Seminar assignments - Assignment 1 - 6 minus 5 PDF

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MAST331 FALL 2016 Assignment 6 Due date October 25 1. Problem 1. Prove that f (x) = x3 (a) has sensitive dependence at −1; (b) has no sensitive dependence at 0 2. Problem 2. p.90 N2. Show that the baker’s function B has sensitive dependence. 3. Problem 3. p.90 N3. (a) Find the Lyapunov exponent associated with the baker’s function B , and show that it is constant (where defined). (b) Find the Lyapunov exponent of B 2 .

4. Problem 4. p.90 N4. Let Bµ (x) =

½

2µx µ(2x − 1)

for 0 ≤ x ≤ 1/2 , for 1/2 < x ≤ 1

where 0 < µ < 1. (a) Find the Lyapunov exponent of Bµ . (b) Determine the values of µ for which Bµ has sensitive dependence on initial conditions. (c) What is happen if µ = 1/2. 5. Problem 5. p.90 N6. Suppose that f is differentiable on the interval J . (a) Suppose that the iterates of x are eventually the fixed point p. Show that λ(x) = ln |f ′ (p)|. (b) Suppose that the iterates of x are eventually the the cycle {p, q, r}. Find a formula for λ(x). 6. Problem 6. Write the binary expansion as an infinite sequence 3 5 7 , (ii) 16 , (iii) 12 . (i) 11 7. Problem 7. Given the logistic map Q4 (x) = 4x(1 − x) and x0 = LLRRRRLRRLR. (a) Is x0 less than, equal to, or greater than 1/2? 1

(b) Is Q8µ (x0 ) less than, equal to, or greater than 1/2? (c) What is the length of the interval LLLLL? 8. Problem 8. Consider the following function f which acts on infinite sequence (”words”) made up of the letters L and R according to the following rules: (a) If the word begins with L, drop the initial L and swap R’s and L’s in the remaining portion. (b) If the word begins with R, just drop the initial R. Find the fixed points and period-2 points for this function.  0 ≤ x < 13  3x, 3x − 1, 1 ≤ x < 23 9. Bonus Problem. Prove that the map f (x) =  3x − 2, 23 ≤x≤1 3 is chaotic.

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