Homework 5

Part A. Question 15 from Part 5 of Dynamical Systems

15. In Chaos Lab #3, somewhere between 2 and 3.2, the orbit diagram curve splits into two branches. How does this relate to what you saw in Chaos Lab #2? Why are there two branches now? What do they correspond to? (Hint: What was the eventual behavior of points in this region?)

    At about lambda = 3, the diagram splits. The part to the left of the split (lambda < 3) represents the single attracting fixed point for those lambda values. The part to the immediate right of the split (lambda > 3) represents the period two orbital points that the those lambda values give. Each branch represents one of the two periodic points. Further to the right the branches split again.

Part B. Question 18 and 20 from General Dynamical Systems Questions of Dynamical Systems

18. In Chaos Lab #2, pick "Quadratic" under the Function menu. Investigate the behavior of the iterates for parameters in a small interval around c = -3/4. Discuss the bifurcation that occurs in the quadratic family f(x) = x² + c for c = -3/4.

    For parameter values less than -.75, the function has an orbit of at least period two. For parameter values greater than -.75, there is only one attracting fixed point.

    At exactly c = -.75, there is an attracting fixed point at -.5 and a repelling fixed point at 1.5. Initial values less than -1.5 or greater than 1.5 go off to positive infinity. Values in between go to the fixed point 0.5.

    At a point slightly to the left of the bifurcation, say at c = -.77, initial values less than -1.51 or greater than 1.51 go off to infinity. Values in between go into a period 2 orbit of -.65 and -.35. There are repelling fixed points at .51 and 1.51.

20. Using Chaos Lab #3, compare the bifurcation diagrams for the logistic family f(x) = lambda * x * (1 - x) and for the quadratic family f(x) = x² + c. Do you see any similarities? Discuss them and point out the differences.

    First of all I notice that for the quadratic family, the bifurcation diagram gets more dense as the parameter gets smaller, while for the logistic family, the bifurcation gets more dense as the parameter gets larger. Both diagrams split with the same pattern: 1, 2, 4, 8, etc.

    What is the range of lambda values in the diagram for the logistic family? What is the range of c values in the diagram for the quadratic map?

    In the logistic family, lambda ranges from 0 to 4. In the quadratic map, c ranges from -2 to -.25 in the diagram.

    For lambda = 4, the graph of the logistic map is special. The interval [0,1/2] is mapped entirely onto [0,1] and also the interval [1/2,1] is mapped entirely onto [0,1]. The logistic map covers the interval [0,1] twice. Notice that this is not true for smaller (positive) values of lambda.

    Find the value for c such that the graph of f(x) = x² + c covers the interval [-2,2] exactly twice, as described above for the logistic map.

    That value for c is -2. The graph of the parabola is always symmetry across the X-axis, so it's always covering some interval twice. Setting c to -2 shifts the curve to just the right position so that the bottom touches the line x = -2 and the curve also runs through the two points (-2,2) and (2,2). Thus, the interval [0,-2] is mapped entirely onto [-2,2] and also the interval [0,2] is mapped entirely onto [-2,2]. At c = -2, the quadratic map covers the interval [-2,2] twice.

Part C. Question 1 from Combining Symmetries

First I created 3 general categories:

1. Decals. Patterns that have symmetry only around a single point.

   (3a, 3b, 3c, 3d, and possibly 12)

2. Borders. Patterns that have symmetry only down a line.

   (2)

3. Wallpaper. Patterns that have symmetry only across a plane.

   (1, 4-10, and possibly 12)

Pattern #11 and #3e don't fit into any of these categories. For pattern #11, I don't see any precise symmetry, though I do see approximate symmetry. If I had to classify it into one of my three categories, I would pick Decals.

I'm not sure how to deal with pattern #11. It does have symmetry, but not the same way the other patterns do, so I chose to leave it out.

#12 fits into Decals, if you assume that the whole theoretical pattern is displayed in the picture we see. But if you assume that the pattern continues (as a fractal) in all directions, theoretically you could cover the entire plane with the pattern, so it would fit into the Wallpaper category.

When would it be useful to organize pictures according to your system of classification?

This classification is useful for deciding which patterns you might want to cover your walls with, which ones will work for making a border, and which ones will work as little decals everywhere.

It still doesn't tell about what types of symmetry are within the pattern. Is it reflection? rotation? translation?

To make the classification more useful, I have made an additional set of categories:

1. Translation. (1-2 and 4-10)
2. Rotation. (1-2, 3a-d, 4-10 and 12)
3. Reflection. (1-2, 3c, 3d, 5-7, 12 and possibly 11)
4. Dihedral. (1, 3c, 3d, 5, 7 and 12)
5. Dilation. (11 and 12)
6. Glide Reflection. (2 and 10)

A pattern belongs to a category if it contains an example of that type of symmetry. Most of the patterns given fit into more than one category.

This additional classification tells us much more about the type of symmetry in the picture. However, we still don't know how multiple symmetries in a pattern are arranged. We don't know about how many folds of symmetry are at a point of rotation. We don't know how many unique points or lines of each type of symmetry there are. For this we need to subcategorize (say rotation gets split into 2-fold rotation, 3-fold rotation, 4-fold rotation, etc.). Then we need to find all possible combinations of categories that a pattern can fit into. Once all combinations of categories have been found, new categories can be made for each possible combination.

This would tell us a lot about the type of symmetry each pattern has as well as whether it could be used as a decal, a border or wallpaper.

However, it still doesn't tell us what color scheme it would fit nicely into.

Part D. Describe your proposed final project in an outline or in a detailed flowchart.

See http://www.geom.umn.edu/~lori/math5337/finalproj/outline.html