CHAOS: An Introduction to Dynamical Systems
by K. Alligood, T. Sauer, J.A. Yorke
Springer-Verlag 1997
Table of Contents
- CHAPTER 1. One-Dimensional Maps
- 1.1 One-dimensional maps
- 1.2 Cobweb plot: Graphical representation of an orbit
- 1.3 Stability of fixed points
- 1.4 Periodic points
- 1.5 The family of logistic maps
- 1.6 The logistic map G(x)=4x(1-x)
- 1.7 Sensitive dependence on initial conditions
- 1.8 Itineraries
- Challenge 1: Period three implies chaos
- Exercises
- Lab Visit 1: Boom, bust, and chaos in the beetle census
- CHAPTER 2. Two-Dimensional Maps
- 2.1 Mathematical models
- 2.2 Sinks, sources, and saddles
- 2.3 Linear maps
- 2.4 Coordinate changes
- 2.5 Nonlinear maps and the Jacobian matrix
- 2.6 Stable and unstable manifolds
- 2.7 Matrix times circle equals ellipse
- Challenge 2: Counting the periodic orbits of linear maps on a torus
- Exercises
- Lab Visit 2: Is the solar system stable?
- CHAPTER 3. Chaos
- 3.1 Lyapunov Exponents
- 3.2 Chaotic orbits
- 3.3 Conjugacy and the logistic map
- 3.4 Transition graphs and fixed points
- 3.5 Basins of attraction
- Challenge 3: Sharkovsky's Theorem
- Exercises
- Lab Visit 3: Periodicity and chaos in a chemical reaction
- CHAPTER 4. Fractals
- 4.1 Cantor sets
- 4.2 Probabilistic constructions of fractals
- 4.3 Fractals from deterministic systems
- 4.4 Fractal basin boundaries
- 4.5 Fractal dimension
- 4.6 Computing the box-counting dimension
- 4.7 Correlation dimension
- Challenge 4: Fractal basin boundaries and the uncertainty exponent
- Exercises
- Lab Visit 4: Fractal dimension in experiments
- CHAPTER 5. Chaos in Two-Dimensional Maps
- 5.1 Lyapunov exponents
- 5.2 Numerical calculation of Lyapunov exponents
- 5.3 Lyapunov dimension
- 5.4 A two-dimensional fixed-point theorem
- 5.5 Markov partitions
- 5.6 The horseshoe map
- Challenge 5: Computer calculations and shadowing
- Exercises
- Lab Visit 5: Chaos in simple mechanical devices
- CHAPTER 6. Chaotic Attractors
- 6.1 Forward limit sets
- 6.2 Chaotic attractors
- 6.3 Chaotic attractors of expanding interval maps
- 6.4 Measure
- 6.5 Natural measure
- 6.6 Invariant measure for one-dimensional maps
- Challenge 6: Invariant measure for the logistic map
- Exercises
- Lab Visit 6: Fractal scum
- CHAPTER 7. Differential Equations
- 7.1 One-dimensional linear differential equations
- 7.2 One-dimensional nonlinear differential equations
- 7.3 Linear differential equations in more than one dimension
- 7.4 Nonlinear systems
- 7.5 Motion in a potential field
- 7.6 Lyapunov functions
- 7.7 Lotka-Volterra models
- Challenge 7: A limit cycle in the Van der Pol system
- Exercises
- Lab Visit 7: Fly vs. fly
- CHAPTER 8. Periodic Orbits and Limit Sets
- 8.1 Limit sets for planar differential equations
- 8.2 Properties of omega-limit sets
- 8.3 Proof of the Poincare-Bendixson Theorem
- Challenge 8: Two incommensurate frequencies form a torus
- Exercises
- Lab Visit 8: Steady states and periodicity in a squid neuron
- CHAPTER 9. Chaos in Differential Equations
- 9.1 The Lorenz attractor
- 9.2 Stability in the large, instability in the small
- 9.3 The Rossler attractor
- 9.4 Chua's circuit
- 9.5 Forced oscillators
- 9.6 Lyapunov exponents in flows
- Challenge 9: Synchronization of chaotic orbits
- Exercises
- Lab Visit 9: Lasers in synchronization
- CHAPTER 10. Stable Manifolds and Crises
- 10.1 The Stable Manifold Theorem
- 10.2 Homoclinic and heteroclinic points
- 10.3 Crises
- 10.4 Proof of the Stable Manifold Theorem
- 10.5 Stable and unstable manifolds for higher dimensional maps
- Challenge 10: The lakes of Wada
- Exercises
- Lab Visit 10: The leaky faucet: minor irritation or crisis?
- CHAPTER 11. Bifurcations
- 11.1 Saddle-node and period-doubling bifurcations
- 11.2 Bifurcation diagrams
- 11.3 Continuability
- 11.4 Bifurcations of one-dimensional maps
- 11.5 Bifurcations in plane maps: Area-contracting case
- 11.6 Bifurcations in plane maps: Area-preserving case
- 11.7 Bifurcations in differential equations
- 11.8 Hopf bifurcations
- Challenge 11: Hamiltonian systems and the Lyapunov Center Theorem
- Exercises
- Lab Visit 11: Iron + sulfuric acid = Hopf bifurcation
- CHAPTER 12. Cascades
- 12.1 Cascades and 4.66920169...
- 12.2 Schematic bifurcation diagrams
- 12.3 Generic bifurcations
- 12.4 The cascade theorem
- Challenge 12: Universality in bifurcation diagrams
- Exercises
- Lab Visit 12: Experimental cascades
- CHAPTER 13. State reconstruction from data
- 13.1 Delay plots and time series
- 13.2 Delay coordinates
- 13.3 Embedology
- Challenge 13: Box-counting dimension and intersection
- APPENDIX A. Matrix Algebra
- A.1 Eigenvalues and eigenvectors
- A.2 Coordinate changes
- A.3 Matrix times circle equals ellipse
- APPENDIX B. Computer Solution of ODEs
- B.1 ODE solvers
- B.2 Error in numerical integration
- B.3 Adaptive step-size methods
- HINTS AND ANSWERS TO SELECTED EXERCISES
- BIBLIOGRAPHY