Problem Sheet 8 MT1100 From Euclid to Mandelbrot Hand in your solutions, with this coversheet, for feedback on Thursday 6 December 2018 YOUR FEEDBACK TO THE LECTURER (e.g., what was hard, interesting, fun, etc. this week): FEEDBACK ON YOUR SOLUTIONS FROM THE MARKER: 1
1. The graph y = f(x) of a function f : [0, 1] -> [0, 1] is given below, plotted against the line y = x. How many fixed points does f have? Which if any of these fixed points are stable? 1.0} 0.8 0.6 A 0.4 0.2 0.2 0.4 0.6 0.8 1.0 2. Let a > 0 be a constant and let f : R -> R be defined by f(x) = x3 + ax. (i) What are the fixed points of f? Justify your answer. [There are either one or three fixed points, depending on a. Mathematica (sketching y = f(x) and y = x for a few values of a) might help you see what happens, though I don't want you to submit Mathematica output for this question.] (ii) Compute the stability of the fixed points when a ¥ 1. (iii) A fixed point u is superstable when f'(u) = 0. For which values of a does f have superstable fixed points? (iv) Plot the positions of the fixed points versus a, as a varies between 0 and 2. [This plot is called a pitchfork diagram. Google "pitchfork diagram" to get some hints as to what your diagram should look like.] 3. Let f : R -> R be given by f (x) = 2 - (x-1)2. (i) What are the fixed points of f? Justify your answer. (ii) Write down an explicit formula for f(2)(x). Find the fixed points of f(2), and hence find the limit cycles of period 2 for f. (iii) Which of these fixed points or limit cycles are stable? Justify your answers. 4. Let g : R -> R be defined by g(x) = (x - 2)2 + 2. (a) Prove, by induction on n, that g(n) (s) = (s - 2)2" + 2. 2
(b) Find the Julia set Jg. Justify your answer. You can use the following fact without proof: For c E R with c ? 0, lim n- 00 C2" = 3 0 when c < 1, 1 when c = 1, 00 when c > 1.