Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 31B (Dynamical Systems)

The Jacobian at the origin iswhose eigenvalues areTheir real parts are negative because $a>0$. The linearization stability theorem therefore makes the origin a locally asymptotically stable hyperbolic sink.

A Lyapunov function for an equilibrium is a continuously differentiable function that is positive definite there and whose derivative along nonconstant trajectories is negative definite in a neighbourhood.

For $V=x^2+k{y}^2$, positivity requires $k>0$, andThe quartic term is negligible sufficiently close to the origin, so $V$ is a local Lyapunov function exactly whenis positive definite. Its determinant condition isThe two roots of equality areTherefore

For $k=1$, the interval in part (b) contains one exactly when $a>1$. With $V=x^2+y^2=r^2$,because $2xy\le r^2$. Hence $\dot{V}<0$ throughoutEvery trajectory starting in the disc $x^2+y^2<a-1$ remains there and approaches the origin. Thus this disc is contained in the basin of attraction of the origin.

The Poincare-Bendixson theorem states that a nonempty compact $\omega$-limit set of a planar $C^1$ flow which contains no equilibrium is a periodic orbit.

The origin is the only equilibrium: at a nonzero equilibrium, putting $q=x^2+y^2-a$ would requirebut its determinant $q^2+3$ is positive. By local asymptotic stability, choose a small simple closed Lyapunov level curve on which the original vector field points inward. On a sufficiently large circle,so the original vector field points outward.

Reverse time. The annulus between these two curves is then a compact positively invariant trapping region: the reversed field points into it at both boundaries. It contains no equilibrium. The Poincaré--Bendixson theorem applied to any reversed trajectory in the annulus produces a periodic orbit. Reversing time does not change its image, so the original system also has a periodic orbit.