Chaos in Kuramoto oscillator networks

Bick, Christian; Panaggio, Mark J.; Martens, Erik Andreas

doi:10.1063/1.5041444

Cited by 48 publications

(52 citation statements)

References 34 publications

(53 reference statements)

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“…Fig. 4(a)] for which the transition occurs at lower values of K. This discrepancy is due to the fact that the collective coordinate models (19)- (20) and (21)-(22) do not account for partial synchronization of the clusters.…”

Section: Four Clusters: Collective Phase Chaosmentioning

confidence: 99%

“…We now numerically explore these cases for the full Kuramoto model (1) with N = 100 oscillators; and shall compare our results with the reduced collective coordinate description (19)- (20) for N = 100 oscillators as well as with the reduced collective coordinate description (21)-(22) in the thermodynamic limit of infinitely many oscillators. The collective coordinate systems involve 7 degrees of freedom: four shape parameters β m and three phase-difference variables f m+1 − f m (the evolution equations, however, are written for f m and hence are 8-dimensional).…”

Section: Four Clusters: Collective Phase Chaosmentioning

confidence: 99%

“…Shown is the order parameterr for the full Kuramoto model (1) with N = 100 oscillators [ Fig. 4(a)], the 8D collective coordinate model (19)- (20) with N = 100 oscillators [ Fig. 4(c)], and the 8D collective coordinate model with infinitely many oscillators (21)- (22) [Fig.…”

Section: Four Clusters: Collective Phase Chaosmentioning

confidence: 99%

“…Such collective chaos has been studied for systems with symmetric bimodal natural frequency distributions which were subjected to a timeperiodic coupling strength, 19 or for different inter-and intra-cluster coupling strengths as well as a phase lag. 20 However, for the classical Kuramoto model, it has been shown that in the thermodynamic limit with bimodal arXiv:1905.02859v2 [nlin.AO] 4 Oct 2019 natural frequency distributions chaos is impossible. 19 For trimodal frequency distributions, which yield three synchronized subpopulations, chaos has been observed for superposed Lorentzian natural frequency distributions, but only in the partially synchronized state, which involves microscopic chaos of incoherent oscillators.…”

Section: Introductionmentioning

confidence: 99%

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Chaos in networks of coupled oscillators with multimodal natural frequency distributions

Smith

Gottwald

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We explore chaos in the Kuramoto model with multimodal distributions of the natural frequencies of oscillators and provide a comprehensive description under what conditions chaos occurs. For a natural frequency distribution with M peaks it is typical that there is a range of coupling strengths such that oscillators belonging to each peak form a synchronized cluster, but the clusters do not globally synchronize. We use collective coordinates to describe the inter-and intra-cluster dynamics, which reduces the Kuramoto model to 2M − 1 degrees of freedom. We show that under some assumptions, there is a time-scale splitting between the slow intracluster dynamics and fast intercluster dynamics, which reduces the collective coordinate model to an M − 1 degree of freedom rescaled Kuramoto model. Therefore, four or more clusters are required to yield the three degrees of freedom necessary for chaos. However, the time-scale splitting breaks down if a cluster intermittently desynchronizes. We show that this intermittent desynchronization provides a mechanism for chaos for trimodal natural frequency distributions. In addition, we use collective coordinates to show analytically that chaos cannot occur for bimodal frequency distributions, even if they are asymmetric and if intermittent desynchronization occurs.

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Section: Four Clusters: Collective Phase Chaosmentioning

confidence: 99%

Section: Four Clusters: Collective Phase Chaosmentioning

confidence: 99%

Section: Four Clusters: Collective Phase Chaosmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Chaos in networks of coupled oscillators with multimodal natural frequency distributions

Smith

Gottwald

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Does this say something about the order of the multibody interactions needed in the phase reduction? Is this chaos connected to collective chaos in the thermodynamic limit, as in [60]?…”

Section: Towards a Minimal Phase Model Of Pure Collective Chaosmentioning

confidence: 99%

Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation

León

Pazó

2019

Phys. Rev. E

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Phase reduction is a powerful technique that makes possible to describe the dynamics of a weakly perturbed limit-cycle oscillator in terms of its phase. For ensembles of oscillators, a classical example of phase reduction is the derivation of the Kuramoto model from the mean-field complex Ginzburg-Landau equation (MF-CGLE). Still, the Kuramoto model is a first-order phase approximation that displays either full synchronization or incoherence, but none of the nontrivial dynamics of the MF-CGLE. This fact calls for an expansion beyond the first order in the coupling constant. We develop an isochron-based scheme to obtain the second-order phase approximation, which reproduces the weak coupling dynamics of the MF-CGLE. The practicality of our method is evidenced by extending the calculation up to third order. Each new term of the power series expansion contributes with additional higher-order multi-body (i.e. non-pairwise) interactions. This points to intricate multi-body phase interactions as the source of pure collective chaos in the MF-CGLE at moderate coupling. II. MEAN-FIELD COMPLEX GINZBURG-LANDAU EQUATIONThe MF-CGLE consists of N diffusively coupled Stuart-Landau oscillators governed by N coupled (complex-valued)

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Fixed‐time Synchronization of Stochastic Kuramoto‐Oscillator Networks With Switching Topology Under Dynamic Event‐triggered Scheme

Wang,

Sun,

et al. 2024

Intl J Robust & Nonlinear

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This paper proposes a dynamic event‐triggered fixed‐time control scheme for the phase synchronization of Kuramoto networks with switching topology and stochastic coupling noise. To deal with the problems of switching topology and stochastic coupling noise, a multilayered control structure with a specially designed noise reduction layer is developed. The effectiveness is proved based on stochastic differential theory. To save communication resources and reduce control frequency, a dynamic event‐triggered mechanism is proposed. By introducing a dynamic variable into the triggering condition, the triggered instants are further reduced compared to the traditional static event‐triggered method. The synchronizing criteria and parameter conditions are established with stochastic Lyapunov stability theory. The analysis of Zeno behavior is also given. Finally, numerical simulation experiments are conducted to demonstrate the effectiveness of the proposed dynamic event‐triggered control scheme.

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Chaos in Kuramoto oscillator networks

Cited by 48 publications

References 34 publications

Chaos in networks of coupled oscillators with multimodal natural frequency distributions

Chaos in networks of coupled oscillators with multimodal natural frequency distributions

Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation

Fixed‐time Synchronization of Stochastic Kuramoto‐Oscillator Networks With Switching Topology Under Dynamic Event‐triggered Scheme

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