Chapter 9.3, Problem 8E

Interpretation Introduction

Interpretation:

To find if $\text{D}$ is an invariant set, if $\text{D}$ is considered to be the disk ${\text{x}}^{\text{2}}{\text{ + y}}^{\text{2}}\le \text{1}$, for the system in polar coordinates given by $\dot{\text{r}}\text{ = r}\left({\text{1 - r}}^{\text{2}}\right)$, $\dot{\text{θ}}\text{ = 1}$. To find if $\text{D}$ attracts an open set of initial conditions. To find if $\text{D}$ an attractor; if yes, to find its basin of attraction. To find if the circle ${\text{x}}^{\text{2}}{\text{ + y}}^{\text{2}}=\text{1}$ is an attractor; if yes, to find its basin of attraction.

Concept Introduction:

• ➢ An attractor is a closed set $\text{A}$ such that with the following properties:

  Any trajectory that starts in $\text{A}$ stays in $\text{A}$ for all time.

  $\text{A}$ attracts an open set of initial conditions.

  There is no proper subset of $\text{A}$ that fulfills the above two conditions.