Solvable model of the collective motion of heterogeneous particles interacting on a sphere

Tanaka, Takuma

doi:10.1088/1367-2630/16/2/023016

Cited by 36 publications

(39 citation statements)

References 40 publications

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“…This system is a general form of the swarm sphere model(a.k.a. Lohe sphere model) [12,18,20,21], the swarm sphere model with frustration [11] and the Winfree sphere model [19](See Example 2.1 for details).…”

Section: Introductionmentioning

confidence: 99%

The Watanabe-Strogatz transform and constant of motion functionals for kinetic vector models

Park¹

2021

Preprint

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We present a kinetic version of the Watanabe-Strogatz(WS) transform for vector models in this paper. From the generalized WS-transform, we obtain the cross-ratio type constant of motion functionals for kinetic vector models under suitable conditions. We present the sufficient and necessary conditions for the existence of the suggested constant of motion functional. As an application of the constant of motion functional, we provide the instability of bipolar states of the kinetic swarm sphere model. We also provide the WS-transform and constant of motion functional for non-identical kinetic vector models.

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Section: Introductionmentioning

confidence: 99%

The Watanabe-Strogatz transform and constant of motion functionals for kinetic vector models

Park¹

2021

Preprint

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“…Many papers address the case of heterogeneous frequencies [Chi et al, 2014;Chandra et al, 2019;Ha et al, 2018]. Some concern the thermodynamic limit N → ∞ [Chi et al, 2014;Tanaka, 2014;Ha et al, 2018;Frouvelle and Liu, 2018]. There is also a discretetime model [Li, 2015].…”

Section: Introductionmentioning

confidence: 99%

High-dimensional Kuramoto models on Stiefel manifolds synchronize complex networks almost globally

2020

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The Kuramoto model of a system of coupled phase oscillators describe synchronization phenomena in nature. We propose a generalization of the Kuramoto model where each oscillator state lives on the compact, real Stiefel manifold St(p, n). Previous work on high-dimensional Kuramoto models have largely been influenced by results and techniques that pertain to the original model. This paper uses optimization and control theory to prove that the generalized Kuramoto model on St(p, n) converges to a completely synchronized state for any connected graph from almost all initial conditions provided (p, n) satisfies p ≤ 2 3 n − 1 and all oscillator frequencies are equal. This result could not have been predicted based on knowledge of the Kuramoto model in complex networks on the circle with homogeneous oscillator frequencies. In that case, almost global synchronization is graph dependent; it applies if the network is acyclic or sufficiently dense. The problem of characterizing all such graphs is still open. This paper hence identifies a property that distinguishes many highdimensional generalizations of the Kuramoto model from the original model. It should therefore have important implications for modeling of synchronization phenomena in physics and control of multi-agent systems in engineering applications.

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“…The noise free case with heterogenous frequencies has recently been proposed as a special case of a generalized Kuramoto model [8]. Watanabe-Strogatz theory [9] and the Ott-Antonsen ansatz [10] for the mean field of noise free oscillators have been shown to generalize to higher dimensions [11,12] as well. However, as a ubiquitous influence in nature, noise plays an important role in the dynamics of the colletive motion.…”

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confidence: 99%

Transition to synchrony in a three-dimensional swarming model with helical trajectories

2021

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We investigate the transition from incoherence to global collective motion in a three dimensional swarming model of agents with helical trajectories, subject to noise and global coupling. Without noise this model was recently proposed as a generalization of the Kuramoto model and it was found, that alignment of the velocities occurs for arbitrary small attractive coupling. Adding noise to the system resolves this singular limit. The transition occurs continuously at a positive critical coupling strength.

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Solvable model of the collective motion of heterogeneous particles interacting on a sphere

Cited by 36 publications

References 40 publications

The Watanabe-Strogatz transform and constant of motion functionals for kinetic vector models

The Watanabe-Strogatz transform and constant of motion functionals for kinetic vector models

High-dimensional Kuramoto models on Stiefel manifolds synchronize complex networks almost globally

Transition to synchrony in a three-dimensional swarming model with helical trajectories

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