Chapter 10.6, Problem 2E

Interpretation Introduction

Interpretation:

1. To explain that ${\text{R}}_{\text{n}}$ is more easily and accurately computed than ${\text{r}}_{\text{n}}$.

2. To compute the first six ${\text{R}}_{\text{n}}$ and use them for estimating $\text{δ}$.

Concept Introduction:

• If $\text{f(x,r)}$ denotes a uni-modal map that undergoes a period-doubling route to chaos as r increases and ${\text{x}}_{\text{m}}$ is the maxima of $\text{f}$, then ${\text{r}}_{\text{n}}$ denotes the value of $\text{r}$ at which ${\text{2}}^{\text{n}}\text{-cycle}$ is born, and ${\text{R}}_{\text{n}}$ denotes the value of $\text{r}$ at which ${\text{2}}^{\text{n}}\text{-cycle}$ is super-stable.

• The formula for calculating $\text{δ}$ is:-

  $\text{δ =}\frac{R_k-R_{k-1}}{R_{k+1}-R_k}=4.66920$

  The value of $\text{δ = }4.66920$ is universal.