Homework 3

Note that this is a computer week. 1,2,5 should have a computer graph
with the numerical answer.

1. page 88 #3,#4

2. page 89 #10

3. Page 91 #23

4. Page 129 #5,#6

5. Using a combination of algebra, the bifurcation diagram for the
logistics equation, and experimenting with the program which iterates
values of points using the logistics equation for various parameters
estimate the bifurcation   point where a stable orbit of period 2
goes through period doubling to become a stable orbit of period 4.
(page 130 #18).

Also, give your best estimate for the "onset of chaos". Don't print out
all these graphs!

The logistics diagram will be calculated by matlab if you type in
logistic. It takes awhile, as the program actually calculates the diagram
each time.  To see a graph of an  iteration given a single parameter
\mu = 3.6, type logistic1(3.6).

Extra credit:

1. Find the solution to the equation on the interval [0, infinity)

                y' + y =  g(t);   y(0) = 0.

              0   for  0 < t < 1
 Here g(t) =  1  {for  1 < t < 2
              0   for  2 < t.


Make a sketch of both the input function g(t)and your solution.


2. Find the linearization of the equation  y' =  y(-1 +4y - 3y^2)

about each of the equilibrium points.