Chapter 10.4, Problem 7E

Interpretation Introduction

Interpretation:

• a) To show that in the logistic map with $\text{r > 1+}\sqrt{\text{5}}$, the first two letters are always RL.

• b) To find the iteration pattern for the orbit found in exercise $\mathrm{10.4.6}$.

Concept Introduction:

• ➢ By convention, we start the cycle at ${\text{x}}_{\text{0}}\text{=}\frac{\text{1}}{\text{2}}$; then, if the ${\text{n}}^{\text{th}}$ iterate ${\text{x}}_{\text{n}}$ lies to the right of ${\text{x}}_{\text{0}}\text{=}\frac{\text{1}}{\text{2}}$, the ${\text{n}}^{\text{th}}$ letter in the string is an R, otherwise it is an L.

• ➢ The string is called the symbol sequence or iteration pattern for the superstable cycle.

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