Chapter 7.3, Problem 5E

Interpretation Introduction

Interpretation:

To show by using Poincare-Bendixson Theorem that the system has at least one periodic solution.

Concept Introduction:

Poincare-Bendixson Theorem: If

R is a closed, bounded subset of the plane

$\dot{\text{x}}\text{=f(x)}$ is continuously differentiable vector field on an open set containing R

R doesn’t contain any fixed points

There exists a trajectory C that is confined in R, in the sense that it starts in R and stays in R for all future time whether C is closed orbit, or it spirals towards a closed orbit as $t\to \infty$. In any case, R contains a closed orbit

To check the stability of fixed point use Jacobian matrix

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

The point $(x^{*},y^{*})$ is said to be a stable fixed point if eigenvalues of Jacobian matrix evaluated at this point have negative real parts, and the point is said to be unstable if one of its eigenvalues has positive real part. If both eigenvalues are purely real, the fixed point is saddle point. If the eigenvalues are purely imaginary, the fixed point has oscillating nature.

Relation between polar and Cartesian coordinates system is

$\text{x = rcosθ}$

$\text{y = rsinθ}$

${\text{r}}^{\text{2}}{\text{= x}}^{\text{2}}{\text{+ y}}^{\text{2}}$