Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 32E / b / iii / Solution

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