Resilience as a Dynamical Property of Risk Trajectories in CPSoS

Abstract

Resilience in cyber-physical systems of systems (CPSoS) is often assessed using static indices or point-in-time metrics that do not adequately account for the temporal evolution of risk following a disruption. This paper formalizes resilience as a functional of the risk trajectory by modelling risk as a dynamic state variable. It is analytically shown that key resilience properties are structurally determined by maximum deviation (peak) and effective damping, and that cumulative risk exposure depends on their ratio. A simplified energy-dependent system illustrates the resulting differences in peak magnitude, recovery dynamics, and cumulative impact. The proposed approach links resilience assessment to stability properties of dynamic systems and provides a system-theoretically consistent foundation for the analysis of time-dependent resilience in CPSoS.