Stability indices of non-hyperbolic equilibria in two-dimensional systems of ODEs

Abstract

We consider families of systems of two-dimensional ordinary differential equations with the origin 0 as a non-hyperbolic equilibrium. For any number $s\in (-\mathrm{\infty },+\mathrm{\infty })$, we show that it is possible to choose a parameter in these equations such that the stability index $\sigma \left(0\right)$ is precisely $\sigma \left(0\right)=s$. In contrast to that, for a hyperbolic equilibrium x it is known that either $\sigma \left(x\right)=-\mathrm{\infty }$ or $\sigma \left(x\right)=+\mathrm{\infty }$. 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.

Keywords:

• Stability
• attraction
• non-hyperbolic equilibrium

AMS classifications:

• 34D20
• 37C25
• 37C75

Acknowledgments

The author is grateful for insightful comments by two anonymous reviewers of an earlier version of this work.

Disclosure statement

No potential conflict of interest was reported by the author(s).