06 - Bendixson-Dulac Theorem

Bendixson-Dulac Theorem

Consider system $\dot{x}=X(x,y)$ and $\dot{y}=Y(x,y)$ with $X,Y\in C^1$
If there exists a function $\phi (x,y)\in C^1$ with

$$\rho :=\frac{\partial }{\partial x}(\phi X)+\frac{\partial }{\partial y}(\phi Y)>0$$

in a simply connected region $R$ then

There can be no non-trivial closed trajectory lying entirely in $R$

Proof

Note that a non-trivial trajectory that is not just a fixed point

Suppose $C$ is a closed trajectory lying entirely in $R$
Let $D$ be the disc (lies entirely in $R$, as $R$ is simply connected) whose boundary is $C$

Applying Green’s Theorem in the Plane then consider integral

$$\begin{array}{rl}{\iint }_D\rho dxdy& ={\iint }_D\left[\frac{\partial }{\partial x}(\phi X)+\frac{\partial }{\partial y}(\phi Y)\right]dxdy\\ & ={\oint }_C-\phi Ydx+\phi Xdy\\ & ={\oint }_C-\phi (-\dot{y}dx+\dot{x}dy)\end{array}$$

However on $C$ then $dx=\dot{x}dt$ and $dy=\dot{y}dt$ which is zero
This contradicts positivity of $\rho$ so there can be no such $C$