Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 1 / 7B (Dynamical Systems)

The Bendixson-Dulac criterion says that if a $C^1$ planar field admits a $C^1$ multiplier $B$ on a simply connected region such that $\nabla \cdot (Bf)$ has one strict sign there, no nonconstant periodic orbit lies wholly in that region. The Poincaré-Bendixson theorem says that a nonempty compact omega-limit set of a planar $C^1$ flow which contains no equilibria is a periodic orbit.

For $\mu <0$, the divergence of the given field is $\mu -2x^2-3y^2<0$ throughout the plane. Taking $B=1$ proves there are no periodic orbits.

For $\mu >0$, use $E=(x^2+y^2)/2$. Along trajectories,On a circle $r=r_{-}$ with $2r_{-}^2<\mu$ the radial flow is outward or tangent, and on a circle $r=r_{+}$ with $r_{+}^2>\mu$ it is inward or tangent. Their intervening closed annulus is positively invariant. At the tangent points $y=0$, the field has $\dot{y}=-x\ne 0$, so no equilibrium sits on its boundaries. Indeed the only equilibrium is the origin, excluded from the annulus. Every forward trajectory in the annulus has a nonempty compact omega-limit set without equilibria. The Poincaré-Bendixson theorem consequently proves at least one periodic orbit exists for every $\mu >0$. No claim of uniqueness follows from this trapping argument.