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Lee S, Braun L, Bönisch F, Schröder M, Thümler M, Timme M. Complexified synchrony. CHAOS (WOODBURY, N.Y.) 2024; 34:053141. [PMID: 38814675 DOI: 10.1063/5.0205897] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2024] [Accepted: 05/06/2024] [Indexed: 05/31/2024]
Abstract
The Kuramoto model and its generalizations have been broadly employed to characterize and mechanistically understand various collective dynamical phenomena, especially the emergence of synchrony among coupled oscillators. Despite almost five decades of research, many questions remain open, in particular, for finite-size systems. Here, we generalize recent work [Thümler et al., Phys. Rev. Lett. 130, 187201 (2023)] on the finite-size Kuramoto model with its state variables analytically continued to the complex domain and also complexify its system parameters. Intriguingly, systems of two units with purely imaginary coupling do not actively synchronize even for arbitrarily large magnitudes of the coupling strengths, |K|→∞, but exhibit conservative dynamics with asynchronous rotations or librations for all |K|. For generic complex coupling, both traditional phase-locked states and asynchronous states generalize to complex locked states, fixed points off the real subspace that exist even for arbitrarily weak coupling. We analyze a new collective mode of rotations exhibiting finite, yet arbitrarily large rotation numbers. Numerical simulations for large networks indicate a novel form of discontinuous phase transition. We close by pointing to a range of exciting questions for future research.
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Affiliation(s)
- Seungjae Lee
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
| | - Lucas Braun
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
- Schülerforschungszentrum Südwürttemberg (SFZ), 88348 Bad Saulgau, Germany
- Gymnasium Wilhelmsdorf, Pfrunger Straße 4/2, 88271 Wilhelmsdorf, Germany
| | - Frieder Bönisch
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
| | - Malte Schröder
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
| | - Moritz Thümler
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
| | - Marc Timme
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
- Cluster of Excellence Physics of Life, TU Dresden, 01062 Dresden, Germany
- Lakeside Labs, 9020 Klagenfurt, Austria
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2
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Zou W. Solvable dynamics of the three-dimensional Kuramoto model with frequency-weighted coupling. Phys Rev E 2024; 109:034215. [PMID: 38632753 DOI: 10.1103/physreve.109.034215] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2024] [Accepted: 03/06/2024] [Indexed: 04/19/2024]
Abstract
The presence of coupling heterogeneity is deemed to be a natural attribute in realistic systems comprised of many interacting agents. In this work, we study dynamics of the 3D Kuramoto model with heterogeneous couplings, where the strength of coupling for each agent is weighted by its intrinsic rotation frequency. The critical coupling strength for the instability of incoherence is rigorously derived to be zero by carrying out a linear stability analysis of an incoherent state. For positive values of the coupling strength, at which the incoherence turns out to be unstable, a self-consistency approach is developed to theoretically predict the degree of global coherence of the model. Our theoretical analyses match well with numerical simulations, which helps us to deepen the understanding of collective behaviors spontaneously emerged in heterogeneously coupled high-dimensional dynamical networks.
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Affiliation(s)
- Wei Zou
- School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
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3
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Zheng Z, Xu C, Fan J, Liu M, Chen X. Order parameter dynamics in complex systems: From models to data. CHAOS (WOODBURY, N.Y.) 2024; 34:022101. [PMID: 38341762 DOI: 10.1063/5.0180340] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2023] [Accepted: 12/14/2023] [Indexed: 02/13/2024]
Abstract
Collective ordering behaviors are typical macroscopic manifestations embedded in complex systems and can be ubiquitously observed across various physical backgrounds. Elements in complex systems may self-organize via mutual or external couplings to achieve diverse spatiotemporal coordinations. The order parameter, as a powerful quantity in describing the transition to collective states, may emerge spontaneously from large numbers of degrees of freedom through competitions. In this minireview, we extensively discussed the collective dynamics of complex systems from the viewpoint of order-parameter dynamics. A synergetic theory is adopted as the foundation of order-parameter dynamics, and it focuses on the self-organization and collective behaviors of complex systems. At the onset of macroscopic transitions, slow modes are distinguished from fast modes and act as order parameters, whose evolution can be established in terms of the slaving principle. We explore order-parameter dynamics in both model-based and data-based scenarios. For situations where microscopic dynamics modeling is available, as prototype examples, synchronization of coupled phase oscillators, chimera states, and neuron network dynamics are analytically studied, and the order-parameter dynamics is constructed in terms of reduction procedures such as the Ott-Antonsen ansatz, the Lorentz ansatz, and so on. For complicated systems highly challenging to be well modeled, we proposed the eigen-microstate approach (EMP) to reconstruct the macroscopic order-parameter dynamics, where the spatiotemporal evolution brought by big data can be well decomposed into eigenmodes, and the macroscopic collective behavior can be traced by Bose-Einstein condensation-like transitions and the emergence of dominant eigenmodes. The EMP is successfully applied to some typical examples, such as phase transitions in the Ising model, climate dynamics in earth systems, fluctuation patterns in stock markets, and collective motion in living systems.
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Affiliation(s)
- Zhigang Zheng
- Institute of Systems Science, Huaqiao University, Xiamen 361021, China and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
| | - Can Xu
- Institute of Systems Science, Huaqiao University, Xiamen 361021, China and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
| | - Jingfang Fan
- School of Systems Science, Beijing Normal University, Beijing 100875, China and Institute of Nonequilibrium Systems, Beijing Normal University, Beijing 100875, China
| | - Maoxin Liu
- School of Systems Science, Beijing Normal University, Beijing 100875, China and Institute of Nonequilibrium Systems, Beijing Normal University, Beijing 100875, China
| | - Xiaosong Chen
- School of Systems Science, Beijing Normal University, Beijing 100875, China and Institute of Nonequilibrium Systems, Beijing Normal University, Beijing 100875, China
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Zou W, He S, Senthilkumar DV, Kurths J. Solvable Dynamics of Coupled High-Dimensional Generalized Limit-Cycle Oscillators. PHYSICAL REVIEW LETTERS 2023; 130:107202. [PMID: 36962012 DOI: 10.1103/physrevlett.130.107202] [Citation(s) in RCA: 8] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/29/2022] [Accepted: 02/10/2023] [Indexed: 06/18/2023]
Abstract
We introduce a new model consisting of globally coupled high-dimensional generalized limit-cycle oscillators, which explicitly incorporates the role of amplitude dynamics of individual units in the collective dynamics. In the limit of weak coupling, our model reduces to the D-dimensional Kuramoto phase model, akin to a similar classic construction of the well-known Kuramoto phase model from weakly coupled two-dimensional limit-cycle oscillators. For the practically important case of D=3, the incoherence of the model is rigorously proved to be stable for negative coupling (K<0) but unstable for positive coupling (K>0); the locked states are shown to exist if K>0; in particular, the onset of amplitude death is theoretically predicted. For D≥2, the discrete and continuous spectra for both locked states and amplitude death are governed by two general formulas. Our proposed D-dimensional model is physically more reasonable, because it is no longer constrained by fixed amplitude dynamics, which puts the recent studies of the D-dimensional Kuramoto phase model on a stronger footing by providing a more general framework for D-dimensional limit-cycle oscillators.
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Affiliation(s)
- Wei Zou
- School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
| | - Sujuan He
- School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
| | - D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695551, Kerala, India
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, Telegraphenberg, Potsdam D-14415, Germany
- Institute of Physics, Humboldt University Berlin, Berlin D-12489, Germany
- Research Institute of Intelligent Complex Systems, Fudan University, Shanghai 200433, China
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Zheng C, Tönjes R. Noise-induced swarming of active particles. Phys Rev E 2022; 106:064601. [PMID: 36671170 DOI: 10.1103/physreve.106.064601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/03/2022] [Accepted: 11/09/2022] [Indexed: 06/17/2023]
Abstract
We report on the effect of spatially correlated noise on the velocities of self-propelled particles. Correlations in the random forces acting on self-propelled particles can induce directed collective motion, i.e., swarming. Even with repulsive coupling in the velocity directions, which favors a disordered state, strong correlations in the fluctuations can align the velocities locally leading to a macroscopic, turbulent velocity field. On the other hand, while spatially correlated noise is aligning the velocities locally, the swarming transition to globally directed motion is inhibited when the correlation length of the noise is nonzero, but smaller than the system size. We analyze the swarming transition in d-dimensional space in a mean field model of globally coupled velocity vectors.
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Affiliation(s)
- Chunming Zheng
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany
| | - Ralf Tönjes
- Institute of Physics and Astronomy, Potsdam University, 14476 Potsdam-Golm, Germany
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Buzanello GL, Barioni AED, de Aguiar MAM. Matrix coupling and generalized frustration in Kuramoto oscillators. CHAOS (WOODBURY, N.Y.) 2022; 32:093130. [PMID: 36182358 DOI: 10.1063/5.0108672] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2022] [Accepted: 08/29/2022] [Indexed: 06/16/2023]
Abstract
The Kuramoto model describes the synchronization of coupled oscillators that have different natural frequencies. Among the many generalizations of the original model, Kuramoto and Sakaguchi (KS) proposed a frustrated version that resulted in dynamic behavior of the order parameter, even when the average natural frequency of the oscillators is zero. Here, we consider a generalization of the frustrated KS model that exhibits new transitions to synchronization. The model is identical in form to the original Kuramoto model but written in terms of unit vectors and with the coupling constant replaced by a coupling matrix. The matrix breaks the rotational symmetry and forces the order parameter to point in the direction of the eigenvector with the highest eigenvalue, when the eigenvalues are real. For complex eigenvalues, the module of order parameter oscillates while it rotates around the unit circle, creating active states. We derive the complete phase diagram for the Lorentzian distribution of frequencies using the Ott-Antonsen ansatz. We also show that changing the average value of the natural frequencies leads to further phase transitions where the module of the order parameter goes from oscillatory to static.
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Affiliation(s)
- Guilhermo L Buzanello
- Instituto de Física "Gleb Wataghin," Universidade Estadual de Campinas, Unicamp, 13083-970 Campinas, SP, Brazil
| | - Ana Elisa D Barioni
- Instituto de Física "Gleb Wataghin," Universidade Estadual de Campinas, Unicamp, 13083-970 Campinas, SP, Brazil
| | - Marcus A M de Aguiar
- Instituto de Física "Gleb Wataghin," Universidade Estadual de Campinas, Unicamp, 13083-970 Campinas, SP, Brazil
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7
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Barioni AED, de Aguiar MAM. Ott-Antonsen ansatz for the D-dimensional Kuramoto model: A constructive approach. CHAOS (WOODBURY, N.Y.) 2021; 31:113141. [PMID: 34881619 DOI: 10.1063/5.0069350] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/30/2021] [Accepted: 11/03/2021] [Indexed: 06/13/2023]
Abstract
Kuramoto's original model describes the dynamics and synchronization behavior of a set of interacting oscillators represented by their phases. The system can also be pictured as a set of particles moving on a circle in two dimensions, which allows a direct generalization to particles moving on the surface of higher dimensional spheres. One of the key features of the 2D system is the presence of a continuous phase transition to synchronization as the coupling intensity increases. Ott and Antonsen proposed an ansatz for the distribution of oscillators that allowed them to describe the dynamics of the order parameter with a single differential equation. A similar ansatz was later proposed for the D-dimensional model by using the same functional form of the 2D ansatz and adjusting its parameters. In this article, we develop a constructive method to find the ansatz, similarly to the procedure used in 2D. The method is based on our previous work for the 3D Kuramoto model where the ansatz was constructed using the spherical harmonics decomposition of the distribution function. In the case of motion in a D-dimensional sphere, the ansatz is based on the hyperspherical harmonics decomposition. Our result differs from the previously proposed ansatz and provides a simpler and more direct connection between the order parameter and the ansatz.
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Affiliation(s)
- Ana Elisa D Barioni
- Instituto de Física "Gleb Wataghin," Universidade Estadual de Campinas, Unicamp, 13083-970 Campinas, São Paulo, Brazil
| | - Marcus A M de Aguiar
- Instituto de Física "Gleb Wataghin," Universidade Estadual de Campinas, Unicamp, 13083-970 Campinas, São Paulo, Brazil
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Lipton M, Mirollo R, Strogatz SH. The Kuramoto model on a sphere: Explaining its low-dimensional dynamics with group theory and hyperbolic geometry. CHAOS (WOODBURY, N.Y.) 2021; 31:093113. [PMID: 34598458 DOI: 10.1063/5.0060233] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2021] [Accepted: 08/23/2021] [Indexed: 06/13/2023]
Abstract
We study a system of N identical interacting particles moving on the unit sphere in d-dimensional space. The particles are self-propelled and coupled all to all, and their motion is heavily overdamped. For d=2, the system reduces to the classic Kuramoto model of coupled oscillators; for d=3, it has been proposed to describe the orientation dynamics of swarms of drones or other entities moving about in three-dimensional space. Here, we use group theory to explain the recent discovery that the model shows low-dimensional dynamics for all N≥3 and to clarify why it admits the analog of the Ott-Antonsen ansatz in the continuum limit N→∞. The underlying reason is that the system is intimately connected to the natural hyperbolic geometry on the unit ball Bd. In this geometry, the isometries form a Lie group consisting of higher-dimensional generalizations of the Möbius transformations used in complex analysis. Once these connections are realized, the reduced dynamics and the generalized Ott-Antonsen ansatz follow immediately. This framework also reveals the seamless connection between the finite and infinite- N cases. Finally, we show that special forms of coupling yield gradient dynamics with respect to the hyperbolic metric and use that fact to obtain global stability results about convergence to the synchronized state.
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Affiliation(s)
- Max Lipton
- Department of Mathematics, Cornell University, Ithaca, New York 14853, USA
| | - Renato Mirollo
- Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467, USA
| | - Steven H Strogatz
- Department of Mathematics, Cornell University, Ithaca, New York 14853, USA
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9
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Zheng C, Toenjes R, Pikovsky A. Transition to synchrony in a three-dimensional swarming model with helical trajectories. Phys Rev E 2021; 104:014216. [PMID: 34412260 DOI: 10.1103/physreve.104.014216] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2020] [Accepted: 06/30/2021] [Indexed: 11/07/2022]
Abstract
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 discontinuously for arbitrarily small attractive coupling. Adding noise to the system resolves this singular limit and leads to a continuous transition, either to a directed collective motion or to center-of-mass rotations.
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Affiliation(s)
- Chunming Zheng
- Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
| | - Ralf Toenjes
- Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
| | - Arkady Pikovsky
- Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany.,Department of Control Theory, Nizhny Novgorod State University, Gagarin Avenue 23, 606950 Nizhny Novgorod, Russia
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10
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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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