Working with Dynamical Systems: A Toolbox for Scientists and Engineers

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
1 Whence Dynamical Systems
1.1 Classical Mechanics
1.1.1 Conservative Equations of Motion
1.1.2 Systems with One Degree of Freedom
1.1.3 Symmetries and Conservation Laws
1.1.4 Interacting Particles
1.1.5 Dissipative Motion
1.2 Chemical Kinetics
1.2.1 Mass Action
1.2.2 Adsorption and Catalysis
1.2.3 Autocatalysis and Self-Inhibition
1.2.4 Thermal Effects
1.3 Biological Models
1.3.1 Population Dynamics
1.3.2 Epidemiological Models
1.3.3 Neural and Genetic Networks
1.4 Electric Currents
1.4.1 Electric Circuits
1.4.2 Electrochemical Reactions
1.4.3 Membrane Transport
1.5 Spatially Extended Systems
1.5.1 From Time to Coordinate Dependence
1.5.2 Fourier Decomposition
1.6 Continuous vs. Discrete
1.6.1 Iterative Systems
1.6.2 From Continuous to Discrete
1.6.3 Poincaré Maps
2 Local Bifurcations
2.1 Bifurcation of Stationary States
2.1.1 Branches of Stationary States
2.1.2 Bifurcation Expansion
2.1.3 Fold and Transcritical Bifurcations
2.1.4 Cusp Singularity
2.1.5 Higher Singularities
2.2 Stability and Slow Dynamics
2.2.1 Linear Stability Analysis
2.2.2 Stable and Unstable Manifolds
2.2.3 Exchange of Stability
2.2.4 Amplitude Equations
2.3 Bifurcations of Periodic Orbits
2.3.1 Hopf Bifurcation
2.3.2 Derivation of the Amplitude Equation
2.3.3 Instabilities of Periodic Orbits
2.4 Example: Exothermic Reaction
2.4.1 Bifurcation of Stationary States
2.4.2 Hopf Bifurcation
2.4.3 Branches of Periodic Orbits
2.5 Example: Population Dynamics
2.5.1 Prey–Predator Models
2.5.2 Stability and Bifurcations
2.5.3 Periodic Orbits
3 Global Bifurcations
3.1 Topology of Bifurcations
3.1.1 More Ways to Create and Break Periodic Orbits
3.1.2 Bifurcations in a System with Three Stationary States
3.2 Global Bifurcations in the Exothermic Reaction
3.2.1 Basin Boundaries
3.2.2 Saddle-Loop Bifurcations
3.2.3 Sniper Bifurcation
3.3 Bifurcation at Double-Zero Eigenvalue
3.3.1 Locating a Double Zero
3.3.2 Quadratic Normal Form
3.3.3 Expansion in the Vicinity of Cusp Singularity
3.4 Almost Hamiltonian Dynamics
3.4.1 Weak Dissipation
3.4.2 Hopf and Saddle-Loop Bifurcations
3.4.3 Bifurcation Diagrams
3.4.4 Basin Boundaries
3.5 Systems with Separated Time Scales
3.5.1 Fast and Slow Variables
3.5.2 Van der Pol Oscillator
3.5.3 FitzHugh–Nagumo Equation
3.5.4 Canards
3.6 Venturing to Higher Dimensions
3.6.1 Dynamics Near Triple-Zero Eigenvalue
3.6.2 Double Hopf Bifurcation
3.6.3 Blue Sky Catastrophe
4 Chaotic, Forced, and Coupled Oscillators
4.1 Approaches to Hamiltonian Chaos
4.1.1 Hiding in Plain Sight
4.1.2 Resonances and Small Divisors
4.1.3 Example: Hénon-Heiles Model
4.1.4 Quantitative Measures of Chaos
4.2 Approaches to Dissipative Chaos
4.2.1 Distilling Turbulence into Simple Models
4.2.2 Chaotic Attractors
4.2.3 Period-Doubling Cascade
4.2.4 Strange, Chaotic, or Both?
4.3 Chaos Near a Homoclinic
4.3.1 Shilnikov's Snake
4.3.2 Complexity in Chaotic Models
4.3.3 Lorenz Model
4.4 Weakly Forced Oscillators
4.4.1 Phase Perturbations
4.4.2 Forced Harmonic Oscillator
4.4.3 Weakly Forced Hamiltonian System
4.5 Effects of Strong Forcing
4.5.1 Universal and Standard Mappings
4.5.2 Forced Dissipative Oscillators
4.5.3 Forced Relaxation Oscillator
4.6 Coupled Oscillators
4.6.1 Phase Dynamics
4.6.2 Coupled Pendulums
4.6.3 Coupled Relaxation Oscillators
4.6.4 Synchronization in Large Ensembles
5 Dynamical Systems in Space
5.1 Space-Dependent Equilibria
5.1.1 Basic Equations
5.1.2 Stationary Solution in One Dimension
5.1.3 Systems with Mass Conservation
5.2 Propagating Fronts
5.2.1 Advance into a Metastable State
5.2.2 Propagation into an Unstable State
5.2.3 Pushed Fronts
5.3 Separated Time and Length Scales
5.3.1 Two-Component Reaction–Diffusion System
5.3.2 Stationary and Mobile Fronts
5.3.3 Stationary and Mobile Bands
5.3.4 Wave Trains
5.4 Symmetry-Breaking Bifurcations
5.4.1 Amplitude Equations
5.4.2 Bifurcation Expansion
5.4.3 Interacting Modes
5.4.4 Plane Waves and their Stability
5.5 Resonant Interactions
5.5.1 Triplet Resonance
5.5.2 Stripes–Hexagons Competition
5.5.3 Standing Waves
5.6 Nonuniform Patterns
5.6.1 Propagation of a Stationary Pattern
5.6.2 Self-Induced Pinning
5.6.3 Propagating Wave Pattern
5.6.4 Nonuniform Wave Patterns
Bibliography
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