# Chapter 8, Exploration 1

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.- Using the tool on the website, set \(\epsilon = 0.0001.\) Try to find an initial condition where the distance between iterates remains less than \(\frac{1}{2}\) for all iterates displayed on the tool. Repeat this with \(\epsilon = 0.00001.\)

Initial Condition: