Due March 23

Please do not postpone the assignment until the last minute! Remember you also need to identify a Challenge problem (one for undergraduates and two for graduate students) to work before the end of the semester. * marks problems for graduate students (extra credit for undergraduates).

1. Reading. By popular demand, we will move on to discuss continuous problems (ODEs). We will cover all of Chapter 7; in fact, much of it should be review, particularly for those who had the modelng course. I will survey the chapter and point out the highlights in class. Please read Chapter 7 in its entirety.

   Please do the following problems.

2. Preservation of containment. A key result in using transition graphs is that maps preserve set containment. Prove that if f is a continuous map on the real line, then implies that . (Keep it short and use the definition of the image of a set: where .)
3. Periodic points. Consider the graph of a map given at the end of the asignment. Use a transition graph to determine the order of all possible periodic points of the map.
4. Period three and chaos. Consider the transition graph that allows only the path , and hence a period three orbit. Does period three imply chaos in this case?
5. *You knew it was coming. Derive the conjugate map for the tent map T and the logistic map . First, try every approach you can think of; perhaps you can devise a method different than the one below.
   1. Consider the interval and use the definition of the conjugate map, G(C(x))=C(T(x)). Write the functional equation for C(x) that results.
   2. There is no standard approach to solving functional equations. One can look for special forms of the solution that exploit apparent properties of the solution (for example, polynomials, power series, rational functions, exponentials, Fourier series). Notice that the functional equation for C(x) involves and C(2x), where now means C squared, not C composed with itself. (If your equation doesn't have these terms, return to part (a).) What common family of functions has the property that ?
   3. Given the insight from part (b), you might be tempted to look for solutions of the form , where and are to be determined. This looks like one term of a Fourier series and you will see that it's easier to work with than sines and cosines. In order to make C(x) real-valued, it would be best to use , where is the complex conjugate of a. Show that when this trial solution is substituted into the functional equation, no useful information can be obtained about a and k.
   4. Why would real exponentials, , make matters even worse than complex exponentials ?
   5. The approach in part (c) almost worked and suggests that the trial solution may need another term. We have included sines and cosines of arbitrary frequency k, so a qualitatively different term is a constant. Try a trial solution of the form

      

      Now determine a and b. Can k be determined?

   6. We know that C must be one-to-one and map [0,1] onto [0,1], so it is reasonable to ask that C(0)=0 and C(1)=1. Furthermore, to impose some symmetry, let's ask that . What constraints do these conditions place on k? Find the most general form of C(x).
6. * Explicit formula for the logistic map. Recall that in Chapter 1, problem 1.15, an explicit formula was given for the sequences generated by the logistic map. Use the conjugate map to derive this formula for the case that .
7. Phase plane problems. Please do problem 7.2 (Chapter 7).
8. Phase plane problems. Please do problem 7.5(c). (The sketch can be fairly rough.)
9. Predator-prey model. Please do problem 7.13(a).