Do calculations by hand unless otherwise specified. You may of course use Maple to check your work if you wish. Useful commands: solve, evalf, expand, factor. Type ?factor, etc. for help on commands you don't know, and it's often quickest to scroll down to the bottom for the examples, rather than reading the explanation.
(1) Sketch (justqualitatively accurate) cobweb diagrams for the logistic map f(x) = rx(1-x), that show a period-2 orbit, a period-3 orbit, and a period-4 orbit. (In each case you'll have to choose the "r-value" so that the map does have an orbit of the specified period.) NOTE: no calculations are required in this problem! Hint: It's a lot easier to draw the cobweb first and then the map than vice-versa. (2) Consider the map f(x) = 2x2 - 5x. (a) Find all the fixed points, i.e. points such that f(x)=x (or, to put it another way, f(x)-x=0). (b) For each fixed point, by considering the value of the derivative of f, determine if it is attracting or repelling, and whether orbits approach or move away in a monotonic or alternating fashion. (Explain your calculations and reasoning.) (c) Find all the period-2 points of the map, i.e. points such that f2(x)=x and that are NOT fixed points of f. HINT: The period-2 condition f2(x) - x =0 is a quartic equation with 4 roots. Two of these are the fixed points of f (which we're not interested in for this part). You can reduce the quartic to a quadratic by using long division to divide out the factors (x - fixed_point_1) and (x - fixed_point_2). The two roots of the remaining quadratic are the two points on the period-2 orbit. (d) Using the result of (c), and considering the derivative of f2, determine if the period-2 orbit is attracting or repelling. (Explain your calculations and reasoning.) (e) Sketch the map f, and mark the fixed points and the period-2 orbit. (3) (a) Sketch a cobweb diagram showing the orbit in the Mohammed-Awel exponential map (see handout) corresponding to the reclamation of abandoned farmland by forest, as we see in many places in NY southwest of Buffalo. (The quantity being modelled is the number of trees.) (b) Can you identify at least one shortcoming of this model as applied to this reclamation process? (4) In the last sentence of Ch. 3 of the Origin of Species, Darwin seems to try to take the harsh edge off the conclusions he has developed in the preceding. What do you think of this last sentence? Do you buy it?