Introduction to Mathematical Modeling and Chaotic Dynamics

This book is aimed at senior undergraduate and/or beginning graduate students who may be interested in mathematical modeling and applications. The presentation is so clear that anyone with even a basic mathematical background can study it and get a clear picture.

The book begins with an introduction to mathematical modeling, with lots of worked out motivational examples and exercises. Various approaches to modeling, such as empirical, theoretical, stochastic, deterministic, statistical, simulation, discrete, and continuous are discussed. Methods of testing stability are also discussed.

The book then goes to deal with modeling systems that appear in natural science. Many common mathematical models in natural science are included. One- and two-dimensional continuous and discrete time models come first, and then an introduction to chaotic dynamics is given. Some methods of investigation and detection of chaos, such as selection for parameter values, calculation of the basin boundary structures, 2D parameter scans, bifurcation diagrams and bifurcation types, time-series analysis, the Poincaré map and Poincaré section, are discussed. Single and multiple species systems in biology are covered. The Rosenzweig-MacArthur model with diffusion and its variants, the DeAngelis model with diffusion, and the Hastings and Powell model for chaotic dynamics are presented. A chapter on engineering systems, such as mechanical systems and electric circuits, is very interesting for examples of chaotic behaviors.

Unlike many other similar textbooks, a rich reference section is given at the end of each chapter. The cautious selection of worked out examples and exercises throughout the book is superb.

For anyone with previous experience of having run into books in mathematical modeling and chaotic dynamics that rapidly move into advanced mathematical content, the book offers a pleasant recourse at an introductory level and therefore can be very inspirational.

Dhruba Adhikari is an assistant professor of mathematics at Southern Polytechnic State University, Marietta, Georgia.