Bendixson–Dulac theorem

Remarks

Suppose a periodic orbit $\Gamma \subseteq D$ exists, and let $R$ be the region inside $\Gamma$; since $D$ is simply connected, $R \subseteq D$. By Green's theorem, $\iint_R \nabla \cdot (\varphi f) \, dx \, dy$ equals the flux $\oint_\Gamma \varphi \, (P \, dy - Q \, dx)$. The flux is $0$, because along $\Gamma$ one has $dx = P \, dt$ and $dy = Q \, dt$; but the double integral is positive, since $R$ is a nonempty open set. The case $\varphi = 1$ is Bendixson's criterion, and a function $\varphi$ that works is called a Dulac function. Example: for $x' = y$, $y' = -x - y + x^2$ one has $\nabla \cdot f = -1$, so the system has no periodic orbit in $\R^2$. Simple connectivity is needed: the system $x' = x(1 - x^2 - y^2) - y$, $y' = y(1 - x^2 - y^2) + x$ has $\nabla \cdot f = 2 - 4(x^2 + y^2) < 0$ on the annulus $0.8 < \sqrt{x^2 + y^2} < 2$, yet the unit circle in this annulus is a periodic orbit.

Going deeper

For a student

Bendixson–Dulac is the standard negative criterion for periodic orbits in the plane, and it complements the Poincaré–Bendixson theorem, which proves their existence. Finding a good Dulac function is an art; for Lotka–Volterra type systems, $\varphi = 1 / (x y)$ is a common choice. On a domain with one hole, the same argument shows that there is at most one periodic orbit that goes around the hole.