Chapter 10.3, Problem 9E

Interpretation Introduction

Interpretation:

To show that the binary shift map ${\text{x}}_{\text{n}}{\text{+ 1 = 2x}}_{\text{n}}\left(\mathrm{mod}1\right)$ has sensitive dependence on initial conditions, infinitely many periodic and aperiodic orbits and a dense orbit.

1. Draw the graph of the map.

2. Find all fixed points.

3. Show that the map has periodic points of all periods but all of them are unstable.

4. Show that the map has infinitely many aperiodic orbits.

5. By considering the rate of separation of two nearby orbits show that the map has sensitive dependence on initial conditions.

Concept Introduction:

• The logistic map is a second degree equivalent mapping. It has mention how its complex points and chaotic logistic map is used to arise into a very straight forward non-linear dynamic equation.

• The logistic map function is ${\text{x}}_{\text{n+1}}{\text{= f(x}}_{\text{n}}\text{)}$.