Constrained basin stability for studying transient phenomena in dynamical systems

Kan, Adrian van; Jegminat, Jannes; Donges, Jonathan F.; Kurths, Jüergen

doi:10.1103/physreve.93.042205

Cited by 21 publications

(12 citation statements)

References 23 publications

(40 reference statements)

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“…As we showed here, the phenomenon of a local peak in synchronization stability is related to (global) bifurcations unrelated to the state of interest. Indeed, multistability and the different basins of attraction are at the core of the notion of synchronization or basin stability in various research topics including carbon cycle dynamics [25], disease spreading [38], and forestry [39] such that this phenomenon is not surprising from a general point of view. Yet, our study highlights the dramatic extent this phenomenon can take on in power grid graphlets with at least three nodes.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…The main question is whether the power-grid system retains synchronization after being perturbed or falls into desynchronization. The volume of the basin of attraction to synchronization in the phase space of perturbation is interpreted as the probability of synchronization recovery, so-called 'basin stability', which has been widely used to quantify the synchronization stability of power grids [16,17,20,21,[24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

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Multistability and variations in basin of attraction in power-grid systems

et al. 2018

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Power grids sustain modern society by supplying electricity and thus their stability is a crucial factor for our civilization. The dynamic stability of a power grid is usually quantified by the probability of its nodes' recovery to phase synchronization of the alternating current it carries, in response to external perturbation. Intuitively, the stability of nodes in power grids is supposed to become more robust as the coupling strength between the nodes increases. However, we find a counterintuitive range of coupling strength values where the synchronization stability suddenly droops as the coupling strength increases, on a number of simple graph structures. Since power grids are designed to fulfill both local and long-range power demands, such simple graph structures or graphlets for local power transmission are indeed relevant in reality. We show that the observed nonmonotonic behavior is a consequence of transitions in multistability, which are related to changes in stability of the unsynchronized states. Therefore, our findings suggest that a comprehensive understanding of changes in multistability are necessary to prevent the unexpected catastrophic instability in the building blocks of power grids.

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Section: Summary and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multistability and variations in basin of attraction in power-grid systems

et al. 2018

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“…In contrast, CSD is usually done with (local) noise only. The usage of shocks has been developed in the context of Basin stability [30,31] and its extensions [23,[39][40][41][42].…”

Section: Discussionmentioning

confidence: 99%

“…An important emphasis on long transients has been made in [10,15,23]. With this term, they refer to trajectories where the relevant and observable phenomena/states, e.g.…”

Section: Introductionmentioning

confidence: 99%

Timing of transients: quantifying reaching times and transient behavior in complex systems

et al. 2017

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In dynamical systems, one may ask how long it takes for a trajectory to reach the attractor, i.e. how long it spends in the transient phase. Although for a single trajectory the mathematically precise answer may be infinity, it still makes sense to compare different trajectories and quantify which of them approaches the attractor earlier. In this article, we categorize several problems of quantifying such transient times. To treat them, we propose two metrics, area under distance curve and regularized reaching time, that capture two complementary aspects of transient dynamics. The first, area under distance curve, is the distance of the trajectory to the attractor integrated over time. It measures which trajectories are 'reluctant', i.e. stay distant from the attractor for long, or 'eager' to approach it right away. Regularized reaching time, on the other hand, quantifies the additional time (positive or negative) that a trajectory starting at a chosen initial condition needs to approach the attractor as compared to some reference trajectory. A positive or negative value means that it approaches the attractor by this much 'earlier' or 'later' than the reference, respectively. We demonstrated their substantial potential for application with multiple paradigmatic examples uncovering new features.

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“…Hellmann et al [31] consider the nonlocal measure survivability, reflecting the likelihood that the trajectory of a random initial condition leaves a certain desired (or allowed) subset of the systems state-space. A related paper by van Kan et al [32] considers constrained basin stability which discusses measures related to P and P τ . Interesting topics for future research are to consider nonlocal versions of existing local measures and also to compare and discuss relations between existing measures and those considered here.…”

Section: Alternative Measures and Topics For Future Researchmentioning

confidence: 99%

How to find simple nonlocal stability and resilience measures

Lundström

2018

Nonlinear Dyn

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Stability of dynamical systems is a central topic with applications in widespread areas such as economy, biology, physics and mechanical engineering. The dynamics of nonlinear systems may completely change due to perturbations forcing the solution to jump from a safe state into another, possibly dangerous, attractor. Such phenomena cannot be traced by the widespread local stability and resilience measures, based on linearizations, accounting only for arbitrary small perturbations. Using numerical estimates of the size and shape of the basin of attraction, as well as the systems returntime to the attractor after given a perturbation, we construct simple nonlocal stability and resilience measures that record a systems ability to tackle both large and small perturbations. We demonstrate our approach on the Solow-Swan model of economic growth, an electro-mechanical system, a stage-structured population model as well as on a highdimensional system, and conclude that the suggested measures detect dynamic behavior, crucial for a systems stability and resilience, which can be completely missed by local measures. The presented measures are also easy to implement on a standard laptop computer. We believe that our approach will constitute an important step toward filling a current gap in the literature by putting forward and explaining simple ideas and meth-N. L. P. Lundström (B) Department of Mathematics and Mathematical Statistics, Umeå University, 90187 Umeå, Sweden e-mail: niklas.lundstrom@umu.se ods, and by delivering explicit constructions of several promising nonlocal stability and resilience measures.

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Constrained basin stability for studying transient phenomena in dynamical systems

Cited by 21 publications

References 23 publications

Multistability and variations in basin of attraction in power-grid systems

Multistability and variations in basin of attraction in power-grid systems

Timing of transients: quantifying reaching times and transient behavior in complex systems

How to find simple nonlocal stability and resilience measures

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