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Bolotov MI, Munyayev VO, Smirnov LA, Osipov GV, Belykh I. Breathing and switching cyclops states in Kuramoto networks with higher-mode coupling. Phys Rev E 2024; 109:054202. [PMID: 38907462 DOI: 10.1103/physreve.109.054202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/08/2024] [Accepted: 04/10/2024] [Indexed: 06/24/2024]
Abstract
Cyclops states are intriguing cluster patterns observed in oscillator networks, including neuronal ensembles. The concept of cyclops states formed by two distinct, coherent clusters and a solitary oscillator was introduced by Munyaev et al. [Phys. Rev. Lett. 130, 107201 (2023)0031-900710.1103/PhysRevLett.130.107201], where we explored the surprising prevalence of such states in repulsive Kuramoto networks of rotators with higher-mode harmonics in the coupling. This paper extends our analysis to understand the mechanisms responsible for destroying the cyclops' states and inducing dynamical patterns called breathing and switching cyclops states. We first analytically study the existence and stability of cyclops states in the Kuramoto-Sakaguchi networks of two-dimensional oscillators with inertia as a function of the second coupling harmonic. We then describe two bifurcation scenarios that give birth to breathing and switching cyclops states. We demonstrate that these states and their hybrids are prevalent across a wide coupling range and are robust against a relatively large intrinsic frequency detuning. Beyond the Kuramoto networks, breathing and switching cyclops states promise to strongly manifest in other physical and biological networks, including coupled theta neurons.
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Affiliation(s)
- Maxim I Bolotov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod, 603022, Russia
| | - Vyacheslav O Munyayev
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod, 603022, Russia
| | - Lev A Smirnov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod, 603022, Russia
| | - Grigory V Osipov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod, 603022, Russia
| | - Igor Belykh
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA
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Munyayev VO, Bolotov MI, Smirnov LA, Osipov GV, Belykh I. Cyclops States in Repulsive Kuramoto Networks: The Role of Higher-Order Coupling. PHYSICAL REVIEW LETTERS 2023; 130:107201. [PMID: 36962033 DOI: 10.1103/physrevlett.130.107201] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2022] [Accepted: 02/09/2023] [Indexed: 06/18/2023]
Abstract
Repulsive oscillator networks can exhibit multiple cooperative rhythms, including chimera and cluster splay states. Yet, understanding which rhythm prevails remains challenging. Here, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of rotators with higher-order Fourier modes in the coupling. Through analysis and numerics, we show that three-cluster splay states with two distinct coherent clusters and a solitary oscillator are the prevalent rhythms in networks with an odd number of units. We denote such tripod patterns cyclops states with the solitary oscillator reminiscent of the Cyclops' eye. As their mythological counterparts, the cyclops states are giants that dominate the system's phase space in weakly repulsive networks with first-order coupling. Astonishingly, the addition of the second or third harmonics to the Kuramoto coupling function makes the cyclops states global attractors practically across the full range of coupling's repulsion. Beyond the Kuramoto oscillators, we show that this effect is robustly present in networks of canonical theta neurons with adaptive coupling. At a more general level, our results suggest clues for finding dominant rhythms in repulsive physical and biological networks.
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Affiliation(s)
- Vyacheslav O Munyayev
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Maxim I Bolotov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Lev A Smirnov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Grigory V Osipov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Igor Belykh
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA
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3
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Munyayev VO, Bolotov MI, Smirnov LA, Osipov GV, Belykh IV. Stability of rotatory solitary states in Kuramoto networks with inertia. Phys Rev E 2022; 105:024203. [PMID: 35291064 DOI: 10.1103/physreve.105.024203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2021] [Accepted: 02/03/2022] [Indexed: 06/14/2023]
Abstract
Solitary states emerge in oscillator networks when one oscillator separates from the fully synchronized cluster and oscillates with a different frequency. Such chimera-type patterns with an incoherent state formed by a single oscillator were observed in various oscillator networks; however, there is still a lack of understanding of how such states can stably appear. Here, we study the stability of solitary states in Kuramoto networks of identical two-dimensional phase oscillators with inertia and a phase-lagged coupling. The presence of inertia can induce rotatory dynamics of the phase difference between the solitary oscillator and the coherent cluster. We derive asymptotic stability conditions for such a solitary state as a function of inertia, network size, and phase lag that may yield either attractive or repulsive coupling. Counterintuitively, our analysis demonstrates that (1) increasing the size of the coherent cluster can promote the stability of the solitary state in the attractive coupling case and (2) the solitary state can be stable in small-size networks with all repulsive coupling. We also discuss the implications of our stability analysis for the emergence of rotatory chimeras.
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Affiliation(s)
- Vyacheslav O Munyayev
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Maxim I Bolotov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Lev A Smirnov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
- Institute of Applied Physics, Russian Academy of Sciences, Ul'yanova Str. 46, Nizhny Novgorod 603950, Russia
| | - Grigory V Osipov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
| | - Igor V Belykh
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603022, Russia
- Department of Mathematics and Statistics, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302, USA
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4
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Barabash NV, Belykh VN, Osipov GV, Belykh IV. Partial synchronization in the second-order Kuramoto model: An auxiliary system method. CHAOS (WOODBURY, N.Y.) 2021; 31:113113. [PMID: 34881584 DOI: 10.1063/5.0066663] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/11/2021] [Accepted: 10/11/2021] [Indexed: 06/13/2023]
Abstract
Partial synchronization emerges in an oscillator network when the network splits into clusters of coherent and incoherent oscillators. Here, we analyze the stability of partial synchronization in the second-order finite-dimensional Kuramoto model of heterogeneous oscillators with inertia. Toward this goal, we develop an auxiliary system method that is based on the analysis of a two-dimensional piecewise-smooth system whose trajectories govern oscillating dynamics of phase differences between oscillators in the coherent cluster. Through a qualitative bifurcation analysis of the auxiliary system, we derive explicit bounds that relate the maximum natural frequency mismatch, inertia, and the network size that can support stable partial synchronization. In particular, we predict threshold-like stability loss of partial synchronization caused by increasing inertia. Our auxiliary system method is potentially applicable to cluster synchronization with multiple coherent clusters and more complex network topology.
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Affiliation(s)
- Nikita V Barabash
- Department of Mathematics, Volga State University of Water Transport, 5A, Nesterov str., Nizhny Novgorod 603950, Russia
| | - Vladimir N Belykh
- Department of Mathematics, Volga State University of Water Transport, 5A, Nesterov str., Nizhny Novgorod 603950, Russia
| | - Grigory V Osipov
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23, Gagarin Ave., 603950 Nizhny Novgorod, Russia
| | - Igor V Belykh
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23, Gagarin Ave., 603950 Nizhny Novgorod, Russia
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Phogat R, Ray A, Parmananda P, Ghosh D. Phase coalescence in a population of heterogeneous Kuramoto oscillators. CHAOS (WOODBURY, N.Y.) 2021; 31:041104. [PMID: 34251244 DOI: 10.1063/5.0050451] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2021] [Accepted: 04/01/2021] [Indexed: 06/13/2023]
Abstract
Phase coalescence (PC) is an emerging phenomenon in an ensemble of oscillators that manifests itself as a spontaneous rise in the order parameter. This increment in the order parameter is due to the overlaying of oscillator phases to a pre-existing system state. In the current work, we present a comprehensive analysis of the phenomenon of phase coalescence observed in a population of Kuramoto phase oscillators. The given population is divided into responsive and non-responsive oscillators depending on the position of the phases of the oscillators. The responsive set of oscillators is then reset by a pulse perturbation. This resetting leads to a temporary rise in a macroscopic observable, namely, order parameter. The provoked rise thus induced in the order parameter is followed by unprovoked increments separated by a constant time τPC. These unprovoked increments in the order parameter are caused due to a temporary gathering of the oscillator phases in a configuration similar to the initial system state, i.e., the state of the network immediately following the perturbation. A theoretical framework corroborating this phenomenon as well as the corresponding simulation results are presented. Dependence of τPC and the magnitude of spontaneous order parameter augmentation on various network parameters such as coupling strength, network size, degree of the network, and frequency distribution are then explored. The size of the phase resetting region would also affect the magnitude of the order parameter at τPC since it directly affects the number of oscillators reset by the perturbation. Therefore, the dependence of order parameter on the size of the phase resetting region is also analyzed.
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Affiliation(s)
- Richa Phogat
- Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India
| | - Arnob Ray
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - P Parmananda
- Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
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Njougouo T, Camargo V, Louodop P, Fagundes Ferreira F, Talla PK, Cerdeira HA. Dynamics of multilayer networks with amplification. CHAOS (WOODBURY, N.Y.) 2020; 30:123136. [PMID: 33380025 DOI: 10.1063/5.0025529] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2020] [Accepted: 12/01/2020] [Indexed: 06/12/2023]
Abstract
We study the dynamics of a multilayer network of chaotic oscillators subject to amplification. Previous studies have proven that multilayer networks present phenomena such as synchronization, cluster, and chimera states. Here, we consider a network with two layers of Rössler chaotic oscillators as well as applications to multilayer networks of the chaotic jerk and Liénard oscillators. Intra-layer coupling is considered to be all to all in the case of Rössler oscillators, a ring for jerk oscillators and global mean field coupling in the case of Liénard, inter-layer coupling is unidirectional in all these three cases. The second layer has an amplification coefficient. An in-depth study on the case of a network of Rössler oscillators using a master stability function and order parameter leads to several phenomena such as complete synchronization, generalized, cluster, and phase synchronization with amplification. For the case of Rössler oscillators, we note that there are also certain values of coupling parameters and amplification where the synchronization does not exist or the synchronization can exist but without amplification. Using other systems with different topologies, we obtain some interesting results such as chimera state with amplification, cluster state with amplification, and complete synchronization with amplification.
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Affiliation(s)
- Thierry Njougouo
- Research Unit Condensed Matter, Electronics and Signal Processing, University of Dschang, P.O. Box 67, Dschang, Cameroon
| | - Victor Camargo
- Center for Interdisciplinary Research on Complex Systems, University of Sao Paulo, Av. Arlindo Bettio 1000, 03828-000 São Paulo, Brazil
| | - Patrick Louodop
- Research Unit Condensed Matter, Electronics and Signal Processing, University of Dschang, P.O. Box 67, Dschang, Cameroon
| | - Fernando Fagundes Ferreira
- Center for Interdisciplinary Research on Complex Systems, University of Sao Paulo, Av. Arlindo Bettio 1000, 03828-000 São Paulo, Brazil
| | - Pierre K Talla
- L2MSP, University of Dschang, P.O. Box 67, Dschang, Cameroon
| | - Hilda A Cerdeira
- São Paulo State University (UNESP), Instituto de Física Teórica, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, 01140-070 São Paulo, Brazil
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7
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Brister BN, Belykh VN, Belykh IV. When three is a crowd: Chaos from clusters of Kuramoto oscillators with inertia. Phys Rev E 2020; 101:062206. [PMID: 32688588 DOI: 10.1103/physreve.101.062206] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2020] [Accepted: 06/01/2020] [Indexed: 06/11/2023]
Abstract
Modeling cooperative dynamics using networks of phase oscillators is common practice for a wide spectrum of biological and technological networks, ranging from neuronal populations to power grids. In this paper we study the emergence of stable clusters of synchrony with complex intercluster dynamics in a three-population network of identical Kuramoto oscillators with inertia. The populations have different sizes and can split into clusters where the oscillators synchronize within a cluster, but notably, there is a phase shift between the dynamics of the clusters. We extend our previous results on the bistability of synchronized clusters in a two-population network [I. V. Belykh et al., Chaos 26, 094822 (2016)CHAOEH1054-150010.1063/1.4961435] and demonstrate that the addition of a third population can induce chaotic intercluster dynamics. This effect can be captured by the old adage "two is company, three is a crowd," which suggests that the delicate dynamics of a romantic relationship may be destabilized by the addition of a third party, leading to chaos. Through rigorous analysis and numerics, we demonstrate that the intercluster phase shifts can stably coexist and exhibit different forms of chaotic behavior, including oscillatory, rotatory, and mixed-mode oscillations. We also discuss the implications of our stability results for predicting the emergence of chimeras and solitary states.
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Affiliation(s)
- Barrett N Brister
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA
| | - Vladimir N Belykh
- Department of Mathematics, Volga State University of Water Transport, 5A Nesterov street, Nizhny Novgorod 603950, Russia
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603950, Russia
| | - Igor V Belykh
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, P.O. Box 4110, Atlanta, Georgia 30302-410, USA
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603950, Russia
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8
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Eydam S, Wolfrum M. The link between coherence echoes and mode locking. CHAOS (WOODBURY, N.Y.) 2019; 29:103114. [PMID: 31675832 DOI: 10.1063/1.5114699] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2019] [Accepted: 09/18/2019] [Indexed: 06/10/2023]
Abstract
We investigate the appearance of sharp pulses in the mean field of Kuramoto-type globally-coupled phase oscillator systems. In systems with exactly equidistant natural frequencies, self-organized periodic pulsations of the mean field, called mode locking, have been described recently as a new collective dynamics below the synchronization threshold. We show here that mode locking can appear also for frequency combs with modes of finite width, where the natural frequencies are randomly chosen from equidistant frequency intervals. In contrast to that, so-called coherence echoes, which manifest themselves also as pulses in the mean field, have been found in systems with completely disordered natural frequencies as a result of two consecutive stimulations applied to the system. We show that such echo pulses can be explained by a stimulation induced mode locking of a subpopulation representing a frequency comb. Moreover, we find that the presence of a second harmonic in the interaction function, which can lead to the global stability of the mode-locking regime for equidistant natural frequencies, can enhance the echo phenomenon significantly. The nonmonotonic behavior of echo amplitudes can be explained as a result of the linear dispersion within the self-organized mode-locked frequency comb. Finally, we investigate the effect of small periodic stimulations on oscillator systems with disordered natural frequencies and show how the global coupling can support the stimulated pulsation by increasing the width of locking plateaus.
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Affiliation(s)
- Sebastian Eydam
- Weierstrass Institute for Applied Analysis and Stochastic, Mohrenstrasse 39, 10117 Berlin, Germany
| | - Matthias Wolfrum
- Weierstrass Institute for Applied Analysis and Stochastic, Mohrenstrasse 39, 10117 Berlin, Germany
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9
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Eydam S, Wolfrum M. Mode locking in systems of globally coupled phase oscillators. Phys Rev E 2017; 96:052205. [PMID: 29347775 DOI: 10.1103/physreve.96.052205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/24/2017] [Indexed: 06/07/2023]
Abstract
We investigate the dynamics of a Kuramoto-type system of globally coupled phase oscillators with equidistant natural frequencies and a coupling strength below the synchronization threshold. It turns out that in such cases one can observe a stable regime of sharp pulses in the mean field amplitude with a pulsation frequency given by spacing of the natural frequencies. This resembles a process known as mode locking in lasers and relies on the emergence of a phase relation induced by the nonlinear coupling. We discuss the role of the first and second harmonics in the phase-interaction function for the stability of the pulsations and present various bifurcating dynamical regimes such as periodically and chaotically modulated mode locking, transitions to phase turbulence, and intermittency. Moreover, we study the role of the system size and show that in certain cases one can observe type II supertransients, where the system reaches the globally stable mode-locking solution only after an exponentially long transient of phase turbulence.
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Affiliation(s)
- Sebastian Eydam
- Weierstrass Institute for Applied Analysis and Stochastic, Mohrenstrasse 39, 10117 Berlin, Germany
| | - Matthias Wolfrum
- Weierstrass Institute for Applied Analysis and Stochastic, Mohrenstrasse 39, 10117 Berlin, Germany
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10
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Bi H, Hu X, Boccaletti S, Wang X, Zou Y, Liu Z, Guan S. Coexistence of Quantized, Time Dependent, Clusters in Globally Coupled Oscillators. PHYSICAL REVIEW LETTERS 2016; 117:204101. [PMID: 27886476 DOI: 10.1103/physrevlett.117.204101] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/13/2015] [Indexed: 06/06/2023]
Abstract
We report on a novel collective state, occurring in globally coupled nonidentical oscillators in the proximity of the point where the transition from the system's incoherent to coherent phase converts from explosive to continuous. In such a state, the oscillators form quantized clusters, where neither their phases nor their instantaneous frequencies are locked. The oscillators' instantaneous speeds are different within the clusters, but they form a characteristic cusped pattern and, more importantly, they behave periodically in time so that their average values are the same. Given its intrinsic specular nature with respect to the recently introduced Chimera states, the phase is termed the Bellerophon state. We provide an analytical and numerical description of Bellerophon states, and furnish practical hints on how to seek them in a variety of experimental and natural systems.
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Affiliation(s)
- Hongjie Bi
- Department of Physics, East China Normal University, Shanghai 200241, China
| | - Xin Hu
- Suzhou Institute of Nano-Tech and Nano-Bionics, Chinese Academy of Sciences, Suzhou 215123, China
| | - S Boccaletti
- CNR-Institute of Complex Systems, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Florence, Italy
- The Embassy of Italy in Tel Aviv, 25 Hamered Street, 68125 Tel Aviv, Israel
| | - Xingang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xian 710062, China
| | - Yong Zou
- Department of Physics, East China Normal University, Shanghai 200241, 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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11
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Martin EA, Hlinka J, Davidsen J. Pairwise network information and nonlinear correlations. Phys Rev E 2016; 94:040301. [PMID: 27841521 DOI: 10.1103/physreve.94.040301] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2015] [Indexed: 12/25/2022]
Abstract
Reconstructing the structural connectivity between interacting units from observed activity is a challenge across many different disciplines. The fundamental first step is to establish whether or to what extent the interactions between the units can be considered pairwise and, thus, can be modeled as an interaction network with simple links corresponding to pairwise interactions. In principle, this can be determined by comparing the maximum entropy given the bivariate probability distributions to the true joint entropy. In many practical cases, this is not an option since the bivariate distributions needed may not be reliably estimated or the optimization is too computationally expensive. Here we present an approach that allows one to use mutual informations as a proxy for the bivariate probability distributions. This has the advantage of being less computationally expensive and easier to estimate. We achieve this by introducing a novel entropy maximization scheme that is based on conditioning on entropies and mutual informations. This renders our approach typically superior to other methods based on linear approximations. The advantages of the proposed method are documented using oscillator networks and a resting-state human brain network as generic relevant examples.
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Affiliation(s)
- Elliot A Martin
- Complexity Science Group, Department of Physics and Astronomy, University of Calgary, Calgary, Alberta, Canada T2N 1N4
| | - Jaroslav Hlinka
- Institute of Computer Science, The Czech Academy of Sciences, Pod vodarenskou vezi 2, 18207 Prague, Czech Republic.,National Institute of Mental Health, Topolová 748, 250 67 Klecany, Czech Republic
| | - Jörn Davidsen
- Complexity Science Group, Department of Physics and Astronomy, University of Calgary, Calgary, Alberta, Canada T2N 1N4
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12
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Belykh IV, Brister BN, Belykh VN. Bistability of patterns of synchrony in Kuramoto oscillators with inertia. CHAOS (WOODBURY, N.Y.) 2016; 26:094822. [PMID: 27781476 DOI: 10.1063/1.4961435] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We study the co-existence of stable patterns of synchrony in two coupled populations of identical Kuramoto oscillators with inertia. The two populations have different sizes and can split into two clusters where the oscillators synchronize within a cluster while there is a phase shift between the dynamics of the two clusters. Due to the presence of inertia, which increases the dimensionality of the oscillator dynamics, this phase shift can oscillate, inducing a breathing cluster pattern. We derive analytical conditions for the co-existence of stable two-cluster patterns with constant and oscillating phase shifts. We demonstrate that the dynamics, that governs the bistability of the phase shifts, is described by a driven pendulum equation. We also discuss the implications of our stability results to the stability of chimeras.
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Affiliation(s)
- Igor V Belykh
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, 30 Pryor Street, Atlanta, Georgia 30303, USA
| | - Barrett N Brister
- Department of Mathematics and Statistics and Neuroscience Institute, Georgia State University, 30 Pryor Street, Atlanta, Georgia 30303, USA
| | - Vladimir N Belykh
- Department of Control Theory, Lobachevsky State University of Nizhny Novgorod, 23, Gagarin Ave., 603950 Nizhny Novgorod, Russia
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13
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Ottino-Löffler B, Strogatz SH. Kuramoto model with uniformly spaced frequencies: Finite-N asymptotics of the locking threshold. Phys Rev E 2016; 93:062220. [PMID: 27415267 DOI: 10.1103/physreve.93.062220] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/11/2016] [Indexed: 11/07/2022]
Abstract
We study phase locking in the Kuramoto model of coupled oscillators in the special case where the number of oscillators, N, is large but finite, and the oscillators' natural frequencies are evenly spaced on a given interval. In this case, stable phase-locked solutions are known to exist if and only if the frequency interval is narrower than a certain critical width, called the locking threshold. For infinite N, the exact value of the locking threshold was calculated 30 years ago; however, the leading corrections to it for finite N have remained unsolved analytically. Here we derive an asymptotic formula for the locking threshold when N≫1. The leading correction to the infinite-N result scales like either N^{-3/2} or N^{-1}, depending on whether the frequencies are evenly spaced according to a midpoint rule or an end-point rule. These scaling laws agree with numerical results obtained by Pazó [D. Pazó, Phys. Rev. E 72, 046211 (2005)PLEEE81539-375510.1103/PhysRevE.72.046211]. Moreover, our analysis yields the exact prefactors in the scaling laws, which also match the numerics.
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Affiliation(s)
| | - Steven H Strogatz
- Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA
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14
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Witthaut D, Timme M. Kuramoto dynamics in Hamiltonian systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032917. [PMID: 25314514 DOI: 10.1103/physreve.90.032917] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/09/2013] [Indexed: 06/04/2023]
Abstract
The Kuramoto model constitutes a paradigmatic model for the dissipative collective dynamics of coupled oscillators, characterizing in particular the emergence of synchrony (phase locking). Here we present a classical Hamiltonian (and thus conservative) system with 2N state variables that in its action-angle representation exactly yields Kuramoto dynamics on N-dimensional invariant manifolds. We show that locking of the phase of one oscillator on a Kuramoto manifold to the average phase emerges where the transverse Hamiltonian action dynamics of that specific oscillator becomes unstable. Moreover, the inverse participation ratio of the Hamiltonian dynamics perturbed off the manifold indicates the global synchronization transition point for finite N more precisely than the standard Kuramoto order parameter. The uncovered Kuramoto dynamics in Hamiltonian systems thus distinctly links dissipative to conservative dynamics.
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Affiliation(s)
- Dirk Witthaut
- Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany
| | - Marc Timme
- Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany and Institute for Nonlinear Dynamics, Faculty of Physics, University of Göttingen, 37077 Göttingen, Germany
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15
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Temirbayev AA, Nalibayev YD, Zhanabaev ZZ, Ponomarenko VI, Rosenblum M. Autonomous and forced dynamics of oscillator ensembles with global nonlinear coupling: an experimental study. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:062917. [PMID: 23848758 DOI: 10.1103/physreve.87.062917] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2013] [Revised: 05/14/2013] [Indexed: 06/02/2023]
Abstract
We perform experiments with 72 electronic limit-cycle oscillators, globally coupled via a linear or nonlinear feedback loop. While in the linear case we observe a standard Kuramoto-like synchronization transition, in the nonlinear case, with increase of the coupling strength, we first observe a transition to full synchrony and then a desynchronization transition to a quasiperiodic state. However, in this state the ensemble remains coherent so that the amplitude of the mean field is nonzero, but the frequency of the mean field is larger than frequencies of all oscillators. Next, we analyze effects of common periodic forcing of the linearly or nonlinearly coupled ensemble and demonstrate regimes when the mean field is entrained by the force whereas the oscillators are not.
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Affiliation(s)
- Amirkhan A Temirbayev
- Physical-Technical Department, al-Farabi Kazakh National University, al-Farabi Avenue 71, 050040 Almaty, Kazakhstan
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16
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Chiba H. Continuous limit and the moments system for the globally coupled phase oscillators. ACTA ACUST UNITED AC 2013. [DOI: 10.3934/dcds.2013.33.1891] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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17
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Temirbayev AA, Zhanabaev ZZ, Tarasov SB, Ponomarenko VI, Rosenblum M. Experiments on oscillator ensembles with global nonlinear coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:015204. [PMID: 22400613 DOI: 10.1103/physreve.85.015204] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/16/2011] [Indexed: 05/31/2023]
Abstract
We experimentally analyze collective dynamics of a population of 20 electronic Wien-bridge limit-cycle oscillators with a nonlinear phase-shifting unit in the global feedback loop. With an increase in the coupling strength we first observe formation and then destruction of a synchronous cluster, so that the dependence of the order parameter on the coupling strength is not monotonic. After destruction of the cluster the ensemble remains nevertheless coherent, i.e., it exhibits an oscillatory collective mode (mean field). We show that the system is now in a self-organized quasiperiodic state, predicted in Rosenblum and Pikovsky [Phys. Rev. Lett. 98, 064101 (2007)]. In this state, frequencies of all oscillators are smaller than the frequency of the mean field, so that the oscillators are not locked to the mean field they create and their dynamics is quasiperiodic. Without a nonlinear phase-shifting unit, the system exhibits a standard Kuramoto-like transition to a fully synchronous state. We demonstrate a good correspondence between the experiment and previously developed theory. We also propose a simple measure which characterizes the macroscopic incoherence-coherence transition in a finite-size ensemble.
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Affiliation(s)
- Amirkhan A Temirbayev
- Physical-Technical Department, al-Farabi Kazakh National University, al-Farabi avenue 71, 050040, Almaty, Kazakhstan
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18
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Omel'chenko OE, Maistrenko YL, Tass PA. Chimera states induced by spatially modulated delayed feedback. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:066201. [PMID: 21230717 DOI: 10.1103/physreve.82.066201] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/12/2010] [Indexed: 05/30/2023]
Abstract
Recently, we have presented spatially modulated delayed feedback as a novel mechanism, which generically generates chimera states, remarkable spatiotemporal patterns in which coherence coexists with incoherence [O. E. Omel'chenko, Phys. Rev. Lett. 100, 044105 (2008)]. Remarkably, such chimera states serve as a natural link between completely coherent states and completely incoherent states. So far, we have studied this mechanism with a self-consistency-based numerical analysis only. In contrast, in this paper we perform a thorough dynamical description and, in particular, a stability analysis of the emerging chimera states. For this, we apply a recently developed reduction procedure [A. Pikovsky and M. Rosenblum, Phys. Rev. Lett. 101, 264103 (2008)]. By combining analytical and numerical approaches, we systematically describe the relationship between the parameters of the delayed feedback on one hand and the properties of the chimera states on the other hand. We provide the general rules for an effective control and manipulation of the chimera states.
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Affiliation(s)
- Oleh E Omel'chenko
- Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany
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19
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Goldobin DS, Teramae JN, Nakao H, Ermentrout GB. Dynamics of limit-cycle oscillators subject to general noise. PHYSICAL REVIEW LETTERS 2010; 105:154101. [PMID: 21230907 DOI: 10.1103/physrevlett.105.154101] [Citation(s) in RCA: 40] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2010] [Indexed: 05/30/2023]
Abstract
The phase description is a powerful tool for analyzing noisy limit-cycle oscillators. The method, however, has found only limited applications so far, because the present theory is applicable only to Gaussian noise while noise in the real world often has non-Gaussian statistics. Here, we provide the phase reduction method for limit-cycle oscillators subject to general, colored and non-Gaussian, noise including a heavy-tailed one. We derive quantifiers like mean frequency, diffusion constant, and the Lyapunov exponent to confirm consistency of the results. Applying our results, we additionally study a resonance between the phase and noise.
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Affiliation(s)
- Denis S Goldobin
- Institute of the Continuous Media Mechanics, UB RAS, Perm 614013, Russia
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20
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Zhang X, Wang R, Zhang Z, Qu J, Cao J, Jiao X. Dynamic phase synchronization characteristics of variable high-order coupled neuronal oscillator population. Neurocomputing 2010. [DOI: 10.1016/j.neucom.2010.05.001] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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21
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Baibolatov Y, Rosenblum M, Zhanabaev ZZ, Pikovsky A. Complex dynamics of an oscillator ensemble with uniformly distributed natural frequencies and global nonlinear coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:016212. [PMID: 20866712 DOI: 10.1103/physreve.82.016212] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2010] [Indexed: 05/29/2023]
Abstract
We consider large populations of phase oscillators with global nonlinear coupling. For identical oscillators such populations are known to demonstrate a transition from completely synchronized state to the state of self-organized quasiperiodicity. In this state phases of all units differ, yet the population is not completely incoherent but produces a nonzero mean field; the frequency of the latter differs from the frequency of individual units. Here we analyze the dynamics of such populations in case of uniformly distributed natural frequencies. We demonstrate numerically and describe theoretically (i) states of complete synchrony, (ii) regimes with coexistence of a synchronous cluster and a drifting subpopulation, and (iii) self-organized quasiperiodic states with nonzero mean field and all oscillators drifting with respect to it. We analyze transitions between different states with the increase of the coupling strength; in particular we show that the mean field arises via a discontinuous transition. For a further illustration we compare the results for the nonlinear model with those for the Kuramoto-Sakaguchi model.
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Affiliation(s)
- Yernur Baibolatov
- Department of Physics and Astronomy, University of Potsdam, Potsdam-Golm, Germany
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22
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Kirst C, Timme M. Partial response to supra-threshold excitation desynchronizes spiking neurons. BMC Neurosci 2009. [DOI: 10.1186/1471-2202-10-s1-o17] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022] Open
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23
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Kirst C, Geisel T, Timme M. Sequential desynchronization in networks of spiking neurons with partial reset. PHYSICAL REVIEW LETTERS 2009; 102:068101. [PMID: 19257635 DOI: 10.1103/physrevlett.102.068101] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/11/2008] [Indexed: 05/27/2023]
Abstract
The response of a neuron to synaptic input strongly depends on whether or not the neuron has just emitted a spike. We propose a neuron model that after spike emission exhibits a partial response to residual input charges and study its collective network dynamics analytically. We uncover a desynchronization mechanism that causes a sequential desynchronization transition: In globally coupled neurons an increase in the strength of the partial response induces a sequence of bifurcations from states with large clusters of synchronously firing neurons, through states with smaller clusters to completely asynchronous spiking. We briefly discuss key consequences of this mechanism for more general networks of biophysical neurons.
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Affiliation(s)
- Christoph Kirst
- Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37073 Göttingen, Germany and Faculty of Physics, Georg August University Göttingen, 37077 Göttingen, Germany
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24
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De Smet F, Aeyels D. Clustering in a network of non-identical and mutually interacting agents. Proc Math Phys Eng Sci 2008. [DOI: 10.1098/rspa.2008.0259] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Clustering is a phenomenon that may emerge in multi-agent systems through self-organization: groups arise consisting of agents with similar dynamic behaviour. It is observed in fields ranging from the exact sciences to social and life sciences; consider, for example, swarm behaviour of animals or social insects, the dynamics of opinion formation or the synchronization (which corresponds to cluster formation in the phase space) of coupled oscillators modelling brain or heart cells. We consider a clustering model with a general network structure and saturating interaction functions. We derive both necessary and sufficient conditions for clustering behaviour of the model and we investigate the cluster structure for varying coupling strength. Generically, each cluster asymptotically reaches a (relative) equilibrium state. We discuss the relationship of the model to swarming, and we explain how the model equations naturally arise in a system of interconnected water basins. We also indicate how the model applies to opinion formation dynamics.
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Affiliation(s)
- Filip De Smet
- SYSTeMS Research Group, Department of Electrical Energy, Systems and Automation, Ghent UniversityTechnologiepark Zwijnaarde 914, 9052 Zwijnaarde, Belgium
| | - Dirk Aeyels
- SYSTeMS Research Group, Department of Electrical Energy, Systems and Automation, Ghent UniversityTechnologiepark Zwijnaarde 914, 9052 Zwijnaarde, Belgium
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25
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De Smet F, Aeyels D. Resonances and entrainment breakup in Kuramoto models with multimodal frequency densities. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:066212. [PMID: 18643355 DOI: 10.1103/physreve.77.066212] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/16/2007] [Revised: 05/05/2008] [Indexed: 05/26/2023]
Abstract
We characterize some intriguing aspects of the entrainment behavior of coupled oscillators by means of a perturbation analysis of the partially synchronized solution of the classical Kuramoto-Sakaguchi model. The analysis reveals that partial entrainment may disappear with increasing coupling strength. It also predicts the occurrence of resonances: partial entrainment is induced in oscillators with natural frequencies in specific intervals not corresponding to high oscillator densities. The results are illustrated by simulations.
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Affiliation(s)
- Filip De Smet
- SYSTeMS Research Group, Department of Electrical Energy, Systems and Automation, Ghent University, Technologiepark Zwijnaarde 914, 9052 Zwijnaarde, Belgium.
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26
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Omel'chenko OE, Maistrenko YL, Tass PA. Chimera states: the natural link between coherence and incoherence. PHYSICAL REVIEW LETTERS 2008; 100:044105. [PMID: 18352280 DOI: 10.1103/physrevlett.100.044105] [Citation(s) in RCA: 88] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2007] [Indexed: 05/26/2023]
Abstract
Chimera states are remarkable spatiotemporal patterns in which coherence coexists with incoherence. As yet, chimera states have been considered as nongeneric, since they emerge only for particular initial conditions. In contrast, we show here that in a network of globally coupled oscillators delayed feedback stimulation with realistic (i.e., spatially decaying) stimulation profile generically induces chimera states. Intriguingly, a bifurcation analysis reveals that these chimera states are the natural link between the coherent and the incoherent states.
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Affiliation(s)
- Oleh E Omel'chenko
- Institute of Neuroscience and Biophysics 3-Medicine, Research Center Jülich, 52425 Jülich, Germany
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27
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Zhai Y, Kiss IZ, Hudson JL. Control of Complex Dynamics with Time-Delayed Feedback in Populations of Chemical Oscillators: Desynchronization and Clustering. Ind Eng Chem Res 2007. [DOI: 10.1021/ie0708632] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Yumei Zhai
- Department of Chemical Engineering, 102 Engineers' Way, University of Virginia, Charlottesville, Virginia 22904-4741
| | - István Z. Kiss
- Department of Chemical Engineering, 102 Engineers' Way, University of Virginia, Charlottesville, Virginia 22904-4741
| | - John L. Hudson
- Department of Chemical Engineering, 102 Engineers' Way, University of Virginia, Charlottesville, Virginia 22904-4741
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28
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Timme M. Revealing network connectivity from response dynamics. PHYSICAL REVIEW LETTERS 2007; 98:224101. [PMID: 17677845 DOI: 10.1103/physrevlett.98.224101] [Citation(s) in RCA: 130] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/18/2006] [Indexed: 05/03/2023]
Abstract
We present a method to infer the complete connectivity of a network from its stable response dynamics. As a paradigmatic example, we consider networks of coupled phase oscillators and explicitly study their long-term stationary response to temporally constant driving. For a given driving condition, measuring the phase differences and the collective frequency reveals information about how the units are interconnected. Sufficiently many repetitions for different driving conditions yield the entire network connectivity (the absence or presence of each connection) from measuring the response dynamics only. For sparsely connected networks, we obtain good predictions of the actual connectivity even for formally underdetermined problems.
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Affiliation(s)
- Marc Timme
- Network Dynamics Group, Max Planck Institute for Dynamics and Self-Organization, and Bernstein Center for Computational Neuroscience, Bunsenstrasse 10, 37073 Göttingen, Germany
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29
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Maistrenko YL, Lysyansky B, Hauptmann C, Burylko O, Tass PA. Multistability in the Kuramoto model with synaptic plasticity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:066207. [PMID: 17677340 DOI: 10.1103/physreve.75.066207] [Citation(s) in RCA: 36] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2006] [Revised: 04/10/2007] [Indexed: 05/08/2023]
Abstract
We present a simplified phase model for neuronal dynamics with spike timing-dependent plasticity (STDP). For asymmetric, experimentally observed STDP we find multistability: a coexistence of a fully synchronized, a fully desynchronized, and a variety of cluster states in a wide enough range of the parameter space. We show that multistability can occur only for asymmetric STDP, and we study how the coexistence of synchronization and desynchronization and clustering depends on the distribution of the eigenfrequencies. We test the efficacy of the proposed method on the Kuramoto model which is, de facto, one of the sample models for a description of the phase dynamics in neuronal ensembles.
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Affiliation(s)
- Yuri L Maistrenko
- Institute of Medicine and Virtual Institute of Neuromodulation, Research Centre Jülich, 52425 Jülich, Germany
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30
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Popovych OV, Hauptmann C, Tass PA. Control of neuronal synchrony by nonlinear delayed feedback. BIOLOGICAL CYBERNETICS 2006; 95:69-85. [PMID: 16614837 DOI: 10.1007/s00422-006-0066-8] [Citation(s) in RCA: 48] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/24/2005] [Accepted: 03/08/2006] [Indexed: 05/08/2023]
Abstract
We present nonlinear delayed feedback stimulation as a technique for effective desynchronization. This method is intriguingly robust with respect to system and stimulation parameter variations. We demonstrate its broad applicability by applying it to different generic oscillator networks as well as to a population of bursting neurons. Nonlinear delayed feedback specifically counteracts abnormal interactions and, thus, restores the natural frequencies of the individual oscillatory units. Nevertheless, nonlinear delayed feedback enables to strongly detune the macroscopic frequency of the collective oscillation. We propose nonlinear delayed feedback stimulation for the therapy of neurological diseases characterized by abnormal synchrony.
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31
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Krachkovskyi V, Popovych OV, Tass PA. Stimulus-locked responses of two phase oscillators coupled with delayed feedback. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:066220. [PMID: 16906959 DOI: 10.1103/physreve.73.066220] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2006] [Indexed: 05/11/2023]
Abstract
For a system of two phase oscillators coupled with delayed self-feedback we study the impact of pulsatile stimulation administered to both oscillators. This system models the dynamics of two coupled phase-locked loops (PLLs) with a finite internal delay within each loop. The delayed self-feedback leads to a rich variety of dynamical regimes, ranging from phase-locked and periodically modulated synchronized states to chaotic phase synchronization and desynchronization. Remarkably, for large coupling strength the two PLLs are completely desynchronized. We study stimulus-locked responses emerging in the different dynamical regimes. Simple phase resets may be followed by a response clustering, which is intimately connected with long poststimulus resynchronization. Intriguingly, a maximal perturbation (i.e., maximal response clustering and maximal resynchronization time) occurs, if the system gets trapped at a stable manifold of an unstable saddle fixed point due to appropriately calibrated stimulus. Also, single stimuli with suitable parameters can shift the system from a stable synchronized state to a stable desynchronized state or vice versa. Our result show that appropriately calibrated single pulse stimuli may cause pronounced transient and/or long-lasting changes of the oscillators' dynamics. Pulse stimulation may, hence, constitute an effective approach for the control of coupled oscillators, which might be relevant to both physical and medical applications.
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Affiliation(s)
- Valerii Krachkovskyi
- Institute of Medicine and Virtual Institute of Neuromodulation, Research Centre Jülich, 52425 Jülich, Germany
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32
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Pazó D, Montbrió E. Universal behavior in populations composed of excitable and self-oscillatory elements. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:055202. [PMID: 16802985 DOI: 10.1103/physreve.73.055202] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/01/2005] [Indexed: 05/10/2023]
Abstract
We study the robustness of self-sustained oscillatory activity in a globally coupled ensemble of excitable and oscillatory units. The critical balance to achieve collective self-sustained oscillations is analytically established. We also report a universal scaling function for the ensemble's mean frequency. Our results extend the framework of the "aging transition" [Phys. Rev. Lett. 93, 104101 (2004)] including a broad class of dynamical systems potentially relevant in biology.
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Affiliation(s)
- Diego Pazó
- Max-Planck-Institut für Physik komplexer Systeme, Dresden, Germany
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33
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Tanguy Y, Houlihan J, Huyet G, Viktorov EA, Mandel P. Synchronization and clustering in a multimode quantum dot laser. PHYSICAL REVIEW LETTERS 2006; 96:053902. [PMID: 16486931 DOI: 10.1103/physrevlett.96.053902] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2005] [Indexed: 05/06/2023]
Abstract
We analyze experimentally the intensity oscillations of the longitudinal modes of quantum dot semiconductor lasers. We show that the modal intensities can oscillate chaotically with different average frequencies, but obey a highly organized antiphase dynamics leading to a constant total output power. The fluctuations are in the MHz range. We report the first experimental observation of frequency clustering associated with synchronization. We also observe the propagation of perturbations across the optical spectrum from blue to red.
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Affiliation(s)
- Yann Tanguy
- Tyndall National Institute and Physics Department, Lee Maltings, National University of Ireland, University College, Cork, Ireland
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34
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Ashwin P, Burylko O, Maistrenko Y, Popovych O. Extreme sensitivity to detuning for globally coupled phase oscillators. PHYSICAL REVIEW LETTERS 2006; 96:054102. [PMID: 16486934 DOI: 10.1103/physrevlett.96.054102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/04/2005] [Indexed: 05/06/2023]
Abstract
We discuss the sensitivity of a population of coupled oscillators to differences in their natural frequencies, i.e., to detuning. We argue that for three or more oscillators, one can get great sensitivity even if the coupling is strong. For N globally coupled phase oscillators we find there can be bifurcation to extreme sensitivity, where frequency locking can be destroyed by arbitrarily small detuning. This extreme sensitivity is absent for N = 2, appears at isolated parameter values for N = 3 and N = 4, and can appear robustly for open sets of parameter values for N > or = 5 oscillators.
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Affiliation(s)
- Peter Ashwin
- Department of Mathematical Sciences, University of Exeter, Exeter EX4 4QE, United Kingdom
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35
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Pazó D. Thermodynamic limit of the first-order phase transition in the Kuramoto model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:046211. [PMID: 16383516 DOI: 10.1103/physreve.72.046211] [Citation(s) in RCA: 89] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/19/2005] [Revised: 07/06/2005] [Indexed: 05/05/2023]
Abstract
In the Kuramoto model, a uniform distribution of the natural frequencies leads to a first-order (i.e., discontinuous) phase transition from incoherence to synchronization, at the critical coupling parameter K(c). We obtain the asymptotic dependence of the order parameter above criticality: r-r(c)alpha(K - K(c))(2/3). For a finite population, we demonstrate that the population size N may be included into a self-consistency equation relating r and K in the synchronized state. We analyze the convergence to the thermodynamic limit of two alternative schemes to set the natural frequencies. Other frequency distributions different from the uniform one are also considered.
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Affiliation(s)
- Diego Pazó
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, 01187 Dresden, Germany.
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36
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Popovych OV, Maistrenko YL, Tass PA. Phase chaos in coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:065201. [PMID: 16089804 DOI: 10.1103/physreve.71.065201] [Citation(s) in RCA: 45] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2004] [Indexed: 05/03/2023]
Abstract
A complex high-dimensional chaotic behavior, phase chaos, is found in the finite-dimensional Kuramoto model of coupled phase oscillators. This type of chaos is characterized by half of the spectrum of Lyapunov exponents being positive and the Lyapunov dimension equaling almost the total system dimension. Intriguingly, the strongest phase chaos occurs for intermediate-size ensembles. Phase chaos is a common property of networks of oscillators of very different natures, such as phase oscillators, limit-cycle oscillators, and chaotic oscillators, e.g., Rössler systems.
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Affiliation(s)
- Oleksandr V Popovych
- Institute of Medicine and Virtual Institute of Neuromodulation, Research Centre Jülich, 52425 Jülich, Germany
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