Dynamical Systems and Feedback

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1. Let \(F(x)=x^2+1\).
   1. Find the first five points in the orbit of \(x_0=0\).
   2. Describe the orbits in \(F(x)\) of any initial condition \(x_0\).
2. Let \(F(x)=x^2 - 2\). Find the first five points in the orbits of:
   1. \(x_0=0\)
   2. \(x_0=\frac{1}{2}\)
   3. \(x_0=1\)
   4. \(x_0=2\)
3. Let \(C(x)=\cos(x)\). For any initial value \(x_0\), the orbit of \(x_0\) in \(C(x)\) converges. Find the value of this convergence.
4. Let \(F(x)=1-x^2\). Determine the \(2\)-cycle reached from \(x_0=\frac{1}{2}\).
5. In the previous problem, the same \(2\)-cycle can be reached for any \(x_0\) where \(-k < x_0 < k\). Find \(k\).
6. Let \(F(x)=\sin(kx)\). Describe the orbit of \(x_0=\frac{1}{2}\) in \(F(x)\) for:
   1. \(k=1\)
   2. \(k=2\)
   3. \(k=3\)
7. Let \(P(x)=2x(1-x)\).
   1. Expand \(P^2(x) = P(P(x)) \).
   2. Determine the first 10 terms in the orbits of \(x_0=0.2, 0.5, 0.8\)
   3. For initial conditions of \(x_0=0.05, 0.1, ..., 0.95\), use a spreadsheet to show that the orbits all converge to the same value. Create a plot of the orbits that depicts them converging.
8. Create a spreadsheet implementing the Babylonian square root algorithm.
   1. Place the value \(17\) in A1, representing the value for which we want to approximate the square root. Then place a guess of \(4\) in A2. The rows below A2 should iteratively improve our approximation of \(\sqrt{17}\text{.}\)
   2. Add more columns to demonstrate the algorithm working for several other examples.

9. Many deployed data models are part of larger feedback systems. Consider a video streaming service which develops a recommendation function \(R\). This function may take many inputs, but one of those inputs is sure to be the list of videos that the user has watched, \(w\). The output of \(R(w)\) is a list of videos for the user, who watches some of those videos, and then gets more recommendations which have been influenced by the earlier recommendations that influenced the user's choices.

   Identify three feedback loops that you interact with, and draw a system diagram depicting the loop. What does "orbit" mean in the context of those systems?

10. Number 11 is a great question!
11. Watch The Equation that Changed How We See the World. Create the bifurcation diagram for the logistic equation in a spreadsheet. Here is one way to do it:
    1. Make column headers for r, x0, x1, etc.
    2. Fill in the A column with r values from 1 to 4.
    3. Fill in the B column with an initial value.
    4. Use the formula for the logistic equation shown in the image below. Note the $ symbol in front of the A. This is so you can drag the equation down, and then right, without that A changing. Each row will contain an orbit.
    5. Once the orbits are expanded to the right, select the last ten columns and create a scatter plot.
    
    You should end up with a chart that looks like this:
    
12. Let \(F(x) = x^2 + \frac{1}{4} \).
    1. Analyze the orbits for real \(|x_{0}| \leq \frac{1}{2}\).
    2. Show that the orbit of any real \(|x_0| > \frac{1}{2}\) will go to \(\infty\).
    3. Is there a value \(x_0 \in \mathbb{C}\) that has a cycle?