Kulenovic / Merino Discrete Dynamical Systems and Difference Equations with Mathematica

Erscheinungsjahr 2010
ISBN: 978-1-4200-3535-3
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 360 Seiten

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.

Zielgruppe

Pure and applied mathematicians, physicists, and mechanical engineers

Autoren/Hrsg.

Kulenovic, Mustafa R.S.

Merino, Orlando

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

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