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Introduction to Mathematical Modeling and Chaotic Dynamics

Ranjit Kumar Upadhyay and Satteluri R. K. Iyengar
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Dhruba Adhikari
, on

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.

Introduction to Mathematical Modeling
What Is Mathematical Modeling?
Classification of Mathematical Models
Limitations Associated with Mathematical Modeling
Modeling Approaches
Modeling/Cyclic Processes
A Modeling Diagram
Compartment Models
Mathematical Preliminaries
Dynamic System and Its Mathematical Model
Numerical Tools and Software Used

Modeling of Systems from Natural Science
Models with Single Population
Two-Dimensional (2D) Continuous Models (Modeling of Population Dynamics of Two Interacting Species)
2D Discrete Models

Introduction to Chaotic Dynamics
Chaos and Chaotic Dynamics
Primary Routes to Study Chaos
Types of Chaos, Transients, and Attractors
Methods of Investigation for Detecting Chaos
Poincaré Map and Poincaré Section
Lyapunov Exponents

Chaotic Dynamics in Model Systems from Natural Science
Chaos in Single Species Model Systems
Chaos in Two Species Model Systems
Chaos in Two Species Model Systems with Diffusion
Chaos in Multi-Species Model Systems

Modeling of Some Engineering Systems
Models in Mechanical Systems
Models in Electronic Circuits
Nonlinear Circuits

Solutions to Odd-Numbered Problems