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Quantifying the stability of an equilibrium is central in the theory ofdynamical systems as well as in engineering and control. A comprehensivepicture must include the response to both small and large perturbations,leading to the concepts of local (linear) and global stability. Here, we showhow systems displaying Hopf bifurcations show contrarian results on these twoaspects of stability: Global stability is large close to the point where thesystem loses its local stability altogether. We demonstrate this effect for anelementary model system, an anharmonic oscillator and a realistic model ofpower system dynamics with delayed control. Detailed investigations of thebifurcation explain the seeming paradox in terms of the location of theattractors relative to the equilibrium.

Local versus global stability in dynamical systems with consecutive Hopf-Bifurcations