Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 31A (Dynamical Systems)

Let $\dot{z}=F(z)$ be a $C^1$ planar system on a simply connected domain $D$. The Bendixson-Dulac criterion states that if some $B\in C^1(D)$ makeshave one sign throughout $D$ and not vanish identically on any open subset, then $D$ contains no periodic orbit.

Indeed, if a periodic orbit $\Gamma$ bounded a region $\Omega \subset D$, then $F$ and hence $BF$ would be tangent to $\Gamma$. Its normal flux would therefore vanish. The divergence theorem would givecontradicting the sign hypothesis. Taking $B=1$ gives the usual Bendixson divergence criterion.

The stability version of the divergence test concerns an existing periodic orbit $\gamma$ of period $T$. Liouville's formula for the variational equation shows that its nontrivial Floquet multiplier isThus the divergence test for a planar periodic orbit says that $\gamma$ is asymptotically stable if the integral is negative and unstable if it is positive.

Put $r^2=x^2+y^2$. The system can be writtenIt therefore vanishes at the origin. The linearization there has eigenvaluesHence the origin is an asymptotically stable focus for $\mu <0$ and an unstable focus for $\mu >0$.

When $\mu =0$ the linearization alone is inconclusive. Direct radial reduction giveswhich is positive for every sufficiently small $r>0$. The origin is therefore unstable at $\mu =0$ as well, although trajectories leave it only at cubic order.

The polar equations areEvery positive zero of the radial factor is therefore a circular periodic orbit of period $2\pi$. Writing $s=r^2$, the equation isThe resulting count isFor the two-orbit range, $s_{-}<\lambda /2<s_{+}$. Differentiating the radial vector field at an orbit givesso the inner orbit is unstable and the outer orbit is stable. Atthey meet in a saddle-node bifurcation of periodic orbits. Atthe inner orbit shrinks into the origin, giving the second bifurcation.

The larger bifurcation value is $\mu =0$. For $\mu <0$ the origin is stable and an unstable periodic orbit of radiussurrounds it. That orbit collapses into the origin as $\mu \uparrow 0$, after which the origin is unstable. This is a subcritical Hopf bifurcation.

After a positive rescaling of radius, its local normal form isor equivalentlyThe full quintic radial Hopf equation also has the stable outer branchIn the $(\mu ,r)$ bifurcation diagram, the branch $r=0$ is stable for $\mu <0$ and unstable for $\mu \ge 0$. The unstable $r_{-}$ branch runs from $(0,0)$ backwards towhere it folds into the stable $r_{+}$ branch; the latter continues to larger $\mu$.

The divergence of the Cartesian vector field isOn a periodic orbit with $r^2=s$ andthis becomes the constantSince the period is $2\pi$, the divergence integral isIt is positive on the inner branch $s_{-}$, negative on the outer branch $s_{+}$, and zero where the branches meet at $s=\lambda /2$. The divergence test therefore classifies the inner orbit as unstable, the outer orbit as stable, and the double orbit as nonhyperbolic, exactly as the radial phase line does.

There is also no conflict with the exclusion form of the divergence criterion. For a circular orbit of radius $R$,so the divergence necessarily changes sign or vanishes somewhere in its interior; it cannot have the strict one-sign property that would exclude the orbit.