PHYSICS 123-INTRODUCTION TO FRACTALS AND CHAOS

TEST 1 3/6/01

NAME.............................................................. PLEDGE...........................................................

CLOSED BOOK AND NOTES.

You may only use files you worked on and were saved on your Phy_Students folder.

PROBLEM 1)

a) Draw by hand (use enclosed ruled paper) the **first two** iterations
(after the generator) of the fractal "Wavy Koch".

Is generated by, starting with a unit line (= initiator) , using the following rules:

- Divide the line in **four **equal pieces.

- Replace the **middle two** segments by a "**square wave**"
made up as in the figure.

- Repeat** on each segment** you find in successive iterations

b) Calculate this fractal's similarity dimension
**DS**, and compare with the Koch curve. Does
the result make sense? Explain.

c) Create and run a TrueBasic Program that calculates at least 5 generations of the "Wavy Koch" .

Print a listing of the statements of the program, and the last generation's picture of it.

d) Repeat c) but using now the IFS rules program instead of TrueBasic. Print the set of rules and the graphics output. Compare with c).

d) Use the Fractal Dimension Calculator ("fdc")
to estimate its box counting dimension **DB** and compare with **DS**** **(part a).

*Hint*: You may use the simultaneous keyboard combination "Command+Shift+3"
to take a picture of the screen obtained on part d).

2) LOGISTIC MAP : **Xn+1 = a*Xn
* (1-Xn)**

a) Calculate X5 for the logistic map with a = 2.5 and X0 = 0.2. Compare with the X5 result obtained (for the same a =2.5) when starting at X0 = 0.1. Do these results make sense? What is the theoretical value predicted for the long term value Xoo?

b) Explain carefully how the bifurcation diagram for the logistic equation is generated. That is, explain what are the variables plotted in the horizontal and vertical axes, and how do you calculate the points to plot in the graph. Then do a rough sketch of the diagram labeling clearly the axes.

c) What do we mean by "period doubling route to chaos"?

d) Explain what is the meaning and importance of the Feigenbaum number d = 4.6692...

3) **GLEICK's BOOK QUESTIONS**

a) Why do you think Mandelbrot's titled his book "**Fractals: The
Geometry of Nature**"?

b) Why do **realistic** population models have to be **nonlinear**?

c) What does S. Ulam refer to when comparing "**nonlinear** science"
to calling zoology "the study of **nonelephants** animals"?

d) How did Lorentz discover the phenomenon called "**sensitive
dependence on initial conditions**"?

4)