Calendar

LEC # TOPICS
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