## Syllabus: Phys 217 – Fall 2000

### Monday, Oct. 16: (Course Organization/Introduction)

- General Info, Home Page Features
- Brief History of Chaos
- Determinism versus Randomness
- Deterministic Chaos
- Symbolic Dynamics
- Reading: Gleick, pp. 1-8; Hall, chapters 1, 14

### Wednesday, Oct. 18: (Mappings)

- The Idea of Feed-Back
- An Experiment with Mathematics
- The Logistic Mapping
- First Encounter with Sensitivity and Chaos
- The Butterfly Effect
- Reading: Hall, chapters 3, 13; Gleick, pp. 11-31

### Friday, Oct. 20: (Programming Techniques)

- Problem Set 1 Due
- Brief History of FORTRAN
- Architecture of the Language
- Types of Variables
- Constructs: Assignments, Loops, Input, Output, Files
- Example Program: Iteration of the Logistic Mapping
- Reading: Hall, chapter 3

### Monday, Oct. 23: (Fixed Points)

- General Definition of N’th Order Fixed Points
- Periodic Orbits

### Wednesday, Oct. 25: (More on Fixed Points)

- First and Second Order Fixed Points in Detail
- Stability of Fixed Points

### Friday, Oct. 27: (Logistic Mapping: Numerical Survey)

- Problem Set 2 Due
- Computer Experiments with the Logistic Mapping
- Searching for Patterns
- Numerical Discovery of Chaos
- Reading: Gleick, pp. 35-56; Hall, chapter 7

### Monday, Oct. 30: (Logistic Mapping: Bifurcation Structure)

- Forward Tree for Positive Control Parameters
- Backward Tree for Negative Control Parameters
- Reading: Gleick, pp. 59-80; Hall, chapter 3

### Wednesday, Nov. 1: (The Transition to Chaos)

- Bifurcation Cascades
- Feigenbaum Number
- Estimate for Lambda Star
- Reading: Gleick, pp. 157-187; Hall, chapter 3

### Friday, Nov. 3: (Binary Shift Map)

- Binary Numbers
- Modulo Operation
- Fixed Points, Cycles
- Liapunov Exponent
- Reading: Gleick, pp. 243-272

### Monday, Nov. 6: (Orders of Infinity, Measure, Dimension)

- Counting of the Rationals
- Irrationals: Uncountable
- Rationals and the Concept of a Dense Set
- Measure
- Paradox of Dimension

### Wednesday, Nov. 8: (Review of Course Material for Midterm)

- Chaos Demonstration
- Devaney’s Definition of Chaos

### Friday, Nov. 10: (Midterm Exam)

- Last Day to Withdraw from 2nd Quarter Classes

### Monday, Nov. 13: (Fractals)

- Tent Map
- Cantor Set
- Fractal Dimension
- Reading: Gleick, pp. 83-118, 303-317; Hall, chapter 10

### Wednesday, Nov. 15: (Chaos Demo and Fractal Dimension)

- Demo of the Kicked Compass Needle
- Introduction to 2D Mappings
- Naturally Occurring Fractals
- Fractal Dimension of a Fern Leaf
- Embedding Dimension
- Formula for Fractal Dimension
- Reading: Hall, chapters 10, 11

### Friday, Nov. 17: (The Chaos Game)

- Problem Set 4 Due
- Rules of the Game
- A Program for the Chaos Game

### Monday, Nov. 27: (Strange Attractors)

- Box Counting Dimension of the Cantor Set
- The Koch Snow Flake
- Attractor of the Chaos Game
- Strange Attractors
- Reading: Gleick, pp. 121-153; Hall, color plates

### Wednesday, Nov. 29: (Julia Sets)

- Complex Numbers
- Curlicues
- Mandelbrot Mapping
- Julia Sets
- Reading: Gleick, pp. 215-240; Hall, chapters 10, 11

### Friday, Dec. 1: (Atomic Ionization and 2D Mappings)

- The Hydrogen Atom
- 1D Version of the Hydrogen Atom
- Splitting the Hydrogen Atom: A Real-Life Julia Set
- Baker’s Mapping
- Phase Portraits
- Cat Map

### Monday, Dec. 4: (Analysis of Kicked Rotor Dynamics)

- Problem Set 5 Due
- Kicked Rotor Dynamics: Standard Mapping
- Stroboscopic Phase Portrait
- Regular Islands
- Chaotic Regions
- KAM Lines
- Julia Set of the Dissociating Rotor

### Dec. 6: (Chaos in the Solar System)

- Is the Solar System Stable?
- Tumbling Hyperion
- Catastrophic vs. Non-Catastrophic Chaos
- The Role of KAM Lines for the Stability of the Solar System
- Reading: Gleick, pp. 121-153; Hall, chapters 8, 18

### Dec. 8: (Last Lecture)

- Structure of the Final Exam
- Review of Course Material
- Modelling with Mappings: Parasite/Host Eco-System
- Reduction of 2D Mappings to 1D Mappings with Memory
- Alternative Ways of Fractal Rendering
- A Method for the Encryption of Images based on Chaos
- Course Evaluation
- Reading: Hall, chapter 4