| 1 |
Introduction |
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| 2 |
Lecture by Prof. Thomas Peacock
Pendulum
Free Oscillator
Global View of Dynamics
Energy in the Plane Pendulum
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| 3 |
Lecture by Prof. Thomas Peacock
Stability of Solutions to ODEs
Linear Systems
Nonlinear Systems
Conservation of Volume in Phase Space
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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
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| 5 |
Parametric Oscillator
Mathieu Equation
Elements of Floquet Theory
Stability of the Parametric Pendulum
Damping
Further Physical Insight
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Problem set 2 due |
| 6 |
Fourier Transforms
Continuous Fourier Transform
Discrete Fourier Transform
Inverse DFT
Autocorrelations, Power Spectra, and the Wiener-Khinitchine Theorem
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| 7 |
Fourier Transforms (cont.)
Power Spectrum of a Periodic Signal
- Sinusoidal Signal
- Non-sinusoidal Signal
- tmax/T ≠ Integer
- Conclusion
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Problem set 3 due |
| 8 |
Fourier Transforms (cont.)
Quasiperiodic Signals
Aperiodic Signals
Poincaré Sections
Construction of Poincaré Sections
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| 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
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| 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
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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
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| 12 |
Fluid Dynamics and Rayleigh-Bénard Convection (cont.)
Rayleigh-Bénard Equations
- Dimensional Form
- Dimensionless Equations
- Bifurcation Diagram
- Pattern Formation
- Convection in the Earth
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Problem set 5 due |
| 13 |
Midterm Exam |
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| 14 |
Introduction to Strange Attractors
Dissipation and Attraction
Attractors with d = 2
Aperiodic Attractors
Example: Rössler Attractor
Conclusion
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| 15 |
Lorenz Equations
Physical Problem and Parametrization
Equations of Motion
- Momentum Equation
- Temperature Equation
Dimensionless Equations
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Problem set 6 due |
| 16 |
Lorenz Equations (cont.)
Stability
Dissipation
Numerical Equations
Conclusion
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| 17 |
Hénon Attractor
The Hénon Map
Dissipation
Numerical Simulations
Experimental Attractors
Rayleigh-Bénard Convection
Belousov-Zhabotinsky Reaction
Fractals
Definition
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| 18 |
Fractals (cont.)
Examples
Correlation Dimention ν
- Definition
- Computation
Relationship of ν to D
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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
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| 20 |
Period Doubling Route to Chaos
Instability of a Limit Cycle
Logistic Map
Fixed Points and Stability
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| 21 |
Period Doubling Route to Chaos (cont.)
Period Doubling Bifurcations
Scaling and Universality
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Problem set 8 due |
| 22 |
Period Doubling Route to Chaos
Universal Limit of Iterated Rescaled ƒ's
Doubling Operator
Computation of α
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| 23 |
Period Doubling Route to Chaos (cont.)
Linearized Doubling Operator
Computation of δ
Comparison to Experiments
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Problem set 9 due |
| 24 |
Guest lecture by Prof. Edward N. Lorenz
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| 25 |
Intermittency (and Quasiperiodicity)
General Characteristics of Intermittency
One-dimensional Map
Average Duration of Laminar Phase
Lyaponov Number
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| 26 |
Intermittency (and Quasiperiodicity) (cont.)
Quasiperiodicity
Special Topic
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Final problem set due |