This introduction to the analysis of discrete dynamical systems is written at the level of a reasonably strong advanced undergraduate. It demands a little bit more sophistication and mathematical background than some comparable texts. The authors suggest that the earlier chapters might be accessible to students with only a bit of calculus, but that would be a stretch.
The focus throughout is largely on the development of concepts and analytical techniques. A number of interesting models are described, but analysis dominates by sheer page count. That seems entirely appropriate considering the skills and background that the authors are trying to convey. But the book’s title suggests a greater emphasis on modeling. It is nonetheless a strong text with a fresh approach, a nice collection of topics and some great examples.
The authors begin with the general idea of recursive phenomena and difference equations and then move on quickly to a fairly complete treatment of linear difference equations. Included here are introductions to both the z-transform and the discrete Fourier transform. Following this, the authors introduce nonlinear first order one-dimensional discrete dynamical systems where they emphasize asymptotic behavior and stability analysis. They make good use of graphical analysis as a companion to a more rigorous treatment, and provide analysis algorithms presented as flowcharts.
The next step takes students into more challenging territory with the qualitative analysis of discrete dynamical systems associated with logistic growth. The authors present Sharkovskii’s theorem (without proof), then introduce topological conjugacy as an analysis tool for studying complicated dynamical systems by identifying simpler but equivalent systems. This is accomplished in the course of introducing the basic ideas of bifurcation and chaos. As if this weren’t enough, this chapter also includes an introduction to dynamical systems in the complex plane.
The following three chapters discuss multidimensional first order discrete systems, Markov chains and then positive matrices and graphs. This final part of the book has some very nice examples. For instance, one chapter is largely devoted to an analysis of Google’s PageRank algorithm. Both the PageRank example and a clever application on demographics from the chapter on Markov chains illustrate novel uses of the Perron-Frobenius theorem on positive matrices. Other notable examples explore the Hardy-Weinberg law of genetics, tennis matches, and “truels” (gunfights involving three people — perhaps the authors have been watching a lot of cowboy movies).
This is a textbook with energy and enthusiasm, full of interesting mathematics and examples. The exercises in the book match the spirit of the rest of the book – challenging and somewhat unusual. Solutions for all of them are provided in the last chapter.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.