Chapter 8.2, Problem 4E

Interpretation Introduction

Interpretation:

To rewrite the system equation $\dot{\text{x}}\text{ = - y + }\mu {\text{x + xy}}^2$, $\dot{\text{y}}\text{ = x + }\mu {\text{y - x}}^{\text{2}}$ in polar coordinate. Show that if $\text{r}<<1$, $\dot{\text{r}}\approx \text{ r +}\frac{1}{8}{\text{r}}^3+\mathrm{...}$, $\dot{\theta }\approx 1$ and have essentially zero time average around one cycle. Suggest the presence of an unstable limit cycle of radius $\text{r}\approx \sqrt{-8\mu }$ for $\mu <0.$

Concept Introduction:

Use polar conversion to rewrite the system equation.