Discrete Dynamical Systems and Difference Equations with Mathematica

Published:
February 27, 2002 by Chapman and Hall/CRC - 360 Pages
Author(s):
Mustafa R.S. Kulenovic, University of Rhode Island, Kingston, USA; Orlando Merino, University of Rhode Island, Kingston, Rhode Island, USA

Purchasing Options

Hardback
$129.95
Add to cart
ISBN 9781584882879
Cat# C2875
 

Features

  • Presents Dynamica-new software that provides tools for studying discrete dynamical systems and difference equations
  • Quickly allows users to study complex difference equations and create state-of-the-art technical analyses and reports
  • Provides insight into difficult problems and concepts, including stability and rational invariants
  • Requires no previous programming experience
  • Includes exercises and research projects based on applications encountered in biology and economics

    • Summary

    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.

    • Table of Contents

    DYNAMICS OF ONE-DIMENSIONAL DYNAMICAL SYSTEMS
    Introduction
    Linear Difference Equations with Constant Coefficients
    Linear Difference Equations with Variable Coefficients
    Stability
    Stability in the Non-Hyperbolic Case
    Bifurcations
    Dynamica Session
    Symbolic Dynamics for One-Dimensional Maps
    Dissipative Maps and Global Attractivity
    Parametrisation and Poincaré Functional Equation
    Exercises
    DYNAMICS OF TWO-DIMENSIONAL DYNAMICAL SYSTEMS
    Introduction
    Linear Theory
    Equilibrium Solutions
    The Riccati Equation
    Linearized Stability Analysis
    Dynamica Session
    Period Doubling Bifurcation
    Lyapunov Numbers
    Box Dimension
    Semicycle Analysis
    Stable and Unstable Manifold
    Dynamica Session on Henon's Equation
    Invariants
    Lyapunov Functions, Stability, and Invariants
    Dynamica Session on Lyness' Map
    Dissipative Maps and Systems
    Dynamica Session on Rational Difference Equations
    Area-preserving Maps and Systems
    Biology Applications Projects
    Applications in Economics
    Exercises
    SYSTEMS OF DIFFERENCE EQUATIONS, STABILITY, AND SEMICYCLES
    Introduction
    Linear Theory
    Stability of Linear Systems
    The Routh-Hurwitz and Schur-Cohn Criterion
    Nonlinear Systems and Stability
    Limit Sets and Invariant Manifolds
    Dissipative Maps
    Stability of Difference Equations
    Semicycle Analysis
    Dynamica Session on Semicycles
    Exercises
    INVARIANTS AND RELATED LYAPUNOV FUNCTIONS
    Introduction
    Invariants for Linear Equations and Systems
    Invariants and Corresponding Lyapunov Functions for Nonlinear Systems
    Invariants of Special Class of Difference Equations
    Applications
    Dynamica Session on Invariants
    Dynamica Session on Lyapunov Functions
    Invariance under Lie Group Transformations
    Exercises
    DYNAMICS OF THREE-DIMENSIONAL DYNAMICAL SYSTEMS
    Introduction
    Dynamica Session on Third Order Difference Equations
    Dissipative Difference Equation of Third Order
    Dynamica Session on Local Asymptotic Stability of Period-Two Solution
    Dynamica Session on Todd's Difference Equation
    Biology Applications Projects
    Exercises
    FRACTALS GENERATED BY ITERATED FUNCTIONS SYSTEMS
    Introduction
    Basic Definitions and Results
    Iterated Function System
    Basic Results on Iterated Functions Systems
    Calculation of Box Dimension for IFS
    Dynamica Session
    Exercises
    BIBLIOGRAPHY
    INDEX

    Editorial Reviews

    "…Several tutorials and example sessions with Dynamica are effectively woven into the main text, and the package is straightforward to use … a good resource and supplementary text…"
    -CHOICE

    Related Titles