# 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?
Explorations 1 through 5 use the same applet. Click here to go to the previous Exploration.

Initial Condition: