Dynamical Systems with Applications using Maple
2nd Edition
Copyright Birkhauser
2009
Stephen Lynch
Contents
Preface
Chapter 0 A Tutorial Introduction to
Maple
0.1 A Quick Tour of Maple
0.2 Tutorial One: The Basics
0.3 Tutorial Two: Plots and Differential Equations
0.4 Simple Maple Programs
0.5 Common Errors
0.6 Maple Exercises
Chapter 1 Differential Equations
1.1 Simple Differential Equations and Applications
1.2 Applications to Chemical Kinetics
1.3 Applications to Electric Circuits
1.4 Existence and Uniqueness Theorem
1.5 Maple Commands in Text Format
1.6 Exercises
Chapter 2 Planar Systems
2.1 Canonical Forms
2.2 Eigenvectors Defining Stable and Unstable Manifolds
2.3 Phase Portraits of Linear Systems in the Plane
2.4 Linearization and Hartman's Theorem
2.5 Constructing Phase Plane Diagrams
2.6 Maple Commands in Text Format
2.7 Exercises
Chapter 3 Interacting Species
3.1 Competing Species
3.2 Predator-Prey Models
3.3 Other Characteristics Affecting Interacting Species
3.4 Maple Commands in Text Format
3.5 Exercises
Chapter 4 Limit Cycles
4.1 Historical Background
4.2 Existence and Uniqueness of Limit Cycles in the Plane
4.3 Nonexistence of Limit Cycles in the Plane
4.4 Perturbation Methods
4.5 Maple Commands in Text Format
4.6 Exercises
Chapter 5 Hamiltonian Systems, Lyapunov Functions, and Stability
5.1 Hamiltonian Systems in the Plane
5.2 Lyapunov Functions and Stability
5.3 Maple Commands in Text Format
5.4 Exercises
Chapter 6 Bifurcation Theory
6.1 Bifurcations of Nonlinear Systems in the Plane
6.2 Normal Forms
6.3 Multistability and Bistability
6.4 Maple Commands in Text Format
6.5 Exercises
Chapter 7 Three-Dimensional Autonomous Systems and Chaos
7.1 Linear Systems and Canonical Forms
7.2 Nonlinear Systems and Stability
7.3 The Rössler System and
Chaos
7.4 The Lorenz Equations, Chua's Circuit, and the Belousov-Zhabotinski
Reaction
7.5 Maple Commands in Text Format
7.6 Exercises
Chapter 8 Poincaré Maps and Nonautonomous Systems in the Plane
8.1 Poincaré Maps
8.2 Hamiltonian Systems with Two Degrees of Freedom
8.3 Nonautonomous Systems in the Plane
8.4 Maple Commands in Text Format
8.5 Exercises
Chapter 9 Local and Global Bifurcations
9.1 Small-Amplitude Limit Cycle Bifurcations
9.2 Gröbner Bases
9.3 Melnikov Integrals and Bifurcating Limit Cycles
from a Centre
9.4 Homoclinic Bifurcations
9.5 Maple Commands in Text Format
9.6 Exercises
Chapter 10 The Second Part of Hilbert's
16th Problem
10.1 Statement of Problem and Main Results
10.2 Poincaré Compactification
10.3 Global Results for Liénard Systems
10.4 Local Results for Liénard Systems
10.5 Maple Commands in Text Format
10.6 Exercises
Chapter 11 Linear Discrete Dynamical
Systems
11.1 Recurrence Relations
11.2 The Leslie Model
11.3 Harvesting and Culling Policies
11.4 Maple Commands in Text Format
11.5 Exercises
Chapter 12 Nonlinear Discrete Dynamical Systems
12.1 The Tent Map and Graphical Iterations
12.2 Fixed Points and Periodic Orbits
12.3 The Logistic Map, Bifurcation Diagram, and Feigenbaum
Number
12.4 The Gaussian and Hénon Maps
12.5 Applications
12.6 Maple Commands in Text Format
12.7 Exercises
Chapter 13 Complex Iterative Maps
13.1 Julia Sets and the Mandelbrot Set
13.2 Boundaries of Periodic Orbits
13.3 Maple Commands in Text Format
13.4 Exercises
Chapter 14 Electromagnetic Waves and Optical Resonators
14.1 Maxwell's Equations and Electromagnetic Waves
14.2 Historical Background
14.3 The Nonlinear Simple Fibre Ring Resonator
14.4 Chaotic Attractors and Bistability
14.5 Linear Stability Analysis
14.6 Instabilities and Bistability
14.7 Maple Commands in Text Format
14.8 Exercises
Chapter 15 Fractals and Multifractals
15.1 Construction of Simple Examples
15.2 Calculating Fractal Dimensions
15.3 A Multifractal Formalism
15.4 Multifractals in the Real World and Some Simple
Examples
15.5 Maple Commands in Text Format
15.6 Exercises
Chapter 16 Controlling Chaos and Synchronization
16.1 Historical Background
16.2 Controlling Chaos in the Logistic Map
16.3 Controlling Chaos in the Hénon Map
16.4 Chaos Synchronization
16.5 Maple Commands in Text Format
16.6 Exercises
Chapter 17 Neural Networks
17.1 Introduction
17.2 The Delta Learning Rule and Backpropagation
17.3 The Hopfield Network and Lyapunov Stability
17.4 Neurodynamics
17.5 Maple Commands in Text Format
17.6 Exercises
Chapter 18 Simulation
18.1 Simulink
18.2 The Connectivity Toolbox
18.3 MapleSim
18.4 Exercises
19. Examination-Type Questions
20. Solutions to Exercises
Bibliography
Maple Program Index
Index