Synchronisation under shocks: The Lévy Kuramoto model

Roberts, Dale; Kalloniatis, Alexander C.

doi:10.1016/j.physd.2017.12.005

Cited by 10 publications

(7 citation statements)

References 34 publications

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“…On this point, firstly there are more elements in this context than have been included in the present model, such as stochasticity [27]; future work will test whether the modified expander hierarchy is also robust against noise. However, there is evidence of a strong anti-correlation between the synchronisation on networks under increasing coupling with increasing noise 'strength' for both uniform [28], Gaussian and Lévy noise [44], which we anticipate would apply here. Secondly, there is the added layer of a mathematical representation of the business of the organisationwhat it is it makes decisions about-in which we are conducting ongoing work.…”

Section: Discussionsupporting

confidence: 52%

Organisational hierarchy constructions with easy Kuramoto synchronisation

Taylor

Kalloniatis

Hoek

2020

J. Phys. A: Math. Theor.

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We provide a graph theoretic construction enabling tree-based hierarchies to be modified into structures with enhanced connectivity and synchronisability. Specifically, the construct transforms trees into members of a family of graphs known as expanders, which we call 'expander-augmented-hierarchies' or trees. We show that this produces graphs with significantly enhanced synchronisation properties in the context of the Kuramoto model of phase oscillators coupled on networks. When considered as organisational structures these networks enjoy both the managability of simple hierarchies with near regular degree distribution, and low critical coupling by the addition of relatively few extra edges. For the expander augmented tree, we examine the synchronisation properties, computed through the time-averaged Kuramoto order parameter over an ensemble of natural frequencies. We compare this with a range of other networks including hierarchies augmented by random matching of the leaf nodes. For these we compute the ratio Q of smallest to largest Laplacian eigenvalues, the smallness of which has been argued to be an indicator of good synchronisability. While not the best of these in Q, the expander-augmentedhierarchy exhibits synchronisability barely distinguishable from others with lower Q within the variance over an ensemble of natural frequencies. However, the expander augmented tree has the advantage that its properties are automatically designed for as opposed to the outcome of a random search for low Q-value graphs that in itself scales very poorly.

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Section: Discussionsupporting

confidence: 52%

Organisational hierarchy constructions with easy Kuramoto synchronisation

Taylor

Kalloniatis

Hoek

2020

J. Phys. A: Math. Theor.

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“…Intuitively, we propose that the underlying reason for this behaviour lies in an interplay of the ratchet mechanism with tempering and the fundamental change in the stable noise probability density for α < 1. As illustrated in [29], for α < 1 the density is no longer centred around zero but with a heavy tail that is increasingly suppressed as λ increases. This centering becomes the dominant effect with tempering, and seemingly acts to counter the deterministic drift in the ratchet potential.…”

Section: Discussionmentioning

confidence: 99%

“…It has been shown that richer behaviour in the dynamics of such models are observed under various forms of Gaussian white or coloured noise perturbations [3,[20][21][22][23][24][25][26][27]. Recent studies have also shown relationships between the heaviness of the noise distribution's tails and synchronisation behaviour of the Kuramoto model [28,29]. The case of tempered stable noise, which we consider here, is particularly interesting due to its relation with tempered fractional diffusion [30][31][32][33][34][35] and are part of broader class of processes called Lévy processes.…”

Section: Introductionmentioning

confidence: 89%

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Two-network Kuramoto-Sakaguchi model under tempered stable Lévy noise

et al. 2019

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We examine a model of two interacting populations of phase oscillators labelled 'Blue' and 'Red'. To this we apply tempered stable Lévy noise, a generalisation of Gaussian noise where the heaviness of the tails parametrised by a power law exponent α can be controlled by a tempering parameter λ . This system models competitive dynamics, where each population seeks both internal phase synchronisation and a phase advantage with respect to the other population, subject to exogenous stochastic shocks. We study the system from an analytic and numerical point of view to understand how the phase lag values and the shape of the noise distribution can lead to steady or noisy behaviour. Comparing the analytic and numerical studies shows that the bulk behaviour of the system can be effectively described by dynamics in the presence of tilted ratchet potentials. Generally, changes in α away from the Gaussian noise limit, 1 < α < 2, disrupts the locking between Blue and Red, while increasing λ acts to restore it. However we observe that with further decreases of α to small values, α 1, with λ = 0, locking between Blue and Red may be restored. This is seen analytically in a restoration of metastability through the ratchet mechanism, and numerically in transitions between periodic and noisy regions in a fitness landscape using a measure of noise. This non-monotonic transition back to an ordered regime is surprising for a linear variation of a parameter such as the power law exponent and provides a novel mechanism for guiding the collective behaviour of such a complex competitive dynamical system.

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“…Since oscillator networks are ubiquitous in many realworld settings where noise and uncertainty play a significant role, there is growing interest in understanding the effects of dynamical perturbations and noise on network synchronization 5,[20][21][22][23][24][25] . Perhaps the most interesting and important effect of noise in nonlinear network dynamics is the tendency to produce large qualitative changes in the behavior over time, called large fluctuations (LFs).…”

Section: Introductionmentioning

confidence: 99%

Rare slips in fluctuating synchronized oscillator networks

Hindes

Schwartz

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We study rare phase slips due to noise in synchronized Kuramoto oscillator networks. In the small-noise limit, we demonstrate that slips occur via large fluctuations to saddle phase-locked states. For tree topologies, slips appear between subgraphs that become disconnected at a saddle-node bifurcation, where phase-locked states lose stability generically. This pattern is demonstrated for sparse networks with several examples. Scaling laws are derived and compared for different tree topologies. On the other hand, for dense networks slips occur between oscillators on the edges of the frequency distribution. If the distribution is discrete, the probability-exponent for large fluctuations to occur scales linearly with the system size. However, if the distribution is continuous, the probability is a constant in the large network limit, as individual oscillators fluctuate to saddles while all others remain fixed. In the latter case, the network's coherence is approximately preserved.

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Synchronisation under shocks: The Lévy Kuramoto model

Cited by 10 publications

References 34 publications

Organisational hierarchy constructions with easy Kuramoto synchronisation

Organisational hierarchy constructions with easy Kuramoto synchronisation

Two-network Kuramoto-Sakaguchi model under tempered stable Lévy noise

Rare slips in fluctuating synchronized oscillator networks

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