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final2 - the order 1 1 1 1 1 1 1 1 1 1 Can you guess...

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Applied Dynamical Systems - ME215A Fall 2010 Take-Home Final Part II - due Wednesday December 8 1. (10 pts) Write a Matlab program to generate a “classic” bifurcation diagram for the logistic map x n +1 = μx n (1 - x n ) . (1) for 2 . 8 < μ < 4. 2. (25 pts total) Consider Equation (1) for μ = 4. (a) (5 pts) This map is more easily studied when written in terms of y given by x = sin 2 ( πy ). Show that the map becomes y n +1 = 2 y n mod 1 . (2) Show that this maps the unit interval onto itself. (b) (5 pts) Divide the interval (0 , 1) into the two parts (0 , 1 / 2) and (1 / 2 , 1), and label these cells 0 and 1, respectively. Suppose y 0 = 3 / 7. Find the relationship between the order of visits to cells 0 and 1 generated by successive iterations starting at this y 0 and the coefficients a j in the expansion y 0 = j =1 a j 2 j . (3) (c) (5 pts) For some initial condition y 0 , suppose the sequence of visits to the cells is in
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Unformatted text preview: the order , , 1 , , , 1 , , , , , 1 , 1 , 1 , 1 , 1 , 1 , , 1 , 1 , , · · · Can you guess what choice of y was used? How sure can you be? (d) (5 pts) For a random y ∈ [0 , 1], are you more likely to get a repeating or a non-repeating sequence of 0’s and 1’s? (e) (5 pts) Consider a perturbation δy of y . Show that for almost all y , | δy n | = e n Λ | δy | , and find the Liapunov exponent Λ. 3. (10 pts) Write a program to simulate the Lorenz equations ˙ X =-PX + PY, ˙ Y =-XZ + rX-Y, ˙ Z = XY-bZ. For the “standard parameters” P = 10, b = 8 / 3, r = 28, convince yourself that a chaotic attractor exists which displays sensitive dependence on initial conditions. 1...
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