Chapter 10.3, Problem 10E

Interpretation Introduction

Interpretation:

• a) Let $\left\{{\text{θ}}_{\text{n}}\right\}$ be an orbit of the binary shift map ${\text{θ}}_{\text{n+1}}{\text{= 2θ}}_{\text{n}}\text{(mod1)}$ and define a new sequence ${\text{{x}}_{\text{n}}\text{}}$ by ${\text{x}}_{\text{n}}{\text{= sin}}^{\text{2}}{\text{(πθ}}_{\text{n}}\text{)}$. To show that $\text{f}\left({\text{x}}_{\text{n}}\right){\text{ = 4x}}_{\text{n}}\left({\text{1 - x}}_{\text{n}}\right)$. Any such orbit is an exact solution of the logistic map with $\text{r = 4}$.

• b) Graph the time series ${\text{x}}_{\text{n}}$ vs n for various choices of ${\text{θ}}_0$.

Concept Introduction:

• ➢ The binary shift map is divided into the minimum number of slots.

• ➢ It is used to determine the approximate or nearest fixed point of the function in the logistic map region.