  Quantifying the stability of an equilibrium is central in the theory of
dynamical systems as well as in engineering and control. A comprehensive
picture must include the response to both small and large perturbations,
leading to the concepts of local (linear) and global stability. Here, we show
how systems displaying Hopf bifurcations show contrarian results on these two
aspects of stability: Global stability is large close to the point where the
system loses its local stability altogether. We demonstrate this effect for an
elementary model system, an anharmonic oscillator and a realistic model of
power system dynamics with delayed control. Detailed investigations of the
bifurcation explain the seeming paradox in terms of the location of the
attractors relative to the equilibrium.
