Stability Indices of Non-Hyperbolic Equilibria

Alexander Lohse

Published 2022-03-18Version 1

We consider families of systems of two-dimensional ordinary differential equations with the origin $0$ as a non-hyperbolic equilibrium. For any number $a \in (-\infty, +\infty)$ we show that it is possible to choose a parameter in these equations such that the stability index $\sigma(0)$ is precisely $\sigma(0)=a$. 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 seems to be globally attracting, highlighting some subtle differences between the local and non-local stability indices.