Chapter 8, Exploration 2

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.

• Try to find a value of \(\epsilon > 0\) where the distance between orbits is always less than \(\frac{1}{2}\) for the 20 iterates shown?

Explorations 1 through 5 use the same applet. Click here to go to the next Exploration, and here to go to the previous Exploration.