Recovery time after localized perturbations in complex dynamical networks

Abstract

Maintaining the synchronous motion of dynamical systems interacting on complex networks is often critical to their functionality. However, real-world networked dynamical systems operating synchronously are prone to random perturbations driving the system to arbitrary states within the corresponding basin of attraction, thereby leading to epochs of desynchronized dynamics with a priori unknown durations. Thus, it is highly relevant to have an estimate of the duration of such transient phases before the system returns to synchrony, following a random perturbation to the dynamical state of any particular node of the network. We address this issue here by proposing the framework of single-node recovery time (SNRT) which provides an estimate of the relative time scales underlying the transient dynamics of the nodes of a network during its restoration to synchrony. We utilize this in differentiating the particularly slow nodes of the network from the relatively fast nodes, thus identifying the critical nodes which when perturbed lead to significantly enlarged recovery time of the system before resuming synchronized operation. Further, we reveal explicit relationships between the SNRT values of a network, and its global relaxation time when starting all the nodes from random initial conditions. Earlier work on relaxation time generally focused on investigating its dependence on macroscopic topological properties of the respective network. However, we employ the proposed concept for deducing microscopic relationships between topological features of nodes and their respective SNRT values. The framework of SNRT is further extended to a measure of resilience of the different nodes of a networked dynamical system. We demonstrate the potential of SNRT in networks of Rössler oscillators on paradigmatic topologies and a model of the power grid of the United Kingdom with second-order Kuramoto-type nodal dynamics illustrating the conceivable practical applicability of the proposed concept.

Power grid stability under perturbation of single nodes: Effects of heterogeneity and internal nodes

“…We therefore characterise the grid stability not only by the probability of return but also by the return times for the perturbations belonging to the attraction basin of the fixed point. A similar concept has been recently introduced in [18].…”

From lakes and glades to viability algorithms: automatic classification of system states according to the topology of sustainable management

“…b indicates strong increases in the size of the glade and dark upstream with stronger management of the energy transition, corresponding to lower relative costs for renewable energy and smaller values of σET This allows us to use the relative volume of each region as an indicator for tsm bifurcations, motivated by the concept of Basin Stability [67,68] and its extensions [69][70][71][72][73][74][75]. We use uniformly distributed points in state space for the Saint-Pierre algorithm.…”

Decreased resilience in power grids under dynamically induced vulnerabilities

“…Recently, the scientific community has put a lot of effort in trying to determine the topological or structural features of those interactions that can enhance or undermine non-linear stability of a power grid, that is, its capability to reject finitesize disturbances, a matter of paramount importance in the design of the future smart grids. This has been approached, for instance, by the means of energy barrier functions [19] and the basin stability concept [20,21,22,23,24,25]. Some interesting findings that deserve to be mentioned, as they provide great insights about the relationship between topology and dynamical stability include: the poor basin stability usually detected on dead-tree arrangements [20], the strong stability found on triangle-shaped motifs [26], the enhanced non-linear stability achieved by increasing global redundancy in the connections [27] or by adding small cyclic motifs [19] and the lower basin stability exhibited in general by high-power generator nodes in [28].…”