Past exam of the topics in applied mathematics course of the University of Oxford / 2011 / B / 4 / a

The Poincaré-Bendixson theorem states that the omega-limit set of a forward orbit confined to a compact region of the plane, containing no equilibrium points, is a periodic orbit.

Put $x=R\cos \theta$, $y=R\sin \theta$. Direct calculation givesHere $1/2\le H\le 17/16$ and $|\cos \theta {\sin }^3\theta |\le 3\sqrt{3}/16$. Thus on the inner circle $R=\sqrt{3}/2$ the radial velocity is positive, and on the outer circle $R=3/2$ it is negative. In their closed annulus,so there is no equilibrium. The theorem gives a periodic orbit in this trapping annulus.

It is asymptotically stable: the vector-field divergence isthroughout the annulus. Its nontrivial Floquet multiplier is $\exp {\int }_0^T\mathrm{div}Fdt<1$. This proves an attracting periodic orbit exists. There can be only one there: two such nested orbits would enclose an annular region with zero outward flux, contradicting the strictly negative divergence via the divergence theorem.