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

TL;DR

This paper explores the stability indices of non-hyperbolic equilibria in 2D ODE systems, showing they can be tuned to any real value, unlike hyperbolic cases, and discusses differences between local and global stability.

Contribution

It demonstrates that stability indices of non-hyperbolic equilibria can be arbitrarily assigned, contrasting with hyperbolic cases, and analyzes local versus global stability in such systems.

Findings

Stability index $\sigma(0)$ can be any real number.

Hyperbolic equilibria have stability indices of $\pm\infty$.

Existence of systems with locally unstable but globally attracting equilibria.

Abstract

We consider families of systems of two-dimensional ordinary differential equations with the origin $0$ as a non-hyperbolic equilibrium. For any number $s \in (-\infty, +\infty)$ 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.