Chap 4 Homework
4: 8, 10, 14, 17, 18, 19, Bonus 22
8
The idea here is to use the same format as figure 4-5, so that
two panels of a stacked plot are shown with the same x-axis (which
is x), and the upper plot has U(x) on the y-axis, and the lower
plot has dx/dt (or x-dot, or v) on the y-axis. Then the limits of
motion along x-axis are the same in both plots.
The period can be calculated from the position as a function of time,
where symmetry suggests that 0 to its max-position will be 1/4 of the
period. Since...
10
This one uses a bit of Matlab code.
14
This can be done most easily in Excel. Each of these
columns has the same formula, repeated n-times. Something
like "= 2.5 * A2 * (1 - A2^2)". The first column uses
a starting point of 0.900000, and the second 0.9000001.
The third column calculates the relative change
"=ABS((A2 - B2)/B2)". A scatterplot finishes off the problem.
(Thanks to Melissa Brown for the graphic)
17
This is also done very easily in Excel, as are most of
these recursion sorts of problems. In this case, the
formula that is repeated is a bit more complicated, something
like: "=2 * $A1 * (A2*(A2<0.5,0,1) + (1-A2)*(A2>.5,0,1))", where
$A1 is the location of α. Note how we used a logical
test to automatically choose the correct part of the tent map.
18
With the results from 17 above, we use the scatter-plot option in
Excel to get the bifurcation map plotted.
19
