After the weird results we observed yesterday with Phi and the golden ratio, and its ubiquity in nature, we doubled-down by showing that we can use Phi to derive an explicit formula for the Fibonacci Sequence. Yes, the most famous recursively defined set of integers can be derived explicitly using an irrational number. Wild. If you missed it, check out this video from Mathologer.
I also demonstrated another Fractal-generating software: IFS Construction Kit. Selecting “rotation” mode in this software allows you to define a piecewise-linear function in complex numbers, one that allows a greater amount of flexibility by using different dilation or rotation factors for the real and imaginary components of the complex number you are iterating. Use the default settings when the software opens up, and you get a very familiar image…
This break, please think about the back of Complex Paint Worksheet 1. It is a summary of the work we’ve been doing with complex linear functions, and completing this will allow us to move on to the final stage of the course: iterations of complex quadratics.
We did some more work collecting information for the Catalog of Behavior for the Logistic Map function we introduced on Monday, getting a clearer idea on the changes in behavior as a increases towards 4.0. The patterns we observed raised a few questions:
- Somewhere within the interval 3.0 < a < 3.1, the single fixed point attractor “bifurcates” (i.e., splits) into a 2 cycle. Where exactly does this happen?
- Later, the 2-cycle becomes a 4-cycle. Do all points of the cycle bifurcate simultaneously, or can one split before another? In other words, is there a 3-cycle within the interval 3.4 < a < 3.5, or does the 2-cycle split directly to a 4-cycle?
- Once chaos appears past a = 3.6, is that it? Or is there anything more to be found there?
Continue your explorations of this function to try and answer these questions. Use the applets posted yesterday. I have a special prize for the first person who can find a 3-cycle…
We took some time today to finish looking at the functions in question 5 of the Iterated Functions sheet, exploring different cases of what happens with the slope of a curve immediately at its fixed point is precisely 1.
From there, we moved to a different family of functions defined by the graph y = ax(1-x). This function is called a logistic map, and it represents the growth of a capped population where x represents the ratio of the current population to a maximum sustainable population (as defined by the carrying capacity), and a is known as a fecundity rate. Naturally, both x and y are bound within the interval (0,1), as a population that reaches 0 is extinct and a population that reaches 1 will necessarily exceed its carrying capacity and become doomed. Your homework tonight is to consider this equation and those limits and to derive the boundaries for a.
We discussed the results of 4g, 4h, and 4i from the Iterated Functions sheet and observed how the discussed the ancient divide and average method of approximating irrational square roots relates to 4i.
We then revisited an observation we made before break: the graph of a function “near” a fixed point will look linear if you magnify it enough, even if the function itself is not linear. As a result, we can use the rules we derived for classifying fixed points based on the slope of a linear function to help us classify fixed points now. For example, if the “nearby” slope around a fixed point is positive and greater than that of y=x (i.e., is greater than 1), then the fixed point will be a “direct attractor.” This raised the question: What if the nearby slope is exactly equal to 1? This occurs when the graph of f(x) is tangent to the graph of y = x at the location of the fixed point.
It turns out that the fixed point has no unique classification, as behavior of seeds to the left of the fixed point differs from the behavior of seeds to the right. A fixed point could be directly attracting on one side, but directly repelling on the other!
Your homework tonight is to finish the Iterated Functions sheet by exploring the equations of part 5. All five pairs of functions have a fixed point at 0, all of which having a nearby slope of 1. Use the Geogebra apps on MathInsight.org to make a graph and test the behavior of seeds on both sides of the fixed point and write down your observations on the Iterated Functions Supplement. All five examples of two equations each: the positive and negative version. Make sure you look at both!
We reviewed results from 4b through 4f on the Iterated Functions sheet, making a brief detour to discuss the Golden Ratio which became surprisingly relevant for 4d. Every example had a single attracting fixed point until 4f, which had two repellers. But what did these repellers repel to? It wasn’t infinity, it was a cycle of values between 0 and -1. This is our first example of a limit cycle, a surprising result in algebraic iterations where a pattern of steps is pushed into a repetitive cycle.
Then we pushed the function a little further, from x^2 – 1 to x^2 – 2. Still with two repellers, this time the system of iterations dissolved into complete chaos, careening randomly within the interval -2 to 2, never settling into one spot. Changing the value of the seed by only one ten-thousandth produced a completely different pattern of iterations, demonstrating the sensitivity to initial conditions that are characteristic of chaos.
We will see plenty more examples of weird behavior in our iterations, but for now please finish question 4, parts h, i, and j. We’ll be discussing these results on Monday when we return.
We observed some patterns that emerge in cobweb diagrams for certain types of fixed points. For an attractor, the pattern of steps in the cobweb diagram will be drawn towards the fixed point; for a repeller they move away. Furthermore, if the fixed point is direct, the pattern of steps will look like actual steps as they move towards or away from the fixed point, trapped between the two lines defined by the function we are iterating and y = x. If the fixed point is alternating, on the other hand, the pattern will spiral around the fixed point with each step, producing a picture that most clearly gives the cobweb diagram its name.
From there, we took our first look at non-linear functions, realizing quickly that in addition to having more than one fixed point, such functions can have different types of fixed points. One could be a direct repeller, while another an alternating attractor. The cobweb diagram remains our best way to observe these differences, and we used the Geogebra-based applications found here to create them.
We looked at 4a (y = x^2) together, classifying the fixed point at 1 as a direct repeller and the fixed point at 0 as a direct attractor. We furthermore observed that -1 is a “pre-image” to the fixed point at 1, and therefore we have different behavior for different seeds. Your homework tonight is to continue and look at 4b-4f, using the cobweb diagram app linked above and the Iterated Functions Supplement to keep track of your results.
We continued our work with the Iterated Functions sheet, making some observations about how we might predict the classification of fixed point we get based on the parameters of the function we are iterating (see a friend in class for the exact notes on this if you missed them!)
We also introduced a new way of visualizing the behavior of functions: the cobweb diagram. In this style of graph, we draw the function we are iterating on the same plot as the line y=x. We pick a seed and move vertically to the graph of y=f(x), then horizontally to the line y=x, resulting in the output we just got becoming the input for the next iteration. We then move vertically again to y=f(x) and horizontally again to y=x, continuing until we get a view of the behavior of the function.
Your homework tonight then is to finish question 1 (parts f and g) and to do parts a, b, and c of question 3. Draw both the time diagram and cobweb diagram for each of these functions, and try to make some observations about how the patterns of the two representations align with each other.
We discussed the reading from Fractals: The Patterns of Chaos and looked at some references to the “Butterfly Effect” in popular culture. We spent most of the rest of the period playing with the Solar System simulator online.
For Monday, please read pages 49-54 from your book, the section on “The Fractals and Chaos of Outer Space.”
We discussed some key terms that we observed in the first part of the Nova documentary (specifically: fixed point, attractor vs repeller, and strange attractor) and we modeled some of these terms in action with some magnet pendulums. At the close of the period, we also looked briefly at a Solar System simulator found here.
For tomorrow, please read the three sections mentioned on yesterday’s post from Fractals: The Patterns of Chaos
We put up the random zig-zags that you created over the weekend and noticed that while the pattern tendered to wander away from the central line, it would eventually trend back as well. The pattern of central line crosses came in “bursts,” and the frequency of these bursts demonstrated fractal-like, self-similar qualities.
We returned to the classroom to examine another version of this, a Chaos Game app designed by Fractals & Chaos alum Istvan Burbank. We saw the impact that different constraints on random movement had on the pattern that movement created, including one very surprising result. We closed out the period investigating and playing with this app some more.
Remember: for tomorrow please read Chaos Hits Wall Street. This is required reading, and I will expect everybody to contribute to tomorrow’s discussion.
Tonight, complete your response to the Chapter 4 Investigative Task. As with the previous task, your response should be completed in a Google Document and submitted to me (email@example.com) by the beginning of class tomorrow. Please name your file appropriately. The file name should have the format “LastName.FirstName.Ch4InvTask”. For example, mine would be “Kirk.Benjamin.Ch4InvTask“. Remember also that you may use no resources except for your own notes (including classwork sheets)
In order to create your graph, I recommend you use the online tools found at Stapplet, specifically the 1 Quantitative Variable, Multiple Groups tool. You can export your graph from the online tool, but you may want to take a screenshot instead. Instructions can be found here (I recommend the “partial screenshot” option).
Finally, we discussed in class how boxplots are great for comparing distributions of data, but maybe aren’t the best option for assessing a distributions shape. We saw this with some interesting visualizations here, which you’re welcome to check out on your own.