# Calendar

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