# Fractals & Chaos Recap for 12/4

We spent the first portion of class today exploring the Feigenbaum Plot today by asking a few supplemental questions to question 3 (is there anything to be found after chaos):

• Which comes first, a 3 cycle (so far found at a = 3.84) or the 5 cycle (found at a = 3.74 but also a = 3.906)?
• Why do some cycles happen more than once (see the 5-cycle above, or the two 6-cycles at a = 3.63 and a = 3.845 or two 4-cycles at a = 3.5 and a = 3.961)?
• In general:
• Where do cycles come from? How are they “born”?
• Is there some order to cycles?

Instructions on using Paul Fischer-York’s Bifurcation Diagram

• Use the Darkness slider to make the image darker and easier to see.
• Left-click and drag a rectangle around any portion of the diagram to zoom in on that portion.
• Right-click anywhere in the diagram to run a series of iterations at that value of a along the x-axis. The software will identify the magnitude of the cycle you’re looking at (though sometimes it makes a mistake, so look at the list as well!)

After some time to explore, we made a few observations about the general questions. The first is that cycles are born in two ways:

1. Bifurcations of “lower” cycles, or
2. Spontaneously arising from chaos

This suggests a certain ordering of cycles: the 2-cycle comes first, then the 4-cycle, the 8-cycle, the 16-cycle, and so on for powers of 2. Eventually, these cycles become so large that the system becomes chaotic. But this only explains cycles that are powers of 2. What about any other? For this, please read this article about the Feigenbaum plot, its constant, and the ordering of cycles.

# Fractals & Chaos Recap for 12/3

We explored cycles a bit more in class today, noting that if {p,q} are the two values of a 2-cycle, then by definition (p) = q and (q) = p. Put another way, ((p)) = p and ((q)) = q. What’s important to note, though, is that the reverse is not true. If ((n)) = n, then n could be a member of a two cycle (n,m,n,m,n,m,…or it could just be a fixed point: (n,n,n,n,n,n,…)

One important takeaway from this is that the first is that the local slope (Instantaneous Rate of Change) to all points of a 2-cycle, or indeed any cycle, is always the same. As a result, all points of a cycle will pass an IRC of -1 simultaneously, meaning all points of a cycle bifurcate simultaneously. This (and the  Chain Rule from calculus) served as our proof of Question 2 posed early in our exploration of this pattern.

The implication of this is clear: after a = 3.0, the pattern of the orbit of the logistic function bifurcates to a two cycle. By a = 3.5 (more specifically at a = 3.449490…), it has bifurcated again to a four-cycle. Soon after (at a = 3.544090…), it bifurcates yet again to an eight-cycle, then to a 16-cycle (at a = 3.564407…), a 32-cycle (at a = 3.568759…), and so on. These bifurcations happen over increasingly shorter intervals for values of a. The fascinating thing is that the ratio between these intervals approaches a constant value, approximately 4.669202…, known as Feigenbaum’s Constant (this link goes to a Numberphile video about this idea that we watched a portion of in class today; the picture of orbit destinations that we have been drawing is known as the Feigenbaum Plot, which we will look at in detail tomorrow).

# Fractals & Chaos Recap for 11/26

We did some more research in the logistic map and the catalog of behavior we’ve been using for the past few days. We’ve answered the first question from yesterday’s list: the first split happens at precisely a = 3.0. With a proof in class, we showed that this is because for a = 3.0, the slope of the curve at the value of the fixed point (which happens to precisely equal 2/3) is exactly -1. Any value of a less than 3.0 will have a slope at its fixed point that is shallower than -1, and as a result the fixed point is an attractor. Any value of a greater than 3.0 will have a slope at its fixed point that is steeper than -1, and as a result the fixed point is a repeller. So when we notice that there’s a 2-cycle at a = 3.1, this is because the fixed point is a repeller, pushing the orbit to the 2-cycle. This 2-cycle exists for all values of 3.0 < a < 3.1.

We also have seen some progress on question 3, finding a 5-cycle at a = 3.906, a 7-cycle at a = 3.702, and our very first 3-cycle at a = 3.83. A follow-up question we could ask here: Can a cycle of any length be found for a > 3.6?

We still haven’t answered question 2: whether or not both points of the 2-cycle at a = 3.4 split simultaneously to form the four cycle found at a = 3.5, or if, for the briefest of moments, one point splits before the other and we can find another 3-cycle somewhere between 3.4 << 3.5. Your homework this break is to investigate this some more.

# Fractals & Chaos Assignment for 11/25

We did some more work collecting information for the Catalog of Behavior for the Logistic Map function Mr. Drix introduced to you last week, getting a clearer idea on the changes in behavior as a increases towards 4.0. The patterns we observed raised a few questions:

1. 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?
2. 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?
3. 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 on the left side of the page. I have a special prize for the first person who can find a 3-cycle…

# Fractals & Chaos Recap for 11/22

We continued our discussion of the logistic map from yesterday by iterating the function for various values of the growth parameter a (which we identified as being bound between 0 and 4). We observed that if a is too low (a < 1), 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 roughly 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).

This weekend, please continue to explore values of a and the destinations of orbits within this graph. Use the apps at MathInsight.org 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 Monday, you should at least have an observation for every tenth value of a between 2.2 and 4.0.