Math 215 - Homework assignments

These assignments can be done by groups of up to three people. Each should result in a printed paper with computer generated illustrations, some of whihc may be animated.

I am open to suggestions for alternative experiments for each of these topics.  Contact me a week or more before an assignment is due if you want to try something different.

• Assignment 1, due Wednesday January 28, 1998
• Assignment 2, due Wednesday February 11, 1998
• Assignment 3 (double credit), due Wednesday March 4, 1998
• Assignment 4 (double credit), due Wednesday April 1, 1998
• Assignment 5, due Wednesday April 15, 1998
• Assignment 6, due Wednesday April 29, 1998
• Assignment 7, due (double credit) Wednesday May 6, 1998

In this and the next assignment we will make use of the SEQUENCES AND DISCRETE DYNAMICS section of PWS OnLine Series - Calculus Modules OnLine. It is an online textbook with exercises (some of which have answers provided) and Mathematica Notebooks which you can download and use.  When you download an Mathematica Notebook, you will be asked if you want to convert it from version 2.2 to 3.0, and you should click on the Convert option. (See the Harkness instructions file from more information. )You should save the Notebooks in your home directory for future use.

• Read Section 1.5 of Gulick.
   
• Read the Learn sections entitled
• Quickstart
• Population Models -- Exponential Models
• Discrete Logistic Models
• Read the Discrete Logistic Models Stretch section and do the first problem, using at least 5 different values of a. You can do this by editing the notebook provided, in particular by changing the line

f[p_]:=3.4 (1-0.001 p) p

in the first cell to

a=3.4; f[p_]:= a (1-0.001 p) p

and then experimenting with various values of a.

 

• Read the Cobweb Diagram Learn section and download the Mathematica Notebook provided.  When you do so you will be asked if you want to create an auto save a package that comes with it and you should say yes. Use the Cobweb procedure there  draw a cobweb diagram showing

p(1), p(2), ... p(20)

for the system

p(n + 1) = a(1 - .001*p(n))p(n)

for the following values of a and p(1):

• a = 2.5, p(1) = 500
• a = 2.5, p(1) = 50
• a = 3.25, p(1) = 500
• a = 3.25, p(1) = 100
• a = 3.54, p(1) = 500
• a = 3.83201, p(1) = 400
• a = 3.9603, p(1) = 50

For the first of these you will need to replace the first line in the cell by

a=2.5; f[p_]:=a* (1- .001*p) p

and the second line by

CobWeb[f,1000,500,20]

• Invariance under scaling. Replace the code above by

a=2.5; k=500; f[p_]:=a*(1- p/k) p;
CobWeb[f, k, 0.5*k,  20]

and experiment with varying the value of k.  Describe what happens and explain why.

• Iterating the logistic function f. Add the following code between the first and second lines above.

f2[p_] := f[f[p]];
f3[p_] := f[f[f[p]]];

Then replace f by f2 and f3 in Cobweb for various values of a and p(1) as above.  Report on what you find.

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Assignment 2, due 2:00 Wednesday February 11, 1998:

• Read Sections 1.1-1.4 and 1.6-1.7 of Gulick.
• Read the Cycles Learn section and download the notebook provided there. Remember to convert it to standard form and to create the auto save package that comes with it. Evaluate the cells as instructed. In the second cell change the constant 3.54 to 4.0 and compare the graphical result with the numerical result of the first cell.  Do the two open ended assignments there, limiting your attention to cycles with periods of 5 or less.  Here are some hints about how to look for cycles:
• It helps to have a definition of the nth iterate of the function f.  You can use

f[x_] := r*x*(1-x);
F[1, x_] := f[x];
F[n_, x_] := f[F[n-1, x]]

Then you can replace f[f[x]] by F[n, x] in the first two cells.

• Cycles can be found either numerically as in the first cell or graphically as in the second cell. Describe and explain how the number of solutions to the equation F[n,x]==x varies with n. You can get Mathematica to count solutions for you by typing Length[Solve[your equation]]. Note that some of these solutions are complex.
• You can use the graphical method of the second cell to find n-cycles by replacing f[f[x]] by F[n, x].  For n=3, describe how that value of the multiplicative constant r (which can range from 0 to 4) effects the number of real solutions. At which values of a does this number change? You can get more precise information about this by plotting over smaller intervals or by using the numerical method above.  Try to give a mathematical proof of your answer.
• Finding critical n-cycles. The function f[x] above has a criticial point (i.e. a point where the derivative is 0) at x = .5.  For theoretical purposes it is useful to study how the iterates of the function behave starting at the critical point for various values of n. For this purpose define

g[r_, x_] := r*x*(1-x);
G[1, r_, x_] := g[r, x];
G[n_, r_, x_] := g[r, G[n-1, r, x]]

Look for solutions to the equation G[n, r, 1/2] == 1/2 for n = 2, 3, 4, and 5. How many real values of r between 0 and 4 are there for each of these n?  How many of them give attractive cycles?  You can solve this numerically with Solve or graphically with

Plot[{.5, G[n, r, .5]},{r, 0, 4}]

• Behavior near the chaotic point r = 4. Experiment with the following command for various values of m.

m=6;
Do[k=10*(2m-1)^2;
   Plot[{G[n, r, .5], .5},{r, 4-k/4^n, 4}],
   {n, m-1, m+3}]

What does this experiment suggest? Can you find a more familiar function of r which approximates G[n, r, .5] for r slightly less than 4?

• Extra credit: Behavior of the criticial orbit for r = 1. Look at the curve produced by the code

f[x_] := r*x*(1-x); F[0,x_] := x; F[n_, x_] := f[F[n-1, x]];

r=1;Points = {}; Do[AppendTo[Points, {n,F[n,.5]}],{n,1,200}]; ListPlot[Points, PlotJoined -> True]

Find a rough formula for this curve, i.e., an approximation for F[n,.5] as a function of n. Give analytic and/or graphical evidence to support your assertion.

Assignment 3 (double credit), due 2:00 Wednesday March 4, 1998:

In this open ended assignment we begin exploring the Mandelbrot set M. Online references include the Encyclopedia of the Mandelbrot Set by Robert P. Munafo.  There are many ways to do this assignment, and the list of suggestions below is far from complete. See also various questions (written in blue) posed in Lectures 7, 8 and 9.

I am open to suggestions for alternative projects. Each of you contact me by Monday, February 23,  to tell me what you plan to do.

You should read Section 4.3 of Gulick. There you will find an introduction to complex numbers and a proof of the Escape Criterion (Theorem 5.4 of Lecture 5) in Theorems 4.9 and 4.15.

• Imbedded Julia sets. Read what Munafo says about imbedded Julia sets and use Fractint to look at his examples.  Compare the actual Julia sets associated with these points M with the imbedded Julia sets.Find some more examples yourself.  What can you say about the dynamics associated with these points?
• Self-similarity and baby M's around pre-periodic points.
• Buds on the main cardioid.
• Why are there sprials in seahorse valley?
• The Fibonacci sequence in the Mandelbrot set.
• Feigenbaum points in the Mandelbrot set.
• Lei's theorem: similarty between neighborhoods of pre-periodic points and their Julia sets.
• Distortions in baby M's.
• Julia sets for baby M's.
• Study chaos in the logistic equation.
• Read the Chaos Learn section.
• Use the notebook of the previous assignment to study the case r = 4, the chaotic case.  Are there any real attractive cycles?  Why or why not?
• Unpredictability or the butterfly effect.  Try varying pop[1] by a small amount and see how this change effects pop[10] or pop[50]. Describe the results of this experiment.
• Write some Mathematica code to see how evenly distributed the values of pop[i] are (between 0 and 1000) over several hundred generations for various values of pop[1] and report your findings.

Assignment 4 (double credit), due 2:00 Wednesday April 1, 1998:

For this assignment you may do another of the project suggested for the previous assignment or study some other fractals similar to the Mandelbrot set. As always I am open to suggestions for alternative projects. Consult the lecture notes for more information on the topics listed below. Consider the posssibility of illustrating your report with an animation.

• Fractals associated with Newton's method.
• The Curry-Garnett-Sullivan experiment.
• Mandelbrot and Julia sets for cubic  and higher degree polynomials.
• Main bodies of Mandelbrot sets for polynomial functions.
• Mandelbrot and Julia sets for transcendental functions.
• The garland fractal.

Assignment 5, due 2:00 Wednesday April 15, 1998:

This assignment concerns iterated functions systems. First you should read the following.

• Pages 127-139 of Gulick, Review of matrices.
• Pages 220-239 of Gulick, Iterated function systems.
• Metric spaces and iterated function systems; hard copy is available.

You should also familiarize yourself with Fractint's ifs and lsystem modes (there is an online tutorial for this) and with the program FDesign in the Math 215 program group.

The following problems should be done individually, not in groups.

• #2, 4, 6, 8 of Section 4.4 of Gulick.
• For each of the following images (which were produced by Fractint), write formulas for the affine maps that generate it. You can assume whatever you like about the coordinate system in each picture.

   

   

Homework images A, B, C, and D.

Assignment 6, due 2:00 Wednesday April 29, 1998:

This assignment concerns fractal dimension.

• Read Section 4.1 of Gulick.

The following problems should be done individually, not in groups. Find the fractal dimension of the following images, giving reasons for your answer.

1. A, B and D from the previous assignment.
2. The images in Figures 4.28 and 4.29 of Gulick.
3. The Menger sponge, a 3-dimensional IFS attractor shown in Figure 4.7 of Gulick.
4. Each of the following IFS attarctors. In the two on the left, it is possible to determine the maps precisely assuming that each of the two affine maps is conformal (angle preserving) and that the two images meet at a single point in the upper attarctor and at exactly two points in the lower one.
   
   

   

  

Homework images E, F, G, and H.

Make an animation of Mandelbrot/Julia sets, IFS attractors or other images of your choice. Your animation should have at least 100 frames of 640x480 pixels or larger, and the motion should be reasonably smooth. Pay attention to color so it is pleasing to the eye. Your paper should indicate clearly how the animation was made, including programs where appropriate. It should also explain what mathematical phenomonon is being illustrated by the animation.

You should confer with me about your plans no later than Friday, May 1. Here are some possible topics.

• Fractint has some 3-dimensional IFS attractors, e.g. the Sierpinski tetrahedron, and a mechanism for viewing one from various angles. Use this to illustrate a 3 dimensional attractor by rotating it, or better yet, by flying through it.
• Animate a sequence of IFS attactors by varying the parameters that define them in some interesting way. Here are two samples, which you will have to download and save before viewing: ifs5x.gif and ifs_sqx.gif. In each case there is a key file (ifs5x.key and ifs_sqx.key) and an ifs file (ifs5.ifs and ifs_sq.ifs) needed to make the images.
• Animate a series of distortions of the Mandelbrot set using linear fractional transformations, as explained at the end of Lecture 12.
• Animate a sequence of Julia sets obtained by moving the parameter c around M in some interesting way.
• Animate a sequence of zooms into the Mandelbrot set.

This page was last revised on April 29, 1998.

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