Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 31B / e / Solution

For $a=-1,b=1$, $\dot{y}=-1+x-y-x^2-y^2\le -1/2<0$ after maximizing the two quadratic expressions. Thus no orbit is periodic. A proper forward bounding region or Lyapunov trapping function would, in a planar flow with no equilibria, force a periodic limit set by Poincare-Bendixson; therefore no such global bounding function exists.