Brusselator periodic-orbit criterion

The spatially homogeneous Brusselator has its unique positive equilibrium at $(r,s/r)$. Its Jacobian there has determinant $r^2$ and trace $s-1-r^2$. If $s-1>r^2$, the equilibrium is a repeller inside the Brusselator trapping region; the Poincare-Bendixson theorem then supplies a periodic orbit.