Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 32E (Dynamical Systems)

The Bendixson-Dulac criterion says the following. Let $D\subseteq {\mathbb{R}}^2$ be a simply connected domain, let $F=(f,g)$ be continuously differentiable, and suppose there is a continuously differentiable function $B$ such thathas one sign throughout $D$ and does not vanish identically on any open subset. Then $\dot{z}=F(z)$ has no periodic orbit contained in $D$.

Indeed, if a periodic orbit $\Gamma$ existed, simple connectivity would put its interior $U$ inside $D$. The vector field $F$, and hence $BF$, is tangent to $\Gamma$, so its outward flux is zero. The divergence theorem would givecontradicting the sign hypothesis.

The Poincare-Bendixson theorem states that a nonempty compact omega-limit set of a continuously differentiable planar flow that contains no equilibrium point is a periodic orbit. More generally, when the omega-limit set contains only finitely many equilibria, it is either an equilibrium, a periodic orbit, or a union of equilibria and connecting trajectories.

Take the Dulac function $B=1$. Fordirect differentiation givesThe quadratic form in parentheses has matrixwhose eigenvalues are $14$ and $40$, so it is positive definite. For $0<k\le 3$ the divergence is nonpositive and vanishes only at the origin when $k=3$. The Bendixson-Dulac criterion therefore gives

Let $R=x^2+y^2$. The cross terms in the linear part cancel, andBoth quadratic forms in the product are positive definite, with least eigenvalue $2$. HenceOn the circle $R=k/2$, the right-hand side is $-k^2<0$, so the disc $R\le k/2$ is forward invariant. While $R\ge k/2$, the same estimate gives $\dot{R}\le -2kR$; therefore every trajectory starting outside the disc enters it in finite time. One suitable choice is

The Jacobian matrix at the origin isIts eigenvalues areThus, when $k>3$, the origin is a hyperbolic repelling spiral. By part (ii), every forward trajectory is eventually trapped in a compact disc. For $k<10$ the hypothesis says that the origin is the only equilibrium; choose any nonstationary trajectory sufficiently close to it. Its omega-limit set is nonempty and compact, and it cannot contain the repelling origin. The Poincare-Bendixson theorem then says that this omega-limit set is a periodic orbit. Combining the restrictions gives