# soln10 - ECE320 Solution Notes 10 Cornell University Spring...

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ECE320 Solution Notes 10 Spring 2006 Cornell University T.L.Fine 1. (a) Using graphical means for a = 1 + 8, and considering the thrice it- erated logistic logistic ( logistic ( logistic ( x ))), determine approximately the fixed points of period 3 of the logistic iteration. Submit your graph. See Figure 1. Estimates from this graph place the three fixed points at about .15, .51, .96. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Thrice Iterated Logistic Function Figure 1: (b) Select a = 3 . 8285 in the logistic to display period 3 behavior and calculate the first 500 terms. Use the Matlab command hist to help you analyze this data, can you verify that this sequence eventually nearly achieves the fixed points calculated in (a)? 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120 140 160 180 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 70 Figure 2: 1

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The left histogram in Figure 2 shows the full 500 iterates and the rarer values occur early in the iteration where it is still dependent upon its initial conditions. The right histogram in Figure 2 shows only the last 200 iterates and better reveals the asymptotic period 3 behavior. Using the rightmost histogram suggests that there is clustering of values about the points .2, .51, .92. As our chosen a is slightly larger than 1 + 8 and our estimates come from eyeing graphs, there is reasonable agreement. Note that the iteration does not settle down to precisely the fixed points. This remains true even if one extends the sequence.
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• Spring '06
• FINE
• Cornell University, limit cycle, Iterated function

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