Chapter 7.3, Problem 1E

Interpretation Introduction

Interpretation:

To classify the fixed point at the origin of the system $\dot{\text{x}}\text{ = x - y - x}\left({\text{x}}^{\text{2}}{\text{ + 5y}}^{\text{2}}\right)$, $\dot{\text{y}}\text{ = x + y - y}\left({\text{x}}^{\text{2}}{\text{ + y}}^{\text{2}}\right)$. Using $\text{r}\dot{\text{r}}\text{ = x}\dot{\text{x}}\text{ + y}\dot{\text{y}}$, and $\dot{\text{θ}}\text{ = }\frac{\left(\text{x}\dot{\text{y}}\text{ - y}\dot{\text{x}}\right)}{{\text{r}}^{\text{2}}}$ the given system should be rewritten in polar coordinates. The circle of maximum radius, ${\text{r}}_{\text{1}}$ and of minimum radius, ${\text{r}}_{\text{2}}$, centered on the origin are to be determined. Also, To prove that the given system has a limit cycle somewhere in the trapping region ${\text{r}}_{\text{1}}\le \text{r}\le {\text{r}}_{\text{2}}$.

Concept Introduction:

The Jacobian matrix at a general point $\left(\text{x, y}\right)$ is given by

$\text{A = }\left(\begin{array}{cc}\frac{\partial \dot{\text{x}}}{\partial \text{x}}& \frac{\partial \dot{\text{x}}}{\partial \text{y}}\\ \frac{\partial \dot{\text{y}}}{\partial \text{x}}& \frac{\partial \dot{\text{y}}}{\partial \text{y}}\end{array}\right)$

The eigenvalue $\text{λ}$ can be calculated using the characteristic equation

$\left|\left(\text{A - λI}\right)\right|\text{ = 0}$

The solution of the quadratic equation is $\text{λ = }\frac{\text{-b ± }\sqrt{{\text{b}}^{\text{2}}\text{- 4ac}}}{\text{2a}}$

The circle of maximum radius centered on the origin with all trajectories having radially outward component on it can be obtained by $\dot{\text{r}}\ge \text{0}$.

The minimum radius circle centered on the origin with all trajectories having component directed radially inward on it can be found by putting $\dot{\text{r}}\le \text{0}$.

Nullclines are the curves in the phase portrait where $\dot{\text{r}}\text{ = 0}$ or $\dot{\text{θ}}\text{ = 0}$.

According to Poincare-Bendixson theorem, if a trapping region for a system does not contain any fixed point, then there must be at least one limit cycle within this trapping region.