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We consider families of systems of two-dimensional ordinary differentialequations with the origin $0$ as a non-hyperbolic equilibrium. For any number$s \in (-\infty, +\infty)$ we show that it is possible to choose a parameter inthese equations such that the stability index $\sigma(0)$ is precisely$\sigma(0)=s$. In contrast to that, for a hyperbolic equilibrium $x$ it isknown that either $\sigma(x)=-\infty$ or $\sigma(x)=+\infty$. Furthermore, wediscuss a system with an equilibrium that is locally unstable but globallyattracting, highlighting some subtle differences between the local andnon-local stability indices.

Stability Indices of Non-Hyperbolic Equilibria in Two-Dimensional Systems of ODEs