Chapter 10.4, Problem 11E

Interpretation Introduction

Interpretation:

By considering ${\text{x}}_{\text{n+1}}{\text{= a}}^{{\text{x}}_{\text{n}}}$ analyze long-term behavior of the sequence $\left\{{\text{ x}}_{\text{n}}\right\}$ as $\text{n}\to \infty$ and check how long-term behavior depends on a.

Concept Introduction:

• ➢ The period-3 window occurs near $\mathrm{3.8284...}\le \text{r}\le \mathrm{3.8415...}$

• ➢ The notation for the logistic map is

  Let, $\text{f(x) = rx}(1-\text{x})$

  Then the logistic map is

  ${\text{x}}_{\text{n+1}}{\text{ = f( x}}_{\text{n}}{\text{), x}}_{\text{n+2}}\text{ = f}\left({\text{f( x}}_{\text{n}}\text{)}\right)\Rightarrow {\text{f}}^{\text{2}}\left({\text{x}}_{\text{n}}\right),{\text{x}}_{\text{n+3}}{\text{ = f}}^2\left({\text{f( x}}_{\text{n}}\text{)}\right)\Rightarrow {\text{f}}^3\left({\text{x}}_{\text{n}}\right)$

  The third iteration map ${\text{f}}^3\left({\text{x}}_{\text{n}}\right)$ is important to understand the birth of the period-3 cycle.