# Chapter 8, Exploration 5

The slider can be used to set an initial condition \(x_0\). Its orbit is shown in blue in the lefthand plot. The second initial condition is \(x_0 + \epsilon\), where the value of \(\epsilon\) can be specified in the dialog box. The orbit of this second initial condition is shown in red. A light blue line connects the corresponding iterates of these points. The righthand plot shows the distance between corresponding orbit points. In other words, the black dots represent the sequence

\( \left\{ \left\lvert f_4^n\left(x_0\right) - f_4^n\left(x_0+\epsilon\right)\right\rvert \right\}. \)

The green, dashed horizontal line in both plots is at height \(\frac{1}{2}\). In the left plot, that is because \(\frac{1}{2}\) is the critical point of the logistic function. In the right plot, it is because \(\frac{1}{2}\) is the value of \(b\) for this particular function.-
Convert your sequences of L's and R's to 0's and 1's by replacing each L with 0 and each R with 1. Call these
sequences \(s\) and \(t\)respectively.
- Find an upper bound for \(d(s,t)\) in \(\Sigma_2\) using the metric given in equation 7.4 in the text.
This is an estimate on how
**close**the points are together initially. - Use the value of \(N\) that you found in Exploration 4 to find a
**lower bound**on \(d(\sigma^N(s),\sigma^N(t)).\) This is an estimate on how far apart these iterates are after \(N\) iterates. - How do you think this relates to sensitive dependence from Chapter 8, Definition 2 in the text?

- Find an upper bound for \(d(s,t)\) in \(\Sigma_2\) using the metric given in equation 7.4 in the text.
This is an estimate on how

Initial Condition: