Consider Xn+1 = f (xn)

Question

(1) 4. Consider Xn+1 = f (xn) so if € (0, 3) where f : [0, 1] → [0, 1], x H if x = [1,1] and c € (0,2). cx с — cx (a) Find the fixed points of eq. 1 in terms of parameter c. (4 marks] (b) Investigate whether the fixed points identified in (a) are locally asymptotically stable. [7 marks] (C) Let 1 < c < 2. Show that 0, 14c 1fc2 and are period-two points of eq. 1. [7 marks] (d) For g e C™ let h(x) = g"(x), n E N, be the n-fold composition of g with itself. Show that +' 1+ 1+c in-1 n- 1'(x) = 4 (g"-(x))! (g"-22)....!'(x), 1 d dr where [2 marks] (e) Investigate whether the non-trivial period-two points identified in (c) are locally asymptotically stable. [5 marks]

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