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Dai X, Li X, Guo H, Jia D, Perc M, Manshour P, Wang Z, Boccaletti S. Discontinuous Transitions and Rhythmic States in the D-Dimensional Kuramoto Model Induced by a Positive Feedback with the Global Order Parameter. PHYSICAL REVIEW LETTERS 2020; 125:194101. [PMID: 33216569 DOI: 10.1103/physrevlett.125.194101] [Citation(s) in RCA: 18] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2020] [Revised: 08/28/2020] [Accepted: 09/01/2020] [Indexed: 06/11/2023]
Abstract
From fireflies to cardiac cells, synchronization governs important aspects of nature, and the Kuramoto model is the staple for research in this area. We show that generalizing the model to oscillators of dimensions higher than 2 and introducing a positive feedback mechanism between the coupling and the global order parameter leads to a rich and novel scenario: the synchronization transition is explosive at all even dimensions, whilst it is mediated by a time-dependent, rhythmic, state at all odd dimensions. Such a latter circumstance, in particular, differs from all other time-dependent states observed so far in the model. We provide the analytic description of this novel state, which is fully corroborated by numerical calculations. Our results can, therefore, help untangle secrets of observed time-dependent swarming and flocking dynamics that unfold in three dimensions, and where this novel state could thus provide a fresh perspective for as yet not understood formations.
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Affiliation(s)
- X Dai
- Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi'an 710072, China
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China
- Unmanned Systems Research Institute, Northwestern Polytechnical University, Xi'an 710072, China
| | - X Li
- Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi'an 710072, China
- School of Computer Science, Northwestern Polytechnical University, Xian 710072, China
| | - H Guo
- Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi'an 710072, China
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China
- Unmanned Systems Research Institute, Northwestern Polytechnical University, Xi'an 710072, China
| | - D Jia
- Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi'an 710072, China
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China
- Unmanned Systems Research Institute, Northwestern Polytechnical University, Xi'an 710072, China
| | - M Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
- Complexity Science Hub Vienna, Josefstädterstraße 39, 1080 Vienna, Austria
- Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 404, Taiwan
| | - P Manshour
- Physics Department, Persian Gulf University, Bushehr 75169, Iran
| | - Z Wang
- Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi'an 710072, China
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China
| | - S Boccaletti
- Unmanned Systems Research Institute, Northwestern Polytechnical University, Xi'an 710072, China
- CNR-Institute of Complex Systems, Via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy
- Moscow Institute of Physics and Technology (National Research University), 9 Institutskiy per., Dolgoprudny, Moscow Region 141701, Russian Federation
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Qiu T, Boccaletti S, Liu Z, Guan S. Characterizing nonstationary coherent states in globally coupled conformist and contrarian oscillators. Phys Rev E 2019; 100:052310. [PMID: 31870024 DOI: 10.1103/physreve.100.052310] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/28/2019] [Indexed: 11/07/2022]
Abstract
For decades, the description and characterization of nonstationary coherent states in coupled oscillators have not been available. We here consider the Kuramoto model consisting of conformist and contrarian oscillators. In the model, contrarians are chosen from a bimodal Lorentzian frequency distribution and flipped into conformists at random. A complete and systematic analytical treatment of the model is provided based on the Ott-Antonsen ansatz. In particular, we predict and analyze not only the stability of all stationary states (such as the incoherent, the π, and the traveling-wave states), but also that of the two nonstationary states: the Bellerophon and the oscillating-π state. The theoretical predictions are fully supported by extensive numerical simulations.
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Affiliation(s)
- Tian Qiu
- Department of Physics, East China Normal University, Shanghai 200241, China.,Institute of Condensed Matter and Material Physics, School of Physics, Peking University, Beijing 100871, China
| | - S Boccaletti
- CNR-Institute of Complex Systems, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Florence, Italy.,Unmanned Systems Research Institute, Northwestern Polytechnical University, Xi'an 710072, China
| | - Zonghua Liu
- Department of Physics, East China Normal University, Shanghai 200241, China
| | - Shuguang Guan
- Department of Physics, East China Normal University, Shanghai 200241, China
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