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Nonlinear Dynamics I: Chaos >> Content Detail



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LEC #TOPICSKEY DATES
1Introduction
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
13Midterm 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

 








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