Fractals and Chaos Recap for 11/28

We continued our discussion of the logistic map from yesterday by iterating the function for various values of the growth parameter a. We observed that if a is too low, the population will die out and the destination of the orbit is zero. Once a passes 1, the population will eventually settle at some proportion of the maximum population; for example at a = 2.6, the orbit of iterations settles on a value of approximately 0.6154 (precisely, this is 8/13 of the possible maximum population). For a = 3.2, we observed a two-cycle of {0.5130,0.7995}, suggesting that the population here will year by year fluctuate between 51% and 80% of its possible maximum population. For a = 3.6, we observed two bands of chaos bound within (0.32,0.6) and (0.79,0.9), suggesting that the population never dies out, but never settles at a stable value (or set of values).

Tonight, please continue to explore values of a and the destinations of orbits within this graph. Use the apps at to help, as well as this Logistic Function Cobweb Diagram I made in Desmos. Keep track of your observations on the Catalog of Behavior you got in class. By tomorrow, you should at least have an observation for every tenth value of a between 2.2 and 4.0.

Fractals and Chaos Recap for 11/26

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 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!

Fractals & Chaos Lesson Recap for 11/20

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.

Fractals & Chaos Recap for 11/15

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.

Fractals & Chaos Recap for 11/14

We’ve started down the road towards understanding the mathematics of chaos by iterating linear functions. Imagine a recursively-defined sequence, where each term of the sequence is defined based on the previous value. That’s what we’re doing here: we start with a seed, then plug that seed into a formula. Each output becomes the next step’s input, and we seek to understand the long-term behavior of formulas.

We are primarily working with this classwork sheet: Iterated Functions. We’ve done parts a, b, and c for part 1, looking at the behavior of seeds both numerically and graphically, creating a plot of steps vs. value called a Time Diagram. In all three examples, we found a fixed point: a value of a seed that is constant through the formula (plugging that value in gets that same value out). Some of the fixed points were attractors, and some were repellers. We discussed how iterating the function repeatedly is a fine way to find an attracting fixed point, but obviously won’t work to find a repeller. The only option there is to solve the equation f(x) = x.

Tonight for homework, please do parts d and e of question 1, then all of question 2. You should work on identifying and classifying the fixed point (as an attractor or repeller).