Forecasting the resilience of networked dynamical systems under environmental perturbation

Abstract

Many real life systems can be viewed as networked systems that are composed by interconnected compartments which exchange mass or energy between each other and with their environment through fluxes. Such interaction with the environment make these systems subject to external perturbations that cause systems parameters to vary away from the nominal values. For nonlinear networked systems, such parameter variations can change the qualitative behaviors of the system (i.e. phase portrait or stability) through a bifurcation [6]. These changes may result in a regime shifts [8] in which the system ”flips” from a nominal operating state to an alternative state. Regime-shifts can be catastrophic for users who have grown accustomed to the quality of services provided by the system prior to the shift. Examples of this can be found in the eutrophication of shallow lakes as a result of human-induced nutrient enrichment or the decline of fisheries due to overfishing practices [8]. Another prime example occurs when voltage collapses cascade through the electric power network [4]. Each of these shifts has the potential to disrupt the services that these systems provide to the society. Forecasting the resilience of these networked systems to parameter variations is therefore crucial for managing their security and sustainability [1, 5]. Consider networked systems ẋ = f(x, μ) whose equilibrium x⇤ depend on parameter μ. The resilience of a system under parameter variation can be measured by the distance = |μ⇤ μ| between the nominal parameter μ and the closest critical paramater μ⇤ at which a bifurcation occur. The quantity , often called distance to closest bifurcation,