1) Your goal today is to make some connections between the web diagrams you drew for linear dynamical systems with the ones you're now making for the logistic model. First, summarize the different categories for the linear case and describe the various behaviors that are possible. Keep in mind that the most important thing about those web diagrams for the linear case was the SLOPE of the line.
2) The quadratic web diagrams are certainly more complicated, but if you look at small parts of them, you might recognize pictures that are similar to the linear ones. Focus in on the neighborhood near P* and see if the picture there looks familiar. Try to match it to one of the linear pictures.
3) For your nonconvergent diagrams (a>3), you might not have anything drawn in P*'s neighborhood because the sequence never got close to equilibrium. If so, then pick a P1 very close to P* and start webbing. What happens?
4) You can use the graphing calculator to help you focus in on P*'s neighborhood. Use the "y=" button to enter the quadratic function, with whatever value of parameter a you're currently considering. Let's start with a = 1.8. You will have to use an x in place of the Pn. Your y2 function would be just y=x. When you go to set up the window, use minimum values of 0 and maximum values of 1000. Set XScl and YScl both at 100. They need to be the same value.
Once you get the graph fitting nicely on the screen, hit ZOOM, then choose 5) ZSquare. This will redraw the picture with the axes scaled equally. You can tell it's working if your x and y tick marks look the same distance apart.5) Next, hit TRACE and use the arrow keys to walk along the function. Continue until you get as close as you can to P* and then hit ZOOM. Choose 2) Zoom In and hit ENTER to see a magnified picture of P*'s neighborhood. Trace again to get even closer to P*, then hit ENTER when you wish to zoom again. Repeat the procedure several times to increase the degree of magnification. What happens to the picture after you magnify it a bunch of times?
6) You should end up with a picture so magnified that the curve begins to look like a line. This is due to a marvelous property known as "local linearity." Certain types of curves look linear under extreme magnification. Why so marvelous, you ask? Well, you already know how linear dynamical systems behave. You can now take that knowledge of linear dynamical systems and apply it to the logistic model.
7) Check magnified pictures for a=1, 2, and 3. Use the graphs to explain why behavior changes at those values of a.
8) There is also a way to get the graphing calculator to draw web diagrams for you. Call me over to try that. It requires using the MODE switch set to "Seq" rather than "Func" and then defining the rule recursively with Un-1.
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