Abstract
Multistability is a common phenomenon which naturally occurs in complex networks. If coexisting attractors are numerous and their basins of attraction are complexly interwoven, the long-term response to a perturbation can be highly uncertain. We examine the uncertainty in the outcome of perturbations to the synchronous state in a Kuramoto-like representation of the British power grid. Based on local basin landscapes which correspond to single-node perturbations, we demonstrate that the uncertainty shows strong spatial variability. While perturbations at many nodes only allow for a few outcomes, other local landscapes show extreme complexity with more than a hundred basins. Particularly complex domains in the latter can be related to unstable invariant chaotic sets of saddle type. Most importantly, we show that the characteristic dynamics on these chaotic saddles can be associated with certain topological structures of the network. We find that one particular tree-like substructure allows for the chaotic response to perturbations at nodes in the north of Great Britain. The interplay with other peripheral motifs increases the uncertainty in the system response even further.
Highlights
The Kuramoto model with inertia (KM+) has been the subject of tremendous research efforts within the last decade [1]
Inspired by findings in an earlier work [26], we examined the uncertainty in the outcome of localized perturbations within a Kuramoto-like representation of the British power grid
Using local basin landscapes corresponding to perturbations at single nodes, we demonstrated that the basin complexity and the uncertainty in the outcome of a perturbation is highly variable with low uncertainty predominantly in the core of the network and high uncertainty concentrated close to peripheral regions
Summary
The Kuramoto model with inertia (KM+) has been the subject of tremendous research efforts within the last decade [1]. If small perturbations are considered, eigenvalues of the operating state can be consulted, whereas properties of its basin of attraction are more suitable if perturbations under consideration are large The latter explicitly takes into account that power grids – and reasonable parametrizations of the KM+ – exhibit multistability. In a recent study [26], we contributed to the field of multistability-based stability analyses by determining the minimal fatal shock in a Kuramoto-like representation of the British power grid (figure 1(a)). A high sensitivity towards small variations in the perturbation is an indicator for complexly interwoven basins of attraction possessing fractal boundaries [36, 37] This impression appears to be confirmed by the erratic – seemingly chaotic – transient dynamics following the fatal shock (figure 1(b),(c)). We provide a short parameter study and conclude with a discussion of our results (section 4)
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