Week #3      Read 61-91

 

Exercises

     T2.5 p.72 First answer the following: If (x,y) is a point of a period two orbit for the Henon Map, must the other point be (y,x)?

     E2.1 p.98

Checkout     T2.8 p.77

Checkout     E2.3 p.98

Problem A. An exercise not in the book:

Let f(x) = 1 - 3x for x in [0, 1/3] and

             = x/2 - 1/6 on [1/3, 1]

[Note f is continuous and its minimum value is 0 at x = 1/3.]

a. Divide [0,1] into 2 intervals intelligently and show the transition graph.

b. Create the periodic table for periods <= 4.

Note: It is important to check for periodic orbits that lie entirely on

the intersecting ends of the intervals you choose. Each such point lies

in 2 intervals simultaneously so they have more than one itinerary.

 

Some software for use in the class (allow time for this not to work smoothly and still be able to recover L).

Get the program from    http://yorke.umd.edu/dynamics

            It has a number of text files describing how to do certain tasks.  I will hand out in class an instruction booklet.

 

            For PCs running Windows 2000 or XP, grab  DynamicsForWinXP.zip

(click on the name, not on the icon). It is a zip file so you have to extract all the files in the collection.

            For UNIX, select the Unix option.

            For Macs, select the Macintosh option.

 

            For older PCs, there is a program in http://yorke.umd.edu/dynamics/class

run INSTALL.EXE

           

            Instruction on how to use this program can be retrieved from the Manual option, a moderately short manual that describes the essentials of the program.

 

will come later, but now I want to establish that you can get the program.

 

The following problems will make use of the program.

    Explain your answers.

     1. Computer assignment: Study the effects of changing parameters in the pendulum map (P in Dynamics).

The command P selects the pendulum from the collection of processes.

Command T allows you to plot a trajectory. Does the resulting trajectory look like it is periodic (aside from the first few iterates?

Try command C (clear screen) while the trajectory is running.

 

Pick pairs of parameters (C1, rho), where C1 is in [0,1] and rho is in [0,3].

Command C1 allows you to set the parameter C1; same for rho.

A grid of a few points is a good way to start. For each choice of C1 and rho, plot a trajectory (the initial point does not matter) on your computer screen.

Create a separate plot by hand (or with another graphics program if you wish) that indicates that behavior of the trajectory. Your plot should have axes C1 and rho. If the trajectory goes to a chaotic attractor, then plot an X at (C1,rho), otherwise plot an O. For the purposes of this assignment, we will say that trajectory goes to a chaotic attractor if the attracting set has lots of points, say, if it appears to have more than 100 points. Using Dynamics, command T allows you to plot a trajectory. It just keeps plotting until you stop it (or until it has plotted 10^8 points). As it continues to plot, the Clear command C erases what has been plotted so far, and plotting continues. You can pick more value of (C1,rho) to make clearer the boundary between chaos and stable periodic orbits.

 

     2. Computer Assignment based on Lab Visit 1, Figure 1.17. Investigate the region of the graph where the parameter (denoted rho below) is greater than .95. Find out what the nature of the attractor is by plotting it. I recommend using Dynamics, choosing "own" for the process. To simplify the task, I am going to provide some of the input at the bottom of this file. Use the BIFS function in the Bifurcation Diagram Menu (BIFM) under the Numerical Explorations Menu (NEM). First use the BIFR function to set the range the parameter rho to go from .95 to 1. For Windows users the TD or PCX functions can be used to save pictures. Other options are listed in the Disk Menu (DM).

 

     3a. Writing F for the Henon Map, (that is, F = f_a,b in Eqn 2.27 on

p. 70), show that the determinant of the Jacobian of the Henon Map F is

constant (independent of x and y) for each a,b.

     b. We say F is "area decreasing" if for any set Q with finite area > 0,

we have

     Area(F(Q)) < Area(Q)

(which holds if |det DF| < 1 ). For which a,b is F area decreasing?

 

     Comp. Example (CE) 2.2 p.76

 

 

-------------- Lab Visit 1 ---------------

After starting Dynamics type "OWN", and create the map below. See p 40 of our book.

 

For the top window, write something like:

"Lab visit 1" (without the quotes)

 

You can wipe out an entire window by entering Control W

 

For the second window, enter something a lot like:

x := 7.48 * z * exp(-.009*z -.012*x)  ! Larvae

y := x * (1-.267)                   ! Pupae

z := y * exp(-.004*z) + z * (1-rho) ! Adults

 

For the third window (initialization)

x := 10

y := 20

z := 30

x_lower := 0       x_upper :=  200 ! X scale

y_lower := 0       y_upper :=  200 ! Y scale

rho := .97

 

Empty out the 4th window by entering Control W

 

When this is done, hit the key F1. If it complains, you have done

something wrong. The cursor will be pointing somewhere near the problem.

=====================================================

For Unix:

Use your favorite web browser (e.g., Netscape or Internet Explorer) and

type in the following URL:

             ftp://jims.umd.edu/pub/unix

See the readme file for instructions.