Homework week 4

Read 92-97 incl. steps 1-6, and 99-109 (through Lyapunov
exponents) & 359-369 (Lorenz equations).

Note on Dynamics: before plotting, use the command WHI to turn the background screen color to white. Later when you use the screen capture and print, you will not be faced with printing an almost solid blue picture.

Do the following problems.

1. You can use the results stated in
the steps on pages 92-97 without proving them> Use these results to make a
periodic table as on p.24 for the "cat" map

( 2 1 )

( 1 1 ) mod 1 in each
coordinate.

for periods 1 thru 6.

Hint: there are 1, 5, 16, and 45 fixed points for iterates 1,2,3,4; you have to figure out the numbers for 5 and 6.

2. (**Check Out Problems**) Prove Step 1 and 2 on p. 93.

3. If (x_1, x_2,
...) is an orbit for f, what is the orbit of f^k

starting with the same point x_1.

T 3.1

T 3.2 Hint: "Most
bounded orbits" converge to a fixed point.

Comp. Exper 3.1
p.109.

Note: Whenever a problem says something like "write a program"

(as this one does) you can instead use Dynamics or any other

program. Using Dynamics you can use BIFS to plot the picture on the

screen -- if you have first set L (the number of exponents) to be

1. If L is the default value of 0, a regular bifurcation diagram is

plotted.) See p. 19 for a "regular" bifurcation diagram of the

logistic map.