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Nakao H, Yamaguchi K, Katayama S, Yanagita T. Sparse optimization of mutual synchronization in collectively oscillating networks. CHAOS (WOODBURY, N.Y.) 2021; 31:063113. [PMID: 34241311 DOI: 10.1063/5.0049091] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2021] [Accepted: 05/14/2021] [Indexed: 06/13/2023]
Abstract
We consider a pair of collectively oscillating networks of dynamical elements and optimize their internetwork coupling for efficient mutual synchronization based on the phase reduction theory developed by Nakao et al. [Chaos 28, 045103 (2018)]. The dynamical equations describing a pair of weakly coupled networks are reduced to a pair of coupled phase equations, and the linear stability of the synchronized state between the networks is represented as a function of the internetwork coupling matrix. We seek the optimal coupling by minimizing the Frobenius and L1 norms of the internetwork coupling matrix for the prescribed linear stability of the synchronized state. Depending on the norm, either a dense or sparse internetwork coupling yielding efficient mutual synchronization of the networks is obtained. In particular, a sparse yet resilient internetwork coupling is obtained by L1-norm optimization with additional constraints on the individual connection weights.
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Affiliation(s)
- Hiroya Nakao
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Katsunori Yamaguchi
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Shingo Katayama
- Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Tatsuo Yanagita
- Department of Engineering Science, Osaka Electro-Communication University, Neyagawa, Osaka 572-8530, Japan
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Achterhof S, Meylahn JM. Two-community noisy Kuramoto model with general interaction strengths. I. CHAOS (WOODBURY, N.Y.) 2021; 31:033115. [PMID: 33810750 DOI: 10.1063/5.0022624] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/22/2020] [Accepted: 01/19/2021] [Indexed: 06/12/2023]
Abstract
We generalize the study of the noisy Kuramoto model, considered on a network of two interacting communities, to the case where the interaction strengths within and across communities are taken to be different in general. By developing a geometric interpretation of the self-consistency equations, we are able to separate the parameter space into ten regions in which we identify the maximum number of solutions in the steady state. Furthermore, we prove that in the steady state, only the angles 0 and π are possible between the average phases of the two communities and derive the solution boundary for the unsynchronized solution. Last, we identify the equivalence class relation in the parameter space corresponding to the symmetrically synchronized solution.
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Affiliation(s)
- S Achterhof
- Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands
| | - J M Meylahn
- Amsterdam Business School, University of Amsterdam, P.O. Box 15953, 1001 NL Amsterdam, The Netherlands
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Sebek M, Kawamura Y, Nott AM, Kiss IZ. Anti-phase collective synchronization with intrinsic in-phase coupling of two groups of electrochemical oscillators. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20190095. [PMID: 31656145 PMCID: PMC6833994 DOI: 10.1098/rsta.2019.0095] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 09/05/2019] [Indexed: 05/02/2023]
Abstract
The synchronization of two groups of electrochemical oscillators is investigated during the electrodissolution of nickel in sulfuric acid. The oscillations are coupled through combined capacitance and resistance, so that in a single pair of oscillators (nearly) in-phase synchronization is obtained. The internal coupling within each group is relatively strong, but there is a phase difference between the fast and slow oscillators. The external coupling between the two groups is weak. The experiments show that the two groups can exhibit (nearly) anti-phase collective synchronization. Such synchronization occurs only when the external coupling is weak, and the interactions are delayed by the capacitance. When the external coupling is restricted to those between the fast and the slow elements, the anti-phase synchronization is more prominent. The results are interpreted with phase models. The theory predicts that, for anti-phase collective synchronization, there must be a minimum internal phase difference for a given shift in the phase coupling function. This condition is less stringent with external fast-to-slow coupling. The results provide a framework for applications of collective phase synchronization in modular networks where weak coupling between the groups can induce synchronization without rearrangements of the phase dynamics within the groups. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.
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Affiliation(s)
- Michael Sebek
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St Louis, MO 63103, USA
| | - Yoji Kawamura
- Center for Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, 236-0001 Yokohama, Japan
| | - Ashley M. Nott
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St Louis, MO 63103, USA
| | - István Z. Kiss
- Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, St Louis, MO 63103, USA
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Knobloch E, Uecker H, Wetzel D. Defectlike structures and localized patterns in the cubic-quintic-septic Swift-Hohenberg equation. Phys Rev E 2019; 100:012204. [PMID: 31499926 DOI: 10.1103/physreve.100.012204] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2019] [Indexed: 11/07/2022]
Abstract
We study numerically the cubic-quintic-septic Swift-Hohenberg (SH357) equation on bounded one-dimensional domains. Under appropriate conditions stripes with wave number k≈1 bifurcate supercritically from the zero state and form S-shaped branches resulting in bistability between small and large amplitude stripes. Within this bistability range we find stationary heteroclinic connections or fronts between small and large amplitude stripes, and demonstrate that the associated spatially localized defectlike structures either snake or fall on isolas. In other parameter regimes we also find heteroclinic connections to spatially homogeneous states and a multitude of dynamically stable steady states consisting of patches of small and large amplitude stripes with different wave numbers or of spatially homogeneous patches. The SH357 equation is thus extremely rich in the types of patterns it exhibits. Some of the features of the bifurcation diagrams obtained by numerical continuation can be understood using a conserved quantity, the spatial Hamiltonian of the system.
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Affiliation(s)
- Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Hannes Uecker
- Institut für Mathematik, Universität Oldenburg, D26111 Oldenburg, Germany
| | - Daniel Wetzel
- Institut für Mathematik, Universität Oldenburg, D26111 Oldenburg, Germany
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Kawamura Y. Collective phase reduction of globally coupled noisy dynamical elements. Phys Rev E 2017; 95:032225. [PMID: 28415367 DOI: 10.1103/physreve.95.032225] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2017] [Indexed: 06/07/2023]
Abstract
We formulate a theory for the collective phase reduction of globally coupled noisy dynamical elements exhibiting macroscopic rhythms. We first transform the Langevin-type equation that represents a group of globally coupled noisy dynamical elements into the corresponding nonlinear Fokker-Planck equation and then develop the phase reduction method for limit-cycle solutions to the nonlinear Fokker-Planck equation. The theory enables us to describe the collective dynamics of a group of globally coupled noisy dynamical elements by a single degree of freedom called the collective phase. As long as the group collectively exhibits macroscopic rhythms, the theory is applicable even when the coupling and noise are strong; it is also independent of the assumption that each element of the group is a self-sustained oscillator. We also provide a simple and accurate numerical algorithm for the collective phase description method and numerically illustrate the theory using a group of globally coupled noisy FitzHugh-Nagumo elements.
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Affiliation(s)
- Yoji Kawamura
- Department of Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan and Research and Development Center for Marine Biosciences, Japan Agency for Marine-Earth Science and Technology, Yokosuka 237-0061, Japan
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Premalatha K, Chandrasekar VK, Senthilvelan M, Lakshmanan M. Chimeralike states in two distinct groups of identical populations of coupled Stuart-Landau oscillators. Phys Rev E 2017; 95:022208. [PMID: 28297891 DOI: 10.1103/physreve.95.022208] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2016] [Indexed: 06/06/2023]
Abstract
We show the existence of chimeralike states in two distinct groups of identical populations of globally coupled Stuart-Landau oscillators. The existence of chimeralike states occurs only for a small range of frequency difference between the two populations, and these states disappear for an increase of mismatch between the frequencies. Here the chimeralike states are characterized by the synchronized oscillations in one population and desynchronized oscillations in another population. We also find that such states observed in two distinct groups of identical populations of nonlocally coupled oscillators are different from the above case in which coexisting domains of synchronized and desynchronized oscillations are observed in one population and the second population exhibits synchronized oscillations for spatially prepared initial conditions. Perturbation from such spatially prepared initial condition leads to the existence of imperfectly synchronized states. An imperfectly synchronized state represents the existence of solitary oscillators which escape from the synchronized group in population I and synchronized oscillations in population II. Also the existence of chimera state is independent of the increase of frequency mismatch between the populations. We also find the coexistence of different dynamical states with respect to different initial conditions, which causes multistability in the globally coupled system. In the case of nonlocal coupling, the system does not show multistability except in the cluster state region.
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Affiliation(s)
- K Premalatha
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
| | - V K Chandrasekar
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA University, Thanjavur 613 401, Tamilnadu, India
| | - M Senthilvelan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
| | - M Lakshmanan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
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Petkoski S, Spiegler A, Proix T, Aram P, Temprado JJ, Jirsa VK. Heterogeneity of time delays determines synchronization of coupled oscillators. Phys Rev E 2016; 94:012209. [PMID: 27575125 DOI: 10.1103/physreve.94.012209] [Citation(s) in RCA: 32] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2015] [Indexed: 05/01/2023]
Abstract
Network couplings of oscillatory large-scale systems, such as the brain, have a space-time structure composed of connection strengths and signal transmission delays. We provide a theoretical framework, which allows treating the spatial distribution of time delays with regard to synchronization, by decomposing it into patterns and therefore reducing the stability analysis into the tractable problem of a finite set of delay-coupled differential equations. We analyze delay-structured networks of phase oscillators and we find that, depending on the heterogeneity of the delays, the oscillators group in phase-shifted, anti-phase, steady, and non-stationary clusters, and analytically compute their stability boundaries. These results find direct application in the study of brain oscillations.
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Affiliation(s)
- Spase Petkoski
- Aix-Marseille Université, Inserm, INS UMR_S 1106, 13005, Marseille, France
- Aix-Marseille Université, CNRS, ISM UMR 7287, 13288, Marseille, France
| | - Andreas Spiegler
- Aix-Marseille Université, Inserm, INS UMR_S 1106, 13005, Marseille, France
| | - Timothée Proix
- Aix-Marseille Université, Inserm, INS UMR_S 1106, 13005, Marseille, France
| | - Parham Aram
- Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield S10 2TN, United Kingdom
| | | | - Viktor K Jirsa
- Aix-Marseille Université, Inserm, INS UMR_S 1106, 13005, Marseille, France
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Burylko O, Kazanovich Y, Borisyuk R. Bifurcation study of phase oscillator systems with attractive and repulsive interaction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:022911. [PMID: 25215803 DOI: 10.1103/physreve.90.022911] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2014] [Indexed: 06/03/2023]
Abstract
We study a model of globally coupled phase oscillators that contains two groups of oscillators with positive (synchronizing) and negative (desynchronizing) incoming connections for the first and second groups, respectively. This model was previously studied by Hong and Strogatz (the Hong-Strogatz model) in the case of a large number of oscillators. We consider a generalized Hong-Strogatz model with a constant phase shift in coupling. Our approach is based on the study of invariant manifolds and bifurcation analysis of the system. In the case of zero phase shift, various invariant manifolds are analytically described and a new dynamical mode is found. In the case of a nonzero phase shift we obtained a set of bifurcation diagrams for various systems with three or four oscillators. It is shown that in these cases system dynamics can be complex enough and include multistability and chaotic oscillations.
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Affiliation(s)
- Oleksandr Burylko
- Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska Strasse 3, 01601 Kyiv, Ukraine
| | - Yakov Kazanovich
- Institute of Mathematical Problems of Biology, Russian Academy of Sciences, 142290 Pushchino, Russia
| | - Roman Borisyuk
- Institute of Mathematical Problems of Biology, Russian Academy of Sciences, 142290 Pushchino, Russia and School of Computing and Mathematics, Plymouth University, Plymouth PL4 8AA, United Kingdom
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Kawamura Y. Phase synchronization between collective rhythms of fully locked oscillator groups. Sci Rep 2014; 4:4832. [PMID: 24776525 PMCID: PMC4003481 DOI: 10.1038/srep04832] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/24/2014] [Accepted: 04/11/2014] [Indexed: 11/16/2022] Open
Abstract
A system of coupled oscillators can exhibit a rich variety of dynamical behaviors. When we investigate the dynamical properties of the system, we first analyze individual oscillators and the microscopic interactions between them. However, the structure of a coupled oscillator system is often hierarchical, so that the collective behaviors of the system cannot be fully clarified by simply analyzing each element of the system. For example, we found that two weakly interacting groups of coupled oscillators can exhibit anti-phase collective synchronization between the groups even though all microscopic interactions are in-phase coupling. This counter-intuitive phenomenon can occur even when the number of oscillators belonging to each group is only two, that is, when the total number of oscillators is only four. In this paper, we clarify the mechanism underlying this counter-intuitive phenomenon for two weakly interacting groups of two oscillators with global sinusoidal coupling.
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Affiliation(s)
- Yoji Kawamura
- Institute for Research on Earth Evolution, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan and Department of Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan
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Komarov M, Pikovsky A. Effects of nonresonant interaction in ensembles of phase oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:016210. [PMID: 21867276 DOI: 10.1103/physreve.84.016210] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2011] [Revised: 06/06/2011] [Indexed: 05/20/2023]
Abstract
We consider general properties of groups of interacting oscillators, for which the natural frequencies are not in resonance. Such groups interact via nonoscillating collective variables like the amplitudes of the order parameters defined for each group. We treat the phase dynamics of the groups using the Ott-Antonsen ansatz and reduce it to a system of coupled equations for the order parameters. We describe different regimes of cosynchrony in the groups. For a large number of groups, heteroclinic cycles, corresponding to a sequential synchronous activity of groups and chaotic states where the order parameters oscillate irregularly, are possible.
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Affiliation(s)
- Maxim Komarov
- Faculty of Computational Mathematics and Cybernetics, Nizhni Novgorod University, Nizhni Novgorod, Russia
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Kawamura Y, Nakao H, Arai K, Kori H, Kuramoto Y. Phase synchronization between collective rhythms of globally coupled oscillator groups: noisy identical case. CHAOS (WOODBURY, N.Y.) 2010; 20:043109. [PMID: 21198079 DOI: 10.1063/1.3491344] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/18/2023]
Abstract
We theoretically investigate the collective phase synchronization between interacting groups of globally coupled noisy identical phase oscillators exhibiting macroscopic rhythms. Using the phase reduction method, we derive coupled collective phase equations describing the macroscopic rhythms of the groups from microscopic Langevin phase equations of the individual oscillators via nonlinear Fokker-Planck equations. For sinusoidal microscopic coupling, we determine the type of the collective phase coupling function, i.e., whether the groups exhibit in-phase or antiphase synchronization. We show that the macroscopic rhythms can exhibit effective antiphase synchronization even if the microscopic phase coupling between the groups is in-phase, and vice versa. Moreover, near the onset of collective oscillations, we analytically obtain the collective phase coupling function using center-manifold and phase reductions of the nonlinear Fokker-Planck equations.
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Affiliation(s)
- Yoji Kawamura
- Institute for Research on Earth Evolution, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan.
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Kawamura Y, Nakao H, Arai K, Kori H, Kuramoto Y. Phase synchronization between collective rhythms of globally coupled oscillator groups: noiseless nonidentical case. CHAOS (WOODBURY, N.Y.) 2010; 20:043110. [PMID: 21198080 DOI: 10.1063/1.3491346] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/18/2023]
Abstract
We theoretically study the synchronization between collective oscillations exhibited by two weakly interacting groups of nonidentical phase oscillators with internal and external global sinusoidal couplings of the groups. Coupled amplitude equations describing the collective oscillations of the oscillator groups are obtained by using the Ott-Antonsen ansatz, and then coupled phase equations for the collective oscillations are derived by phase reduction of the amplitude equations. The collective phase coupling function, which determines the dynamics of macroscopic phase differences between the groups, is calculated analytically. We demonstrate that the groups can exhibit effective antiphase collective synchronization even if the microscopic external coupling between individual oscillator pairs belonging to different groups is in-phase, and similarly effective in-phase collective synchronization in spite of microscopic antiphase external coupling between the groups.
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Affiliation(s)
- Yoji Kawamura
- Institute for Research on Earth Evolution, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan.
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Sheeba JH, Chandrasekar VK, Lakshmanan M. Chimera and globally clustered chimera: impact of time delay. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:046203. [PMID: 20481803 DOI: 10.1103/physreve.81.046203] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2009] [Revised: 01/11/2010] [Indexed: 05/29/2023]
Abstract
Following a short report of our preliminary results [Sheeba, Phys. Rev. E 79, 055203(R) (2009)], we present a more detailed study of the effects of coupling delay in diffusively coupled phase oscillator populations. We find that coupling delay induces chimera and globally clustered chimera (GCC) states in delay coupled populations. We show the existence of multiclustered states that act as link between the chimera and the GCC states. A stable GCC state goes through a variety of GCC states, namely, periodic, aperiodic, long- and short-period breathers and becomes unstable GCC leading to global synchronization in the system, on increasing time delay. We provide numerical evidence and theoretical explanations for the above results and discuss possible applications of the observed phenomena.
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Affiliation(s)
- Jane H Sheeba
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamilnadu, India
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Sheeba JH, Chandrasekar VK, Lakshmanan M. Globally clustered chimera states in delay-coupled populations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:055203. [PMID: 19518512 DOI: 10.1103/physreve.79.055203] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2009] [Revised: 04/02/2009] [Indexed: 05/27/2023]
Abstract
We have identified the existence of globally clustered chimera states in delay-coupled oscillator populations and find that these states can breathe periodically and aperiodically and become unstable depending upon the value of coupling delay. We also find that the coupling delay induces frequency suppression in the desynchronized group. We provide numerical evidence and theoretical explanations for the above results and discuss possible applications of the observed phenomena.
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Affiliation(s)
- Jane H Sheeba
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli, 620024 Tamilnadu, India
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Sheeba JH, Chandrasekar VK, Stefanovska A, McClintock PVE. Asymmetry-induced effects in coupled phase-oscillator ensembles: Routes to synchronization. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:046210. [PMID: 19518315 DOI: 10.1103/physreve.79.046210] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/03/2008] [Indexed: 05/18/2023]
Abstract
A system of two coupled ensembles of phase oscillators can follow different routes to interensemble synchronization. Following a short report of our preliminary results [Phys. Rev. E 78, 025201(R) (2008)], we present a more detailed study of the effects of coupling, noise, and phase asymmetries in coupled phase-oscillator ensembles. We identify five distinct synchronization regions and routes to synchronization that are characteristic of the coupling asymmetry. We show that noise asymmetry induces effects similar to that of coupling asymmetry when the latter is absent. We also find that phase asymmetry controls the probability of occurrence of particular routes to synchronization. Our results suggest that asymmetry plays a crucial role in controlling synchronization within and between oscillator ensembles, and hence that its consideration is vital for modeling real life problems.
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Affiliation(s)
- Jane H Sheeba
- Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom
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Abstract
There is growing evidence in favor of the temporal-coding hypothesis that temporal correlation of neuronal discharges may serve to bind distributed neuronal activity into unique representations and, in particular, that theta (3.5-7.5 Hz) and delta (0.5 < 3.5 Hz) oscillations facilitate information coding. The theta- and delta-rhythms are shown to be involved in various sleep stages, and during anesthesia, they undergo changes with the depth of anesthesia. We introduce a thalamocortical model of interacting neuronal ensembles to describe phase relationships between theta- and delta-oscillations, especially during deep and light anesthesia. Asymmetric and long-range interactions among the thalamocortical neuronal oscillators are taken into account. The model results are compared with experimental observations. The delta- and theta-activities are found to be separately generated and are governed by the thalamus and cortex, respectively. Changes in the degree of intraensemble and interensemble synchrony imply that the neuronal ensembles inhibit information coding during deep anesthesia and facilitate it during light anesthesia.
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