Local versus global stability in dynamical systems with consecutive Hopf-Bifurcations

TL;DR

This paper investigates the contrasting behaviors of local and global stability in dynamical systems undergoing Hopf bifurcations, revealing that global stability can be unexpectedly large near loss of local stability, with insights from models and power systems.

Contribution

It demonstrates the contrasting nature of local and global stability in systems with Hopf bifurcations through theoretical analysis and practical models.

Findings

Global stability peaks near the point of local stability loss.

Contradiction between local and global stability behaviors.

Analysis of attractor locations explains the stability paradox.

Abstract

Quantifying the stability of an equilibrium is central in the theory of dynamical systems as well as in engineering and control. A comprehensive picture must include the response to both small and large perturbations, leading to the concepts of local (linear) and global stability. Here, we show how systems displaying Hopf bifurcations show contrarian results on these two aspects of stability: Global stability is large close to the point where the system loses its local stability altogether. We demonstrate this effect for an elementary model system, an anharmonic oscillator and a realistic model of power system dynamics with delayed control. Detailed investigations of the bifurcation explain the seeming paradox in terms of the location of the attractors relative to the equilibrium.