Chapter 11.4, Problem 6E

Interpretation Introduction

Interpretation:

To analyse the strange repeller for the tent map.

1. To find the set of the initial conditions that escape after one ortwo iterations.

2. To find the set of conditions that neverescape.

3. To find the box dimension of the set of conditions that never escape.

4. To show that the Liapunov exponent is positive at each pointin the invariant set.

Concept Introduction:

• The tent map on the interval $\left[\text{0,1}\right]$ is defined as: ${\text{x}}_{\text{n+1}}\text{= f}\left({\text{x}}_{\text{n}}\right)$, where $\text{r > 0}$

  $\text{f}\left({\text{x}}_{\text{n}}\right)=\{\begin{array}{l}\text{rx, 0}\le \text{x}\le \frac{1}{2}\\ \text{r}\left(\text{1-x}\right),\frac{1}{2}\le \text{x}\le 1\end{array}$

• Here, it is assumed that $\text{r > 2}$. Hence, some points are such that: ${\text{f(x}}_{\text{0}}\text{)>1}$, which means that the points are escaped. Similarly, if ${\text{f}}^{\text{k}}\text{(x0) >1}$ for some finite k, but ${\text{f}}^{\text{n}}\left({\text{x}}_{\text{0}}\right)\in \left[\text{0, 1}\right]$ for all $\text{n< k}$, then we say that ${\text{x}}_{\text{0}}$ has escaped after kiterations.