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 epsilon (-infinity,+infinity), 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) = -infinity or sigma(x) = +infinity. 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.