# Chapter 8, Exploration 4

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.
• Let $$x_0=0.2$$ and $$\epsilon = 0.0001$$. Track the orbits by making a list of L's and R's. Write an L (for left) every time an iterate is less than $$\frac{1}{2}$$ and write an R (right) every time an iterate is greater than $$\frac{1}{2}.$$ This is why the green dashed line is shown. Since the blue orbit starts at 0.2, the first letter in its list is L. Similarly for the red orbit that starts at 0.2001. Note: If two points are very, very close, then you only see the red point on the display.
• How do these lists relate to one another, if at all?
• What is the first entry where one list has an L but the other list has an R? Call this value $$N$$ for future reference.
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: