| LEC # | TOPICS | KEY DATES |
|---|---|---|
| 1 | Introduction | |
| 2 | Lecture by Prof. Thomas Peacock
Pendulum
Free Oscillator Global View of Dynamics Energy in the Plane Pendulum
| |
| 3 | Lecture by Prof. Thomas Peacock
Stability of Solutions to ODEs
Linear Systems Nonlinear Systems
Conservation of Volume in Phase Space
| Problem set 1 due |
| 4 | Damped Oscillators and Dissipative Systems
General Remarks Phase Portrait of Damped Pendulum Summary
Forced Oscillators and Limit Cycles
General Remarks Van der Pol Equation 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
| Problem set 2 due |
| 6 | Fourier Transforms
Continuous Fourier Transform Discrete Fourier Transform Inverse DFT Autocorrelations, Power Spectra, and the Wiener-Khinitchine Theorem | |
| 7 | Fourier Transforms (cont.)
Power Spectrum of a Periodic Signal - Sinusoidal Signal - Non-sinusoidal Signal - tmax/T ≠ Integer - Conclusion | Problem set 3 due |
| 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 | Problem set 4 due |
| 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 | Problem set 5 due |
| 13 | Midterm Exam | |
| 14 | Introduction to Strange Attractors
Dissipation and Attraction Attractors with d = 2 Aperiodic Attractors Example: Rössler Attractor Conclusion | |
| 15 | Lorenz Equations
Physical Problem and Parametrization Equations of Motion
- Momentum Equation - Temperature Equation
Dimensionless Equations | Problem set 6 due |
| 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 Dimention ν
- Definition - Computation
Relationship of ν to D | Problem set 7 due |
| 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 | Problem set 8 due |
| 22 | Period Doubling Route to Chaos
Universal Limit of Iterated Rescaled ƒ's Doubling Operator Computation of α | |
| 23 | Period Doubling Route to Chaos (cont.)
Linearized Doubling Operator Computation of δ Comparison to Experiments | Problem set 9 due |
| 24 | Guest lecture by Prof. Edward N. Lorenz | |
| 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 | Final problem set due |