Homework #2: MAT335 - Winter, 2001

• (1) Make sketches of the Cantor set and the von Koch curve and indicate on your sketch the regions with addresses;
  112, 212, 211(11) for the Cantor set,
  12, 321, 2(22) for the von Koch curve.
  (Recall that (xy) means xyxyxy......)
• (2) Write down, in 'plain english', the rules of the Chaos Game for the von Koch and square von Koch curves.
• (3) Consider the Sierpinski IFS, and set p_1 = 0.5, p_2 = p_3 = 0.25. Suppose the chaos game is played with 10,000 points.
  (a) How many points will lie in the small triangles with address 11 ? Address 12 ? Adress 22 ?
  (b) Among all the small triangles with addresses of length 3 (so, D_ijk), which will have the most points? The least points? Intermediate number of points?
  (c) Based on your answers to parts (a) and (b), and by considering other small triangles, what do you expect the final image to look like? Use the Chaos Game applet to play this game with these probabilities and verify your answer.
• (4) Use the Fractal Movie applet to explain the intermediate patterns that are produced with;
  (a) Start Pattern: Full Square (via drop down menu). End Pattern: Full Square, but with probabilities p_1 = p_2 = p_3 = .3334, p_4 = 0. (Try with m = 200, # of iterations = 50,000.)
  (b) Start Pattern: Full Square (via drop down menu). End Pattern: Full Square, but with probabilities p_1 = 0, p_2 = 0, p_3 = 0.5, p_4 = 0.5. (Try with m = 200, # of iterations = 50,000.)
  Try to explain as many features of the intermediate patterns as you can.
• (5) Consider the Barnsley Fern. Suppose we play the chaos game with 100,000 points.
  (a) How tall is the fern produced by this game when (i) the probabilities are all 1/4?, (ii) when the probabilities are p_1=0.85, p_2=p_3=p_4 = 0.05?
  (b) How long is the first branch on the right side of the fern when (i) the probabilities are all 1/4?, (ii) when the probabilities are p_1=0.85, p_2=p_3=p_4 = 0.05?
  (By 'height of the fern' we mean how many of the segments that are drawn by w_4 appear by the end of the chaos game. (Note that each iteration of the IFS - if you were drawing the fractal by iterating the IFS - produces another segment of the stem. If you are drawing the fractal with the chaos game, and if at least one point in the chaos game appears in a particular segment, then we say that segment appears in the game). By 'length of the first branch' we mean how many of the segments of its stem appear by the end of the chaos game.)
• (6) Use the 'full triangle' IFS to answer the following questions. (The full triangle' IFS has the three transformations that make up the Sierpinski IFS plus a forth one that 'fills' the middle hole of the Sierpinski IFS, i.e., w_4 = (-0.5, 0, 0, -0.5) + (0.75, 0.5) )
  • (a) What image is produced from the chaos game using the sequence {1,2,3,4,1,2,3,4,...} (i.e., 1,2,3,4, repeating)? (Hint: consider the addresses 43214321...., 14321432..., 21432143..., and 32143214... )
  • (b) Suppose the sequence {s_i} is produced by removing every occurence of the string 1,2 from a random sequence of the integers 1,2,3,4. What is the image produced using this sequence for the chaos game? (Hint: think of addresses!)
  • (c) Is the image produced in (b) a fractal?
  • (d) What is the fractal dimension of the images produced in parts (a) and (b)?
• (7) Follow the procedure discussed in class in the case of the Fern, to find 'better' probabilities than the equal ones p_i = 1/4 for the Crystal 4 fractal. That is, determine the probabilities such that there are an equal density of game points in each region with address of length 1. Check your answer by running the chaos game applet first with equal probabilities and then with the 'better' probabilities you found.
• (8) Write down a sequence that contains all strings of length 3 of the integers 1,2,3 that is more 'efficient' than the 'naive' one of length 3*3^3 = 81. (The 'best' such sequence has length 3^3 +1 = 28).
• (9) Use the Chaos Game applet, or better yet the VB program Fractal Pattern, to investigate how 'efficient' a random sequence is when used in the chaos game to draw the Cantor Maze fractal. Follow the procedure described in class;
  • determine the level m; the smallest addresses that can be resolved on your computer screen (these addresses have length m);
  • calculate the length of the 'best' sequence and the worst, 'naive', sequence that will draw the fractal to this level m;
  • show that a string of random numbers of the same length as the 'best' sequence does not draw the fractal well (to level m), while the fractal can be drawn well (to level m) by a string of random numbers that is shorter than the 'naive' sequence.
  Give an explanation of your results above. Based on your results, how much 'redundancy' does a random sequence have, i.e., how many addresses at level m are repeated in a random sequence?
• (10) Sketch the graph of a function that has the period 4 orbit {-1,1,2,3} (see the notes "Graphical Iteration"). Use graphical iteration to verify that {-1,1,2,3} is indeed an orbit of your function.
• (11) Use graphical iteration to study the orbits of the following functions (i.e., find any fixed points and determine their stability and stable sets);
  • g(x) = e^x - 1. Does g(x) have any periodic points of period greater than 1? Why or why not? (answer this question just by considering the orbits of g(x))
  • f(x) = 0.5*x^2 + 2*x - 2 (note this is not the same as I wrote in class). How many points of period 1 does f(x) have (i.e., fixed points)? How many points of prime period 2 does f(x) have? How many points of prime period 3 does f(x) have? What is the stability of these periodic points and what are their stable sets?
    To answer these questions, plot f(x), f^2(x), f^3(x) and the diagonal y = x on the same graph (I will hand out such a plot). Use graphical iteration to determins the stability of the periodic orbits and their stable sets.
• (12) Use the applet Graphical Iteration to study the stability of the orbits of the logistic function f(x) = 3.5*x(1-x) (that is, when the parameter a = 3.5).
  • First study the stability of the fixed point p. Note that by solving the equation f(p) = 3.5*p(1-p) we obtain that p=25/35 = 0.714285714285.... . Use the applet with the function f(x) with initial point x_0 near the fixed point. Note how the orbit moves away from the fixed point. Where does the orbit go?
  • Now study the stability of the period 2 orbit. Use the applet with the function f^2(x). Note that there are 3 fixed points (3 points where the graph of f^2(x) and the diagonal line y = x intersect); the one in the middle is the fixed point of f(x) and the two on either side are the points in the period 2 orbit. If you look carefully at the behavior of orbits near the two period two points you will see that the orbits approach a 'box' around either one or the other of the two points (the orbits do not converge to the two points). That is, the orbits are approaching a period 2 orbit of f^2(x), which means they are approaching a period 4 orbit of f(x). Of course, the fixed point p = 25/35 is unstable under iteration by f^2(x).
  • Now study the stability of the period 4 orbit. Use the applet with f^4(x). If you look closely you will see that there are 7 fixed points of f^4(x); two groups of 3 (one near the middle and one near the upper right), and one fixed point in between (this is the fixed point 25/35 of f(x)). The middle point in each group of 3 is one of the points of the period two orbit. If you make a sketch of the geometry of the graph of f^4(x) and the line y = x near these groups of 3 fixed points, you will see that the middle one is unstable (i.e., the graph of f^4(x) is above the diagonal on the right of the fixed point and below the diagonal on the left of the fixed point); this is the instability of the period 2 orbit. Starting with any point x_0, you will observe that the orbit converges to one of the 4 fixed points that are the outer points of the groups of 3 fixed points.
• (13) Find an orbit of T~ (the transformation on binary sequences) with prime period 3. Use your answer to find a periodic point of f(x) = 4x(1-x) with prime period 3. Use the applet Graphical Iteration to study the stability of this periodic orbit.

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