MATH 178

Math 178 Homework and Assignments

Short writing assignment #1. Draft due Friday (in class), May 18, final version due Friday May 25

Write a one page article describing what fractals are. Imagine that this article will appear in the SFU student newspaper, so your audience is everyone at SFU. You may want to include diagrams or pictures.

Please hand in your draft (outline) in friday's class. I will return them next wednesday and the final version will be due Friday, May 25.

Quantitative assignment #1. Due: Friday June 1

(1) Recall the decomposition of the triangle into 4 smaller triangles of size 1/2 the whole triangle (as when we discussed the Sierpinski triangle). Number the small triangles 1 (bottom left), 2 (bottom right), 3 (top), 4 (centre).
- (a) Make a sketch of the triangle and indicate on the sketch address regions 24, 144, 1234.
- (b) Use the template below to make a sketch of the image of the triangle after removing all address regions UP TO ADDRESS LENGTH 5 that contain a '21'. Please use this template for your sketch (it will be easier for us to mark). Indicate on your sketch the addresses of these regions. It will be easier to shade the regions missing rather than the regions remaining.
  (Start with locating all address regions of length 2 that contain '21', then all address regions of length 3 that contain a '21', for example 121, and 211, then all address regions of length 4, etc.). Is this object self-similar? Why or why not?
  
  Here's an image of the triangle decomposed into smaller triangles; you can use this to make sketches for your answers.
(2) Use the formula for the sum of a geometric series;

to show that all the area of the original triangle is removed during the construction of the Sierpinski triangle.
(3) Recall the construction of the Cantor (middle thirds) set; Construction of the Cantor set, and the description of it in terms of ternary expansions of numbers (i.e., the Cantor set is the numbers in [0,1] whose ternary expansion contains no '1'; Cantor set and ternary expansions.

Also, recall the method to calculate the ternary expansions of numbers;
- (3a) Determine whether the following numbers are in the Cantor set or not; 5/7, 37/40, Pi/4 (for this last one, just follow the same procedure as above).
- (3b) Give a description of how to obtain the Cantor "middle fifths" set by first decomposing [0,1] into 5 pieces (with end points at 0, 1/5, 2/5, 3/5, 4/5, 1), and removing the middle fifth segment (2/5, 3/5). Then decomposing the remaining 4 pieces each into 5 sections and removing the middle fifths from each of these, etc.
  - What is the length of all the intervals removed?
  - Describe the Cantor middle fifths set in terms of expansions of numbers in base 5.
  - Are the following numbers in the Cantor middle fifths set or not; 1/3, (square root of 2)/5.
(4) Suppose the graph of log(u) vs log(1/s) for a curve is a straight line with slope 3/2. If u(1)=100, what is the length of this curve at the scale 1/4? What is the length of this curve?
(5a) Can a curve of finite length have length exponent d>0? Explain.
(5b) Can a curve of infinite length have length exponent d=0? Explain.
(6) Recall the Square von Koch curve. It was constructed by repeating this procedure of adding squares.
Calculate the length exponent d for this curve.
(7) Recall that the covering dimension, or fractal dimension, of a set is denoted by D. Compute the covering dimension D of the following sets.
- The Cantor middle fifths set (see problem (3b) ). Is this set more or less complicated that the Cantor (middle thirds) set?
- The Square von Koch curve. Compare this with your answer for the length exponent d you found in question (6).
- The fractal obtained by following this template (variation on Sierpinski; keep only the middle 4 triangles of the 16 triangle decomposition of the full triangle). Is this object more or less complicated that the Sierpinski triangle fractal?
- This square variation fractal, which was obtained from this template.

Writing assignment #2. Draft due Friday (in class), June 15, final version due Friday June 22

here's a summary of the topics covered

Friday, June 22

Quantitative assignment #2.

Problems 1 - 6 due Friday, June 29 in class, Problems 7,8 due Wednesday July 4. Solutions will be provided at that time no late assignments will be accepted.

(1) Recall our definition of fractal numbers; A number D is a fractal number if there is a set whose fractal dimension is D.
- (1a) What is the smallest and largest fractal number that one may encounter?
- (1b) If a set S is drawn on a blackboard, explain why it's fractal dimension must be between 0 and 2, including perhaps 0 and 2.
- (1c) Construct sets with the following fractal dimensions; log(5)/log(9), 1.5, log(17)/log(4).
- (1d) Use consistent coverings to construct sets which we haven't seen before in class, with the following properties;
  - A set whose fractal dimension is 1 but is entirely dust (i.e., contains no lines).
  - A set with fractal dimension log(5)/log(4) but is not self-similar.
- (1e) Explain in as much detail as you can, what are the possible fractal numbers. (Can any number be a fractal number? Can you give a formula for a possible fractal number? Remember, if you say that D is a fractal number you must find a set S whose fractal dimension is D, which means you have to find a covering for S which gives dimension D.)
(2) Recall that an IFS W is made up of affine transformations w_i = A_i + v_i where A_i is a 2 by 2 matrix and v_i is a (2 dimensional) vector (the 'shift vector').
- Write out the matrix A and shift vectors v for the following transformations (refer to my notes on affine transformations);
  - Rotation 20^o clockwise followed by contraction in the x-direction by 0.6 and contraction in the y-direction by 0.9 (no shift).
  - Dilation by 0.4 followed by reflection across the x-axis and then shifted so that the top right corner of the image of the square lies at the location (1,1).
  Find the fixed points of these transformations (verify that these points p are indeed fixed points by computing w(p) and seeing that this is equal to p.)
  
  Added: Sketch the image of the square under these transformations, and also compute the image of the square by calculating the locations of the vertices (0,0), (1,0), (1,1), (0,1) under these transformations.
(3) Let W be the IFS for the Cantor Maze (this can be found in section 5.2). Make a sketch of the first 3 iterations of this IFS when the initial image S is;
- the diagonal line from the bottom left to upper right corner of the initial square
- a circle centered at the midpoint of the initial square and diameter equal to the diameter of the square (so the circle just fills the initial square)
- an equilateral triangle that fills the initial square (like the outlins of Sierpinski's triangle)
Which one of these initial images do you think would produce the image of the fractal the fastest when the IFS is iterated?
(4) Use the Visual Basic program Fractal Pattern or the Chaos Game Applet that can be found on the software page to create unique (i.e., ones we haven't seen before) fractals with the following properties. Please hand in an image of the fractal along with the parameters of the IFS you used to create them. (Here are instructions for using and producing images of fractals with these programs.)
Here are some notes on affine transformations (handed out in class).
Go here for examples of iterations of IFS created with the Fractal Pattern Visual Basic Program (which you can download), and here for examples of how to create certain affine transformations using this program.

******************************************************************************* In case you can't get the Visual Basic program Fractal Pattern running (eg., your OS is Mac), then you can use the Fractal Movie applet to view the blue print of the IFS and the fractal. Here you just enter the IFS you are interested in as the start pattern and as the end pattern (see the instructions posted with the applet). This way the IFS doesn't change while the program is running. To get an image of the fractal, follow the instructions in here regarding printing the image using the chaos game applet.

********************************************************************************
- Fractals 1,2,3 Reproduce the fractals in Figure 5.28 of the text (Section 5.3). That is, figure out what the IFS is for each and then run the programs to draw.
- Fractal 4 A cloud-like fractal. Try to make one that looks like this cloud. (Hint: start with the spiral fractal (swirls) and add more 'lenses' to this IFS.)
- Fractal 5 A new, interesting, fractal! (More marks will be given the more complicated the fractal is, i.e., number of transformations and if they are more than just dilations. Also, I will display these in class later on.)
(5) Consider these two fractals; Mess1 and Mess2. Here is a blue print (image of the square under one iteration of the IFS). Which one of these fractals is the fixed point of this IFS? Justify your answer. (You may also want to consider this second iterate of the IFS.)
(6) Consider the IFS for the Crystal 4 fractal; W = w₁ U w₂ U w₃ U w₄ (see Figure 5.14 in Section 5.2, or look it up with the applets). The contraction rates for each transformation are; w₁, w₂, w₃ ; 0.255, w₄ ; 0.74. (If the matrix entries are a,b,c,d like we have been denoting, then the rate of contraction for the matrix is square root of ad-bc).

Assume that your computer can draw 10,000 squares per second, and the size of the initial square is 1000 by 1000 pixels. Calculate how long it will take your computer to draw this fractal by iteration of the IFS. Can your computer draw this fractal this way?

(7) Consider the Sierpinski Triangle IFS with probabilities p₁ = p₂ = 0.2, p₃=0.6.

(7a)

(7b)

Run the Chaos Game applet with these probabilities and comment on how many game points are needed there and compare to your answer.

The sizes (in pixels) of the canvases that contain the image produced by the chaos game applet are;

Version of applet	Canvas size (pixels)
640 X 480 (VGA)	304 X 304
800 X 600 (SVGA)	380 X 380
1024 X 768 (XVGA)	500 X 500
1280 X 1024 (UVGA)	645 X 645
1400 X 1050	800 X 800

(8) Consider the Full Triangle IFS for this question (that is, the same IFS as for Sierpinski but add in the middle, 4th (upside down), triangle).

Writing assignment #3. Due Wednesday, June 27.

mathematically precise

Here is a list of important points and ideas you should include

Here's my solution

Midterm test; Friday July 6 (2 hours)
The midterm test will cover all the material we have discusses in the course so far (all about fractals). You can expect quantitative questions similar (but not necessarily identical) to the quantitative assignment problems. In addition, there will be 2 or 3 short written questions (one or two paragraphs each) concerning the theory discussed in the course.

Please bring a calculator and ruler.

Writing assignment #4. Due: July 25.

Group presentations about sections of articles on chaos that were handed out in class. See here for details about this assignment and the evaluation rubric (this has been updated from what was handed out in class wednesday).

Groups will work on this at next weeks' seminar (July 18). Presentation will be due at the seminar on July 25.

I passed around two books that give `popular' accounts of the development of chaos theory, and which may be useful references here;
Chaos: Making a New Science, by James Gleick
Does God Play Dice?, by Ian Stewart.

Quantitative assignment #3. Due: Friday, August 3 in class (no late assignments; solutions will be provided then)

(1) Here's plots of the function g_a(x)=x³+ax for a = -2, 0, 2; plots of g_a(x)=x³+ax (the diagonal line y = x is also plotted).

(1a)

W_s(p)

(1b)

g_a(x)

-2 < a < 2

(2) Sketch the graph of a function that has the period 4 orbit {1,-2,3,0} (in that order!). Verify by graphical iteration that this function does indeed have that period 4 orbit. (Hint: you want to first plot the points 1, -2, 3, 0 along the x-axis and then place the graph above these points at the appropriate location.)

(3) Using the Logistic Movie applet, find a stable (prime) period 6 orbit for the logistic equation near parameter valus a=3.6 (record the exact value of a you used). Include a sketch or screen shot of the histogram of the orbit at this value. Then use the Bifurcation applet to locate (precisely) all points in this stable period 6 orbit (click on with mouse) and all points in the unstable period 6 orbit. Verify, by iterating the logistic equation with your calculator, that these points do form period 6 orbits (include the results from your calculator). Run the Graphical Iteration applet with this a value and with a point in the stable period 6 orbit. Make a sketch or take a screen shot of the 'staircase' path of this period 6 orbit (graphical iteration of the orbit). Label the points in the orbit along the x-axis.

(4) Suppose g_a(x) is a family of functions whose orbits (for various values of the parameter a) go through a sequence of period-doubling bifurcations like we observed for the logistic family of functions. Let a₁ be the parameter value where the fixed point (period 1 orbit) bifurcates into a period 2 orbit, a₂ the parameter value where the period 2 orbit bifurcates into a period 4 orbit, etc.

"Universality" says that period-doubling takes place at a common rate, namely that the ratios of the distances between successive bifurcations is nearly constant at the value d for a wide variety of dynamical systems;

.
Here, d is the Feigenbaum constant; d = 4.6692.... We say that the system has reached chaos when it reaches the "last" bifurcation point a_infinity, also referred to as the Feigenbaum point (at which the periodic orbit now has infinite period). Beyond this value of a orbits have aperiodic, or ergodic, behaviour. (This is mentioned by David Ruelle in his article on turbulence that I handed out for group presentations. See also Sections 11.2 and 11.3 in the text.)

(4a) Verify that the period-doubling rates for the logistic equation nearly satisfy this universality relation (you can look up the 'exact' values of the bifurcations in the text in Section 11.2).
(4b) If we know the values of a₁, a₂ (or any 2 consecutive bifurcation points), we can estimate the parameter value a_infinity where the onset of chaos occurs in the following way;
.
Use this formula to estimate the Feigenbaum point for the logistic equation and compare this value to what you measure on the final state diagram.
(4c) A dynamical system changes as the surrounding temperature T changes. Initially, you observe a steady state of the system. As the temperature increases you observe that the system switches to oscillating between two states at T = 70C. Then at T= 100C you observe that the system begins to oscillate between 4 different states. As the temperature continues to increase the system goes through a period-doubling sequence of bifurcations into more and more complex oscillations. Assuming Universality, at what temperature would you expect to observe the onset of chaos in this system?

(5) Show that the saw tooth transformation has an ergodic orbit.

(6) Use symbolic dynamics to find all period 2 points of the logistic equation f₄(x). (First find the symbol sequences (binary expansions), then transform these to numbers (hint: they are simple fractions), then transform these numbers via the function h(x) into numbers for the logistic equation.)
Verify the answers by iterating the logistic function.

Last updated: July 26