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

We consider families of systems of two-dimensional ordinary differential equations 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 in these equations such that the stability index $σ(0)$ is precisely $σ(0)=s$. In contrast to that, for a hyperbolic equilibrium $x$ it is known that either $σ(x)=-\infty$ or $σ(x)=+\infty$. Furthermore, we discuss a system with an equilibrium that is locally unstable but globally attracting, highlighting some subtle differences between the local and non-local stability indices.