Chapter 12.1, Problem 1E

Interpretation Introduction

Interpretation:

Considering linear map ${\text{x}}_{\text{n+1}}{\text{= ax}}_{\text{n}},{\text{y}}_{\text{n+1}}{\text{= by}}_{\text{n}}$ with a and b are real parameters, draw the possible patterns of orbits near the origin

Concept Introduction:

• The uncoupled linear map equations are

  $\begin{array}{l}{\text{x}}_{\text{n+1}}{\text{= ax}}_{\text{n}}\\ {\text{y}}_{\text{n+1}}{\text{= by}}_{\text{n}}\end{array}$

  Here the equations are independent of each other, so it is easy to analyze them.

  The convergence of the system will depend only on the values of the parameters a and b.

  If

  $\begin{array}{l}\text{a, b}<1\to \text{Converge}\\ \text{a, b}=1\to \text{Remains constant}\\ \text{a, b}>1\to \text{Diverge}\end{array}$

  The sign of a and b decides the behavior of the orbits.