Discrete Dynamical Systems and Difference Equations with Mathematica

This is both an introductory survey on the theory and techniques of discrete dynamical systems and difference equations and a users'' manual for the software package Dynamica, developed by the authors. The Dynamica package is based on the Mathematica software and is designed as a collection of tools for studying discrete dynamical systems and difference equations. Available for download from the Internet, it is simple to use and gives users a series of algebraic, numerical, and graphical techniques they can quickly learn to employ to study complex difference equations. The book includes exercises and research projects based on applications encountered in biology and economics.

Following the work of Yorke and Li in 1975, the theory of discrete dynamical systems and difference equations developed rapidly. The applications of difference equations also grew rapidly, especially with the introduction of graphical-interface software that can plot trajectories, calculate Lyapunov exponents, plot bifurcation diagrams, and find basins of attraction.

Modern computer algebra systems have opened the door to the use of symbolic calculation for studying difference equations. This book offers an introduction to discrete dynamical systems and difference equations and presents the Dynamica software. Developed by the authors and based on Mathematica, Dynamica provides an easy-to-use collection of algebraic, numerical, and graphical tools and techniques that allow users to quickly gain the ability to:

Find and classify the stability character of equilibrium and periodic points

Perform semicycle analysis of solutions

Calculate and visualize invariants

Calculate and visualize Lyapunov functions and numbers

Plot bifurcation diagrams

Visualize stable and unstable manifolds

Calculate Box Dimension

While it presents the essential theoretical concepts and results, the book''s emphasis is on using the software. The authors present two sets of Dynamica sessions: one that serves as a tutorial of the different techniques, the other features case studies of well-known difference equations. Dynamica and notebooks corresponding to particular chapters are available for download from the Internet.