Chapter 10.6, Problem 1E

Interpretation Introduction

Interpretation:

To plot the sine map for random initial condition.

To find ${\text{r}}_{\text{1}}{\text{, r}}_{\text{2}}{\text{, r}}_{\text{3}}{\text{, r}}_{\text{4}}{\text{, r}}_{\text{5}}{\text{, r}}_{\text{6}}$ by resolving the diagram and to find Feigenbaum ratio numerically using ${\text{r}}_{\text{1}}\text{, …}{\text{., r}}_{\text{6}}\text{.}$

Concept Introduction:

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

• 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 λ terms at an ordinary stable point.

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