Chapter 10.7, Problem 2E

Interpretation Introduction

Interpretation:

To rewrite the given map in terms of the rescaled variable ${\text{x}}_{\text{n}}{\text{= αy}}_{\text{n}}$ and to use this to show that rescaling and inversion converts ${\text{f}}^{\text{2}}\left({\text{x,R}}_{\text{1}}\right)\text{ into αf}\left(\frac{\text{x}}{\text{α}}{\text{,R}}_{\text{1}}\right)$

Concept Introduction:

• The map is a difference equation which follows ${\text{x}}_{\text{n}}{}_{\text{+1}}{\text{=f(x}}_{\text{n}}\text{)}$.

• The superstable fixed points are fixed points with multiplier $\text{λ= 0}\text{.}$ They are called superstablebecauseany disturbances applied to them decay much faster thannon-zero $\text{λ}$ terms at an ordinary stable point.

• The plot of the equation is called a sine map when it follows the difference equation -

  ${\text{x}}_{\text{n}}{}_{\text{+1}}{\text{=rsinπx}}_{\text{n}}\text{for 0 }\le \text{r}\le \text{ 1 and 0 }\le \text{x}\le \text{ 1}.$

• The convergence rate for all unimodal maps appears to be constant. This constant is called the Feigenbaum number which is given by, $\text{δ}=\text{ 4}\text{.669}\mathrm{....}$