Chaos Under Control Sample Syllabus 1

David Peak and Michael Frame

1. Background (3 sessions, read Ch 1)

* overview of ideas in course

2. Iterated Function Systems (5 sessions, read Ch 2)

* what affine transformations do to a picture

* making pictures from nature using IFS

(Labs: IFS explorations -- guided parameter play; IFS code for a leaf)

3. Fractal dimensions (5 sessions, read Ch 3)

* dimension as exponent

* information contained in dimension

(Labs: dimensions of paper wads, grain clusters, tear paths, and viscous fingers)

4. Dynamics: linear and nonlinear (6 sessions, read Ch 4)

* what a model is, what it isn't

* exponential growth and decay in linear dynamics

* the tent map

* the characteristics of chaos

* the bifurcation diagram

(Labs: properties of the simple pendulum; exponential growth)

5. Dynamics: period doubling (5 sessions, read Ch 5)

* the logistic map

* fractal character of the bifurcation diagram

* Feigenbaum's number and universality

(Labs: dripping faucet; diode circuit)

6. Noise and detecting chaos (3 sessions, read Ch 6)

* statistical characterization of a noisy signal

* randomness versus chaos

* return maps

* close pairs analysis

* IFS analysis

* relation of noise to art and music

(Lab: return map, close-pairs, and IFS for some real world data)

7. Controlling chaos (3 sessions, read Ch 6)

* controlling the Tent Map

* controlling higher dimensional chaos: magnetic ribbons, lasers, electronic circuits, chemical reactions, flames, hearts, and brains; communications

* synchronization of chaotic systems: encoding communications

8. Cellular automata (3 sessions, read Ch 9)

* introduction to 1- and 2-d CA as models for space-time structures

* Wolfram's classification scheme

(Lab: CA exploration)

9. Game of Life, self-organization (3 sessions, read Ch 9)

* gliders and other complex structures

* self-organized criticality, Life, and colored noise

(Lab: avalanches in grain piles)

10. The Mandelbrot Set (3 sessions, read Ch 7)

* M-map as a transporter machine

* characteristics of the M-set

11. Fractal basin boundaries (3 sessions, read Ch 8)

* fractal entanglement of basins of attraction

(Lab: basins of attraction for the magnetic pendulum toy)


11. Project reports by students & wrap-up (3 sessions, read Ch 10)

Naturally, some of the sessions in this course will have to be devoted to quizzes, tests, problem solving, review, and the like. This material is ideal for student projects and we have had good success in engaging students in creative and novel explorations of the course's ideas.

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