Chapter 10.5, Problem 6E

Interpretation Introduction

Interpretation:

Using a computer, compute and plot the Liapunov exponent as a function of $\text{r}$ for the sine map ${\text{x}}_{\text{n+1}}\text{= r sin}\left({\text{πx}}_{\text{n}}\right)$, for $0\le {\text{x}}_{\text{n}}\le \text{1 and }0\le r\le 1$.

Concept Introduction:

• ➢ A logistic map can exhibit aperiodic orbits for certain parameter values.

• ➢ Given some initial condition ${\text{x}}_{\text{0}}$, consider the nearby point ${\text{x}}_{\text{0}}{\text{+δ}}_{\text{0}}$, where the initial separation ${\text{δ}}_{\text{0}}$ is extremely small. Let ${\text{δ}}_{\text{n}}$ be the separation after $\text{n}$ iterates. If ${\text{|δ}}_{\text{n}}\left|\approx \right|{\text{δ}}_{\text{0}}{\text{| e}}^{\text{n λ}}$, then $\text{λ}$ is called the Liapunov exponent.

• ➢ Apositive Liapunov exponent is a signature of chaos.