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Miranda M, Frasca M, Estrada E. Topologically induced suppression of explosive synchronization. CHAOS (WOODBURY, N.Y.) 2023; 33:2887742. [PMID: 37125934 DOI: 10.1063/5.0142418] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2023] [Accepted: 04/06/2023] [Indexed: 05/03/2023]
Abstract
Nowadays, explosive synchronization is a well-documented phenomenon consisting in a first-order transition that may coexist with classical synchronization. Typically, explosive synchronization occurs when the network structure is represented by the classical graph Laplacian, and the node frequency and its degree are correlated. Here, we answer the question on whether this phenomenon can be observed in networks when the oscillators are coupled via degree-biased Laplacian operators. We not only observe that this is the case but also that this new representation naturally controls the transition from explosive to standard synchronization in a network. We prove analytically that explosive synchronization emerges when using this theoretical setting in star-like networks. As soon as this star-like network is topologically converted into a network containing cycles, the explosive synchronization gives rise to classical synchronization. Finally, we hypothesize that this mechanism may play a role in switching from normal to explosive states in the brain, where explosive synchronization has been proposed to be related to some pathologies like epilepsy and fibromyalgia.
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Affiliation(s)
- Manuel Miranda
- Institute of Cross-Disciplinary Physics and Complex Systems, IFISC (UIB-CSIC), 07122 Palma de Mallorca, Spain
| | - Mattia Frasca
- Department of Electrical, Electronics and Computer Science Engineering, University of Catania, I-95125 Catania, Italy
- Istituto di Analisi dei Sistemi ed Informatica "A. Ruberti", Consiglio Nazionale delle Ricerche (IASI-CNR), 00185 Roma, Italy
| | - Ernesto Estrada
- Institute of Cross-Disciplinary Physics and Complex Systems, IFISC (UIB-CSIC), 07122 Palma de Mallorca, Spain
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Li F, Li X, Ren L. Noise-induced collective dynamics in the small-world network of photosensitive neurons. J Biol Phys 2022; 48:321-338. [PMID: 35879584 PMCID: PMC9411397 DOI: 10.1007/s10867-022-09610-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/21/2022] [Accepted: 06/19/2022] [Indexed: 11/30/2022] Open
Abstract
Photosensitive neurons can capture and convert external optical signals, and then realize the encoding signal. It is confirmed that a variety of firing modes could be induced under optical stimuli. As a result, it is interesting to explore the mode transitions of collective dynamics in the photosensitive neuron network under external stimuli. In this work, the collective dynamics of photosensitive neurons in a small-world network with non-synaptic coupling will be discussed with spatial diversity of noise and uniform noise applied on, respectively. The results prove that a variety of different collective electrical activities could be induced under different conditions. Under spatial diversity of noise applied on, a chimera state could be observed in the evolution, and steady cluster synchronization could be detected in the end; even the nodes in each cluster depend on the degree of each node. Under uniform noise applied on, the complete synchronization window could be observed alternately in the transient process, and steady complete synchronization could be detected finally. The potential mechanism is that continuous energy is pumped in the phototubes, and energy exchange and balance between neurons to form the resonance synchronization in the network with different noise applied on. Furthermore, it is confirmed that the evolution of collective dynamical behaviors in the network depends on the external stimuli on each node. Moreover, the bifurcation analysis for the single neuron model is calculated, and the results confirm that the electrical activities of single neuron are sensitive to different kinds of noise.
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Affiliation(s)
- Fan Li
- College of Energy Engineering, Yulin University, Yulin, 719000, People's Republic of China.
| | - Xiaola Li
- College of Energy Engineering, Yulin University, Yulin, 719000, People's Republic of China
| | - Liqing Ren
- College of Energy Engineering, Yulin University, Yulin, 719000, People's Republic of China
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Gao J, Efstathiou K. Reduction of oscillator dynamics on complex networks to dynamics on complete graphs through virtual frequencies. Phys Rev E 2020; 101:022302. [PMID: 32168684 DOI: 10.1103/physreve.101.022302] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/22/2019] [Accepted: 01/22/2020] [Indexed: 11/06/2022]
Abstract
We consider the synchronization of oscillators in complex networks where there is an interplay between the oscillator dynamics and the network topology. Through a remarkable transformation in parameter space and the introduction of virtual frequencies we show that Kuramoto oscillators on annealed networks, with or without frequency-degree correlation, and Kuramoto oscillators on complete graphs with frequency-weighted coupling can be transformed to Kuramoto oscillators on complete graphs with a rearranged, virtual frequency distribution and uniform coupling. The virtual frequency distribution encodes both the natural frequency distribution (dynamics) and the degree distribution (topology). We apply this transformation to give direct explanations to a variety of phenomena that have been observed in complex networks, such as explosive synchronization and vanishing synchronization onset.
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Affiliation(s)
- Jian Gao
- Bernoulli Institute for Mathematics, Computer Science, and Artificial Intelligence, University of Groningen, P.O. Box 407, 9700 AK, Groningen, The Netherlands
| | - Konstantinos Efstathiou
- Division of Natural and Applied Sciences and Zu Chongzhi Center for Mathematics and Computational Science, Duke Kunshan University, No. 8 Duke Avenue, Kunshan 215316, China
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Xu C, Gao J, Boccaletti S, Zheng Z, Guan S. Synchronization in starlike networks of phase oscillators. Phys Rev E 2019; 100:012212. [PMID: 31499803 DOI: 10.1103/physreve.100.012212] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/28/2019] [Indexed: 11/07/2022]
Abstract
We fully describe the mechanisms underlying synchronization in starlike networks of phase oscillators. In particular, the routes to synchronization and the critical points for the associated phase transitions are determined analytically. In contrast to the classical Kuramoto theory, we unveil that relaxation rates to each equilibrium state indeed exist and remain invariant under three levels of descriptions corresponding to different geometric implications. The special symmetry in the coupling determines a quasi-Hamiltonian property, which is further unveiled on the basis of singular perturbation theory. Since starlike coupling configurations constitute the building blocks of technological and biological real world networks, our paper paves the way towards the understanding of the functioning of such real world systems in many practical situations.
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Affiliation(s)
- Can Xu
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
| | - Jian Gao
- Bernoulli Institute for Mathematics, Computer Science, and Artificial Intelligence, University of Groningen, P.O. Box 407, 9700 AK, Groningen, The Netherlands
| | - Stefano 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
| | - Zhigang Zheng
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
| | - Shuguang Guan
- Department of Physics, East China Normal University, Shanghai 200241, China
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Zhao N, Sun Z, Yang X, Xu W. Explosive death of conjugate coupled Van der Pol oscillators on networks. Phys Rev E 2018; 97:062203. [PMID: 30011432 DOI: 10.1103/physreve.97.062203] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2018] [Indexed: 06/08/2023]
Abstract
Explosive death phenomenon has been gradually gaining attention of researchers due to the research boom of explosive synchronization, and it has been observed recently for the identical or nonidentical coupled systems in all-to-all network. In this work, we investigate the emergence of explosive death in networked Van der Pol (VdP) oscillators with conjugate variables coupling. It is demonstrated that the network structures play a crucial role in identifying the types of explosive death behaviors. We also observe that the damping coefficient of the VdP system not only can determine whether the explosive death state is generated but also can adjust the forward transition point. We further show that the backward transition point is independent of the network topologies and the damping coefficient, which is well confirmed by theoretical analysis. Our results reveal the generality of explosive death phenomenon in different network topologies and are propitious to promote a better comprehension for the oscillation quenching behaviors.
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Affiliation(s)
- Nannan Zhao
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, People's Republic of China
| | - Zhongkui Sun
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, People's Republic of China
| | - Xiaoli Yang
- College of Mathematics and Information Science, Shaan'xi Normal University, Xi'an 710062, People's Republic of China
| | - Wei Xu
- Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, People's Republic of China
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Kundu P, Khanra P, Hens C, Pal P. Transition to synchrony in degree-frequency correlated Sakaguchi-Kuramoto model. Phys Rev E 2017; 96:052216. [PMID: 29347755 DOI: 10.1103/physreve.96.052216] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2017] [Indexed: 06/07/2023]
Abstract
We investigate transition to synchrony in degree-frequency correlated Sakaguchi-Kuramoto (SK) model on complex networks both analytically and numerically. We analytically derive self-consistent equations for group angular velocity and order parameter for the model in the thermodynamic limit. Using the self-consistent equations we investigate transition to synchronization in SK model on uncorrelated scale-free (SF) and Erdős-Rényi (ER) networks in detail. Depending on the degree distribution exponent (γ) of SF networks and phase-frustration parameter, the population undergoes from first-order transition [explosive synchronization (ES)] to second-order transition and vice versa. In ER networks transition is always second order irrespective of the values of the phase-lag parameter. We observe that the critical coupling strength for the onset of synchronization is decreased by phase-frustration parameter in case of SF network where as in ER network, the phase-frustration delays the onset of synchronization. Extensive numerical simulations using SF and ER networks are performed to validate the analytical results. An analytical expression of critical coupling strength for the onset of synchronization is also derived from the self-consistent equations considering the vanishing order parameter limit.
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Affiliation(s)
- Prosenjit Kundu
- Department of Mathematics, National Institute of Technology, Durgapur 713209, India
| | - Pitambar Khanra
- Department of Mathematics, National Institute of Technology, Durgapur 713209, India
| | - Chittaranjan Hens
- Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Pinaki Pal
- Department of Mathematics, National Institute of Technology, Durgapur 713209, India
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Xu C, Xiang H, Gao J, Zheng Z. Collective dynamics of identical phase oscillators with high-order coupling. Sci Rep 2016; 6:31133. [PMID: 27491401 PMCID: PMC4974564 DOI: 10.1038/srep31133] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/27/2016] [Accepted: 07/12/2016] [Indexed: 12/02/2022] Open
Abstract
In this paper, we propose a framework to investigate the collective dynamics in ensembles of globally coupled phase oscillators when higher-order modes dominate the coupling. The spatiotemporal properties of the attractors in various regions of parameter space are analyzed. Furthermore, a detailed linear stability analysis proves that the stationary symmetric distribution is only neutrally stable in the marginal regime which stems from the generalized time-reversal symmetry. Moreover, the critical parameters of the transition among various regimes are determined analytically by both the Ott-Antonsen method and linear stability analysis, the transient dynamics are further revealed in terms of the characteristic curves method. Finally, for the more general initial condition the symmetric dynamics could be reduced to a rigorous three-dimensional manifold which shows that the neutrally stable chaos could also occur in this model for particular parameters. Our theoretical analysis and numerical results are consistent with each other, which can help us understand the dynamical properties in general systems with higher-order harmonics couplings.
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Affiliation(s)
- Can Xu
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China.,Department of Physics, Beijing Normal University, Beijing 100875, China.,Beijing-Hong Kong-Singapore Joint Center for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China
| | - Hairong Xiang
- Department of Physics, Beijing Normal University, Beijing 100875, China.,Beijing-Hong Kong-Singapore Joint Center for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China
| | - Jian Gao
- Department of Physics, Beijing Normal University, Beijing 100875, China.,Beijing-Hong Kong-Singapore Joint Center for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China
| | - Zhigang Zheng
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
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Gao J, Xu C, Sun Y, Zheng Z. Order parameter analysis for low-dimensional behaviors of coupled phase-oscillators. Sci Rep 2016; 6:30184. [PMID: 27443639 PMCID: PMC4957088 DOI: 10.1038/srep30184] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2016] [Accepted: 06/30/2016] [Indexed: 11/09/2022] Open
Abstract
Coupled phase-oscillators are important models related to synchronization. Recently, Ott-Antonsen(OA) ansatz is developed and used to get low-dimensional collective behaviors in coupled oscillator systems. In this paper, we develop a simple and concise approach based on equations of order parameters, namely, order parameter analysis, with which we point out that OA ansatz is rooted in the dynamical symmetry of order parameters. With our approach the scope of OA ansatz is identified as two conditions, i.e., the limit of infinitely many oscillators and only three nonzero Fourier coefficients of the coupling function. Coinciding with each of the conditions, a distinctive system out of the scope is taken into account and discussed with the order parameter analysis. Two approximation methods are introduced respectively, namely the expectation assumption and the dominating-term assumption.
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Affiliation(s)
- Jian Gao
- College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China.,Department of Physics and the Beijing-Hong Kong-Singapore Joint Center for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China
| | - Can Xu
- Department of Physics and the Beijing-Hong Kong-Singapore Joint Center for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China
| | - Yuting Sun
- Department of Physics and the Beijing-Hong Kong-Singapore Joint Center for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China
| | - Zhigang Zheng
- College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
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Synchronization of phase oscillators with frequency-weighted coupling. Sci Rep 2016; 6:21926. [PMID: 26903110 PMCID: PMC4763290 DOI: 10.1038/srep21926] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2015] [Accepted: 02/02/2016] [Indexed: 11/08/2022] Open
Abstract
Recently, the first-order synchronization transition has been studied in systems of coupled phase oscillators. In this paper, we propose a framework to investigate the synchronization in the frequency-weighted Kuramoto model with all-to-all couplings. A rigorous mean-field analysis is implemented to predict the possible steady states. Furthermore, a detailed linear stability analysis proves that the incoherent state is only neutrally stable below the synchronization threshold. Nevertheless, interestingly, the amplitude of the order parameter decays exponentially (at least for short time) in this regime, resembling the Landau damping effect in plasma physics. Moreover, the explicit expression for the critical coupling strength is determined by both the mean-field method and linear operator theory. The mechanism of bifurcation for the incoherent state near the critical point is further revealed by the amplitude expansion theory, which shows that the oscillating standing wave state could also occur in this model for certain frequency distributions. Our theoretical analysis and numerical results are consistent with each other, which can help us understand the synchronization transition in general networks with heterogenous couplings.
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