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 $\sigma(0)$ is precisely $\sigma(0)=s$. In contrast to that, for a hyperbolic equilibrium $x$ it is known that either $\sigma(x)=-\infty$ or $\sigma(x)=+\infty$. 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.