| 1 |
Introduction |
| 2 |
Pendulum
Free Oscillator
Global View of Dynamics
Energy in the Plane Pendulum
Stability of Solutions to ODEs
Linear Systems
Nonlinear Systems |
| 3 |
Conservation of Volume in Phase Space
Damped Oscillators and Dissipative Systems
General Remarks
Phase Portrait of Damped Pendulum
Summary
Forced Oscillators and Limit Cycles
General Remarks
Van der Pol Equation |
| 4 |
Forced Oscillators and Limit Cycles (cont.)
Energy Balance for Small ε
Limit Cycle for ε Large
A Final Note |
| 5 |
Parametric Oscillator
Mathieu Equation
Elements of Floquet Theory
Stability of the Parametric Pendulum
Damping
Further Physical Insight |
| 6 |
Fourier Transforms
Continuous Fourier Transform
Discrete Fourier Transform
Inverse DFT
Autocorrelations, Power Spectra, and the Wiener-Khintchine Theorem |
| 7 |
Fourier Transforms (cont.)
Power Spectrum of a Periodic Signal
- Sinusoidal Signal
- Non-sinusoidal Signal
- tmax/T ≠ Integer
- Conclusion |
| 8 |
Fourier Transforms (cont.)
Quasiperiodic Signals
Aperiodic Signals
Poincaré Sections
Construction of Poincaré Sections |
| 9 |
Poincaré Sections (cont.)
Types of Poincaré Sections
- Periodic
- Quasiperiodic Flows
- Aperiodic Flows
First-return Maps
1-D Flows
Relation of Flows to Maps
- Example 1: The Van der Pol Equation |
| 10 |
Poincaré Sections (cont.)
Relation of Flows to Maps (cont.)
- Example 2: Rössler Attractor
- Example 3: Reconstruction of Phase Space from Experimental Data
Fluid Dynamics and Rayleigh-Bénard Convection
The Concept of a Continuum
Mass Conservation |
| 11 |
Fluid Dynamics and Rayleigh-Bénard Convection (cont.)
Momentum Conservation
- Substantial Derivative
- Forces on Fluid Particle
Nondimensionalization of Navier-Stokes Equations
Rayleigh-Bénard Convection |
| 12 |
Fluid Dynamics and Rayleigh-Bénard Convection (cont.)
Rayleigh-Bénard Equations
- Dimensional Form
- Dimensionless Equations
- Bifurcation Diagram
- Pattern Formation
- Convection in the Earth |
| 13 |
Midterm |
| 14 |
Introduction to Strange Attractors
Dissipation and Attraction
Attractors with d = 2
Aperiodic Attractors
Example: Rössler Attractor
Conclusion |
| 15 |
Lorenz Equations
Physical Problem and Parameterization
Equations of Motion
- Momentum Equation
- Temperature Equation
Dimensionless Equations |
| 16 |
Lorenz Equations (cont.)
Stability
Dissipation
Numerical Equations
Conclusion |
| 17 |
Hénon Attractor
The Hénon Map
Dissipation
Numerical Simulations
Experimental Attractors
Rayleigh-Bénard Convection
Belousov-Zhabotinsky Reaction
Fractals
Definition |
| 18 |
Fractals (cont.)
Examples
Correlation Dimension ν
- Definition
- Computation
Relationship of ν to D |
| 19 |
Lyaponov Exponents
Diverging Trajectories
Example 1: M Independent of Time
Example 2: Time-dependent Eigenvalues
Numerical Evaluation
Lyaponov Exponents and Attractors in 3-D
Smale's Horseshoe Attractor |
| 20 |
Period Doubling Route to Chaos
Instability of a Limit Cycle
Logistic Map
Fixed Points and Stability |
| 21 |
Period Doubling Route to Chaos (cont.)
Period Doubling Bifurcations
Scaling and Universality |
| 22 |
Guest Lecture by Lorenz |
| 23 |
Period Doubling Route to Chaos (cont.)
Universal Limit of Iterated Rescaled ƒ's
Doubling Operator
Computation of α |
| 24 |
Period Doubling Route to Chaos (cont.)
Linearized Doubling Operator
Computation of δ
Comparison to Experiments |
| 25 |
Intermittency (and Quasiperiodicity)
General Characteristics of Intermittency
One-dimensional Map
Average Duration of Laminar Phase
Lyaponov Number |
| 26 |
Intermittency (and Quasiperiodicity) (cont.)
Quasiperiodicity
Special Topic |