Global stability properties of equilibria, periodic, and chaotic solutions

Abstract

Nonlinear systems can exhibit multiple stable steady-state solutions at a fixed set of parameter values. A disturbance to a given attractor can cause a transition across a basin boundary resulting in a qualitative, and often substantial quantitative, change in the long-term response of the system. In this paper we examine how the global stability properties of steady-state solutions may be examined in the context of basins of attraction. Such a global approach, based on geometric considerations, may usefully be combined with a linearized stability analysis. We first examine the stability properties of point attractors and harmonic solutions, and then we extend the analysis to higher periodic and chaotic solutions. In addition we show that in situations where basins of attraction are highly intertwined there may be not only a loss of stability but also a loss of predictability.