Survivability of Deterministic Dynamical Systems

TL;DR

This paper introduces survivability as a new measure of stability for deterministic systems, assessing the likelihood that transient behavior remains within desirable states, with applications across climate, neural, and power grid models.

Contribution

It defines survivability as a novel stability concept and demonstrates its utility through models from various fields, including a semi-analytic lower bound for linear systems.

Findings

Survivability provides insights beyond traditional asymptotic stability.

Semi-analytic bounds enable efficient analysis of power grid models.

Applications demonstrate survivability's relevance in climate, neural, and power systems.

Abstract

The notion of a part of phase space containing desired (or allowed) states of a dynamical system is important in a wide range of complex systems research. It has been called the safe operating space, the viability kernel or the sunny region. In this paper we define the notion of survivability: Given a random initial condition, what is the likelihood that the transient behaviour of a deterministic system does not leave a region of desirable states. We demonstrate the utility of this novel stability measure by considering models from climate science, neuronal networks and power grids. We also show that a semi-analytic lower bound for the survivability of linear systems allows a numerically very efficient survivability analysis in realistic models of power grids. Our numerical and semi-analytic work underlines that the type of stability measured by survivability is not captured by common asymptotic stability measures.