Discrete Dynamical Systems and Difference Equations with Mathematica

Discrete Dynamical Systems and Difference Equations with Mathematica

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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.show more

Product details

  • Hardback | 360 pages
  • 154.4 x 246.4 x 24.6mm | 675.86g
  • Taylor & Francis Inc
  • CRC Press Inc
  • Bosa Roca, United States
  • English
  • 50 black & white illustrations
  • 1584882875
  • 9781584882879

Review quote

"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" -CHOICEshow more

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 Poincare 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 INDEXshow more

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