# Chapter 8, Exploration 3

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.
• If the distance between orbits ever becomes greater than $$\frac{1}{2}$$, does this distance remain so for all iterates thereafter?
Explorations 1 through 5 use the same applet. Click here to go to the next Exploration, and here to go to the previous Exploration.

Initial Condition: