Chapter 10.3, Problem 7E

Interpretation Introduction

Interpretation:

Consider the decimal shift map on the unit interval given by ${\text{x}}_{\text{n+1}}{\text{=10x}}_{\text{n}}\text{(mod1)}$.

• a) The graph of the map is to be drawn.

• b) Find all fixed points.

• c) Prove that the map has periodic points of all periods, but all of them are unstable.

• d) Prove that the map has infinitely many aperiodic orbits.

• e) Show that the map has sensitive dependance on initial conditions by considering the rate of separation of two nearby orbits.

Concept Introduction:

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

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