Chapter 10.3, Problem 12E

Interpretation Introduction

Interpretation:

Let ${\text{R}}_{\text{n}}$ denote the value of r at which the logistic map has a superstable cycle of the period ${\text{2}}^{\text{n}}$.

• a) To write an implicit, but exact, formula for ${\text{R}}_{\text{n}}$ in terms of the point $\text{x =}\frac{\text{1}}{\text{2}}$ and the function $\text{f}\left(\text{x,r}\right)\text{ = rx}\left(\text{1-x}\right)$.

• b) To find ${\text{R}}_{\text{2}}{\text{,R}}_{\text{3}}\mathrm{........}{\text{,R}}_{\text{7}}$ to five significant figures using computer and result of part a).

• c) To find $\frac{R_6-R_5}{R_7-R_6}$.

Concept Introduction:

• ➢ The periodic orbit has a period $\text{p}$ and the fixed point is the intersection point between the line and function which is not greater than the period of the orbit.