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% DYNAMICAL SYSTEMS WITH APPLICATIONS USING MATLAB %
% COPYRIGHT BIRKHAUSER 2004 STEPHEN LYNCH %
% PUBLISHED June 2004 %
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CONTENTS
Preface
0. A Tutorial Introduction to MATLAB and the Symbolic Math Toolbox
0.1 Tutorial One: The Basics and the Symbolic Math Toolbox (1 Hour)
0.2 Tutorial Two: Plots and Differential Equations (1 Hour)
0.3 MATLAB Program Files, or M-Files
0.4 Hints for Programming
0.5 MATLAB Exercises
1. Linear Discrete Dynamical Systems
1.1 Recurrence Relations
1.2 The Leslie Model
1.3 Harvesting and Culling Policies
1.4 MATLAB Commands
1.5 Exercises
2. Nonlinear Discrete Dynamical Systems
2.1 The Tent Map and Graphical Iterations
2.2 Fixed Points and Periodic Orbits
2.3 The Logistic Map, Bifurcation Diagram, and Feigenbaum Number
2.4 Gaussian and Henon Maps
2.5 Applications
2.6 MATLAB Commands
2.7 Exercises
3. Complex Iterative Maps
3.1 Julia Sets and the Mandelbrot Set
3.2 Boundaries of Periodic Orbits
3.3 MATLAB Commands
3.4 Exercises
4. Electromagnetic Waves and Optical Resonators
4.1 Maxwell's Equations and Electromagnetic Waves
4.2 Historical Background of Optical Resonators
4.3 The Nonlinear Simple Fibre Ring Resonator
4.4 Chaotic Attractors and Bistability
4.5 Linear Stability Analysis
4.6 Instabilities and Bistability
4.7 MATLAB Commands
4.8 Exercises
5. Fractals and Multifractals
5.1 Construction of Simple Examples
5.2 Calculating Fractal Dimensions
5.3 A Multifractal Formalism
5.4 Multifractals in the Real World and Some Simple Examples
5.5 MATLAB Commands
5.6 Exercises
6. Controlling Chaos
6.1 Historical Background
6.2 Controlling Chaos in the Logistic Map
6.3 Controlling Chaos in the Henon Map
6.4 MATLAB Commands
6.5 Exercises
7. Differential Equations
7.1 Simple Differential Equations and Applications
7.2 Applications to Chemical Kinetics
7.3 Applications to Electric Circuits
7.4 Existence and Uniqueness Theorem
7.5 MATLAB Commands
7.6 Exercises
8. Planar Systems
8.1 Canonical Forms
8.2 Eigenvectors Defining Stable and Unstable Manifolds
8.3 Phase Portraits of Linear Systems in the Plane
8.4 Linearization and Hartman's Theorem
8.5 Constructing Phase Plane Diagrams
8.6 MATLAB Commands
8.7 Exercises
9. Interacting Species
9.1 Competing Species
9.2 Predator-Prey Models
9.3 Other Characteristics Affecting Interacting Species
9.4 MATLAB Commands
9.5 Exercises
10. Limit Cycles
10.1 Historical Background
10.2 Existence and Uniqueness of Limit Cycles in the Plane
10.3 Non-Existence of Limit Cycles in the Plane
10.4 Exercises
11. Hamiltonian Systems, Lyapunov Functions, and Stability
11.1 Hamiltonian Systems in the Plane
11.2 Lyapunov Functions and Stability
11.3 MATLAB Commands
11.4 Exercises
12. Bifurcation Theory
12.1 Bifurcations of Nonlinear Systems in the Plane
12.2 Multistability and Bistability
12.3 MATLAB Commands
12.4 Exercises
13. Three-Dimensional Autonomous Systems and Chaos
13.1 Linear Systems and Canonical Forms
13.2 Nonlinear Systems and Stability
13.3 The Rossler System and Chaos
13.4 The Lorenz Equations, Chua's Circuit, and the Belousov-Zhabotinski Reaction
13.5 MATLAB Commands
13.6 Exercises
14. Poincare Maps and Nonautonomous Systems in the Plane
14.1 Poincare Maps
14.2 Hamiltonian Systems with Two Degrees of Freedom
14.3 Nonautonomous Systems in the Plane
14.4 MATLAB Commands
14.5 Exercises
15. Local and Global Bifurcations
15.1 Small-Amplitude Limit Cycle Bifurcations
15.2 Melnikov Integrals and Bifurcating Limit Cycles from a Center
15.3 Homoclinic Bifurcations
15.4 MATLAB Commands
15.5 Exercises
16. The Second Part of David Hilbert's 16'th Problem
16.1 Statement of Problem and Main Results
16.2 Poincare Compactification
16.3 Global Results for Lienard Systems
16.4 Local Results for Lienard Systems
16.5 Exercises
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 MATLAB Commands
17.6 Exercises
18. SIMULINK
18.1 Introduction
18.2 Electric Circuits
18.3 Mechanical Systems
18.4 Nonlinear Optics
18.5 The Lorenz System and Chaos Synchronization
18.6 Exercises
19. Solutions to Exercises
References
Textbooks
Research Papers
MATLAB Program File Index
Simulink Model File Index
Index
