An experimental approach to nonlinear dynamics and chaos

by

Nicholas B. Tufillaro, Jeremiah Reilly, and Tyler Abbott

Copyright 1992, also published by Addison-Wesley 1992. Please contact your favoriate book seller to get the complete published version.

Preface

• Flows and Maps
• Binary Arithmetic
• Fractals
• Bouncing Ball
  • Introduction
  • Model
    • Stationary Table
    • Impact Relation for the Oscillating Table
    • The Equations of Motion: . Phase and Velocity Maps
    • Parameters
  • High Bounce Approximation
  • Qualitative Description of Motions
    • Trapping Region
    • Equilibrium Solutions
    • Sticking Solutions
    • Period One Orbits and Period Doubling
    • Chaotic Motions
  • Attractors
  • Bifurcation Diagrams
• Quadratic Map
  • Introduction
  • Iteration and Differentiation
  • Graphical Method
  • Fixed Points
  • Periodic Orbits
    • Graphical Method
    • Period One Orbits
    • Period Two Orbit
    • Stability Diagram
  • Bifurcation Diagram
  • Local Bifurcation Theory
    • Saddle-node
    • Period Doubling
    • Transcritical
  • Period Doubling Ad Infinitum
  • Sarkovskii's Theorem
  • Sensitive Dependence
  • Fully Developed Chaos
    • Hyperbolic Invariant Sets
    • Symbolic Dynamics
    • Topological Conjugacy
  • Symbolic Coordinates
    • What's in a name? Location.
    • Alternating Binary Tree
    • Topological Entropy
• String
  • Introduction
  • Experimental Apparatus
  • Single-Mode Model
  • Planar Vibrations: Duffing Equation
    • Equilibrium States
    • Unforced Phase Plane
      • Conservative Case
      • Dissipative Case
    • Extended Phase Space
    • Global Cross Section
  • Resonance and Hysteresis
    • Linear Resonance
    • Nonlinear Resonance
    • Response Curve
    • Hysteresis
    • Basins of Attraction
  • Homoclinic Tangles
  • Nonplanar Motions
    • Free Whirling
    • Response Curve
    • Torus Attractor
    • Circle Map
    • Torus Doubling
  • Experimental Techniques
    • Experimental Cross Section
      • Planar Cross Section
      • Triggering Circuit
      • Nonplanar Cross Section
    • Embedding
    • Power Spectrum
      • Fourier Series
      • Finite Fourier Series
      • Power Spectrum
      • Spectral Signatures
    • Attractor Identification
    • Correlation Dimension
• Dynamical Systems Theory
  • Introduction
  • Flows and Maps
    • Flows
    • Poincaré Map
    • Suspension of a Map
    • Creed and Quest
  • Asymptotic Behavior and Recurrence
    • Invariant Sets
    • Limit Sets: , , and Nonwandering
    • Chain Recurrence
  • Expansions and Contractions
    • Derivative of a Map
    • Jacobian of a Map
    • Divergence of a Vector Field
    • Dissipative and Conservative
    • Equation of First Variation
  • Fixed Points
    • Stability
    • Linearization
    • Hyperbolic Fixed Points: . Saddles, Sources, and Sinks
  • Invariant Manifolds
    • Center Manifold Theorem
    • Homoclinic and Heteroclinic Points
  • Example: Laser Equations
    • Steady States
    • Eigenvalues of a Matrix
    • Eigenvectors
    • Stable Focus
  • Smale Horseshoe
    • From Tangles to Horseshoes
    • Horseshoe Map
    • Symbolic Dynamics
    • From Horseshoes to Tangles
  • Hyperbolicity
  • Lyapunov Characteristic Exponent
• Knots and Templates
  • Introduction
  • Periodic Orbit Extraction
    • Algorithm
    • Local Torsion
    • Example: Duffing Equation
  • Knot Theory
    • Crossing Convention
    • Reidemeister Moves
    • Invariants and Linking Numbers
    • Braid Group
    • Framed Braids
  • Relative Rotation Rates
  • Templates
    • Motivation and Geometric Description
    • Algebraic Description
      • Drawing Conventions
      • Braid Linking Matrix
    • Location of Knots
      • Trees
      • Kneading Theory
    • Calculation of Relative Rotation Rates
      • Horseshoe Example
      • General Algorithm
  • Intertwining Matrices
    • Horseshoe
    • Lorenz
  • Duffing Template
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