unfortunately, it takes a while for this story to become apparent---the book would be much stronger if the path of the story were more clearly laid out in the introductory chapter. The drawback of his approach is that it is more difficult to lift specific techniques from such a text. However, I much prefer a narrative approach to a more traditional technique-example style. The pace of the text is probably too fast for most undergraduates. It requires the reader to be adept at the techniques of linear algebra, and ordinary and partial differential equations, and to have some knowledge of basic thermodynamics and Newtonian mechanics. For someone interested in seeing non-linear dynamics in a specific context, Instabilities, Chaos and Turbulence is a nice book.
Stephen T. Ahearn teaches at the Department of Mathematics and Computer Science of Macalester College.