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

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern