Chapter 6.3, Problem 15E

Interpretation Introduction

Interpretation:

The phase portrait of the system ${\dot{\text{r}}\text{ = r(1 - r}}^{\text{2}}\text{), }\dot{\text{θ}}\text{ = 1 - cosθ }$ is to be sketched and the fixed point ${\text{r}}^{\text{*}}{\text{ = 1, θ}}^{\text{*}}\text{ = 0}$ is attracting but not Liapunov stable is to be shown.

Concept Introduction:

The trajectories of the dynamic system are called phase portrait.

Phase portrait gives the qualitative behavior of the solutions of a differential equation.

The fixed point is attracting if all the trajectories approach to it.

The fixed point is Liapunov stable if all the trajectories directly end up at it. All the trajectories which start sufficiently close to the fixed point remain close to it all time.