Chapter 10.4, Problem 3E

Interpretation Introduction

Interpretation:

(A super stable $\text{3 -}$ cycle) the map ${\text{x}}_{\text{n+1}}\text{= 1 - r}{\left({\text{x}}_{\text{n}}\right)}^2$ has a super stable $\text{3 -}$ cycle at a certain value of $\text{r}\text{.}$ Find a cubic equation for this $\text{r}\text{.}$

Concept Introduction:

• Determine the fixed points for the given equation.

• Compute ${\text{f}}^{\text{3}}\left(\text{x}\right)\text{ = x}$, and plug the fixed point.

• Obtain the a cubic equation for $\text{r}\text{.}$