Chapter 7.1, Problem 5E

Interpretation Introduction

Interpretation:

To show that the system $\dot{r}=r\left(1-r^2\right)$, $\dot{\text{θ}}\text{=1}$ is equivalent to $\dot{\text{x}}\text{ = x - y - x}\left({\text{x}}^{\text{2}}{\text{+y}}^{\text{2}}\right)$, $\dot{\text{y}}\text{= x + y - y}\left({\text{x}}^{\text{2}}{\text{+y}}^{\text{2}}\right)$ for polar to Cartesian coordinates.

Concept Introduction:

A limit cycle is a closed trajectory. Isolated means that neighboring trajectories are not closed.

A limit cycle is said to be unstable or half stable, if all neighboring trajectories approach the limit cycle.

These systems oscillate even in the absence of external periodic force.