Chapter 7.3, Problem 11E

Interpretation Introduction

Interpretation:

Suppose $\dot{\text{x}}\text{ = f}\left(\text{x}\right)$ is a smooth vector field on ${\text{R}}^2$, plot the phase portrait for the system $\dot{\text{r}}\text{ = r}\left({\text{1 - r}}^{\text{2}}\right)\left[{\text{r}}^{\text{2}}{\text{sin}}^{\text{2}}\text{θ + }{\left({\text{r}}^{\text{2}}{\text{cos}}^{\text{2}}\text{θ - 1}\right)}^{\text{2}}\right]$ and ${\dot{\text{θ}}\text{ = r}}^{\text{2}}{\text{sin}}^{\text{2}}\text{θ + }{\left({\text{r}}^{\text{2}}{\text{cos}}^{\text{2}}\text{θ - 1}\right)}^{\text{2}}$. Sketch $\text{x }$ vs. $\text{t}$ for the trajectory starting away from the unit circle. Explain at $\text{t}\to \infty$.

Concept Introduction:

• ➢ Perform the polar to Cartesian conversion and obtain the system equation in the Cartesian form.

• ➢ Plot the system equation.

• ➢ Plot the $\text{x }$ vs. $\text{t}$ plot.