Time lags occur in a vast range of real-world dynamical systems due to finite reaction times or propagation speeds. Here we derive an analytical approach to determine the asymptotic stability of synchronous states in networks of coupled inertial oscillators with constant delay. Building on the master stability formalism, our technique provides necessary and sufficient delay master stability conditions. We apply it to two classes of potential future power grids, where processing delays in control dynamics will likely pose a challenge as renewable energies proliferate. Distinguishing between phase and frequency delay, our method offers an insight into how bifurcation points depend on the network topology of these system designs.
<p>Following Hasselmann&#8217;s ansatz, the climate system may be viewed as a multistable dynamical system internally driven by noise. Its long-term evolution will then feature noise-induced critical transitions between the competing attracting states. In the weak-noise limit, large deviation theory allows predicting the transition rate and most probable transition path of these tipping events. However, the limit of zero noise is never obtained in reality. In this work we show that, even for weak finite noise, sample transition paths may disagree with the large deviation prediction &#8211; the minimum action path, or instanton &#8211; if multiple timescales are at play. We illustrate this behavior in selected box models of the bistable Atlantic Meridional Overturning Circulation (AMOC), where different restoring times of temperature and salinity induce a fast-slow characteristic. While the minimum action path generally crosses the basin boundary at a saddle point, we demonstrate cases in which ensembles of sample transition paths cross far away from the saddle. We discuss the conditions for saddle avoidance and relate this to the flatness of the quasipotential, a central object of large deviation theory. We further probe the vicinity of the weak-noise limit by applying a pathspace method that generates transition samples for arbitrarily weak noise. Our results highlight that predictions by large deviation theory must be treated cautiously in multiscale dynamical systems.</p>
<p>The multistability of the Atlantic Meridional Overturning Circulation (AMOC) challenges the predictability of long-term climate evolution. In light of an observed weakening in AMOC strength, it is crucial to study the probabilities of noise-induced transitions between the different competing flow regimes. From a dynamical systems perspective, the phase space of a multistable system can be characterised as a non-equilibrium potential landscape, with valleys corresponding to the different basins of attraction. Knowing the potential, one can infer the statistics and pathways of noise-induced transitions. Particularly, in the weak-noise limit, transition paths lead through special regions of the basin boundaries, called Melancholia states. Recent studies have applied these concepts to climate models of low and intermediate complexity. Here, we investigate the quasi-potential landscape of a three-box model of the AMOC, based on the popular model by Rooth. We analyse noise-induced transitions between the two stable circulation states and elucidate the role of the Melancholia state. Forcing the model with different noise laws, which represent fluctuations caused by different physical processes, we discuss how the properties of transitions change when considering non-Gaussian processes, specifically L&#233;vy noise. Simulated transition rates are related to their theoretical values using the quasi-potential landscape. Our results yield a comprehensive picture of the dynamical properties of an inter-hemispheric three-box AMOC model under stochastic forcing. By relating the deterministic structure of this simple model to the statistics of critical transitions, we hope to build a basis for transferring this approach to more complex models of the AMOC.</p>
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