Chapter 7.2, Problem 17E

Interpretation Introduction

Interpretation:

Assuming the hypotheses of Dulac’s criteria and R is topologically equivalent to annulus (exactly one hole in it).By using Green’s theorem, showthat there exists at mostone closed orbit in R.

Concept Introduction:

Dulac’s Criterion: For $\dot{\text{x}}\text{=f(x)}$ is a continuously differentiable vector field on a simply connected subset R of the plane. If there exists a continuously differentiable real valued function $\text{g(x)}$ such that $\nabla .\left(g\dot{x}\right)$ has one sign throughout R, then there are no closed orbits lying entirely in R.

The hypotheses of the Dulac’s criteria are

If the limit cycle doesn’t enclose a hole in the region, then the cycle is not possible.

If the limit cycle encloses a hole in the region, then there is at least one limit cycle.

Green’s theorem: If C is positively oriented, simple curve, piecewise smooth, and D be the region enclosed by the curve. If P and Q have continuous first order partial derivatives on D then

${\displaystyle \underset{C}{\oint }Pdx+Qdy}={\displaystyle \underset{D}{\iint }\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)}dA$