1
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Sabhahit NG, Khurd AS, Jalan S. Prolonged hysteresis in the Kuramoto model with inertia and higher-order interactions. Phys Rev E 2024; 109:024212. [PMID: 38491693 DOI: 10.1103/physreve.109.024212] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/05/2023] [Accepted: 01/08/2024] [Indexed: 03/18/2024]
Abstract
The inclusion of inertia in the Kuramoto model has long been reported to change the nature of a phase transition, providing a fertile ground to model the dynamical behaviors of interacting units. More recently, higher-order interactions have been realized as essential for the functioning of real-world complex systems ranging from the brain to disease spreading. Yet analytical insights to decipher the role of inertia with higher-order interactions remain challenging. Here, we study the Kuramoto model with inertia on simplicial complexes, merging two research domains. We develop an analytical framework in a mean-field setting using self-consistent equations to describe the steady-state behavior, which reveals a prolonged hysteresis in the synchronization profile. Inertia and triadic interaction strength exhibit isolated influence on system dynamics by predominantly governing, respectively, the forward and backward transition points. This paper sets a paradigm to deepen our understanding of real-world complex systems such as power grids modeled as the Kuramoto model with inertia.
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Affiliation(s)
- Narayan G Sabhahit
- Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, India
| | - Akanksha S Khurd
- Department of Physics, Indian Institute of Science Education and Research (IISER) Tirupati, Tirupati 517507, India
| | - Sarika Jalan
- Complex Systems Lab, Department of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore-453552, India
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2
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Smith LD, Liu P. Determining bifurcations to explosive synchronization for networks of coupled oscillators with higher-order interactions. Phys Rev E 2024; 109:L022202. [PMID: 38491677 DOI: 10.1103/physreve.109.l022202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2023] [Accepted: 01/22/2024] [Indexed: 03/18/2024]
Abstract
We determine bifurcations from gradual to explosive synchronization in coupled oscillator networks with higher-order coupling using self-consistency analysis. We obtain analytic bifurcation values for generic symmetric natural frequency distributions. We show that nonsynchronized, drifting, oscillators are non-negligible and play a crucial role in bifurcation. As such, the entire natural frequency distribution must be accounted for, rather than just the shape at the center. We verify our results for Lorentzian- and Gaussian-distributed natural frequencies.
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Affiliation(s)
- Lauren D Smith
- Department of Mathematics, University of Auckland, Auckland 1142, New Zealand
| | - Penghao Liu
- Department of Mathematics, University of Auckland, Auckland 1142, New Zealand
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3
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Berner R, Lu A, Sokolov IM. Synchronization transitions in Kuramoto networks with higher-mode interaction. CHAOS (WOODBURY, N.Y.) 2023; 33:073138. [PMID: 37463093 DOI: 10.1063/5.0151038] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2023] [Accepted: 06/21/2023] [Indexed: 07/20/2023]
Abstract
Synchronization is an omnipresent collective phenomenon in nature and technology, whose understanding is still elusive for real-world systems in particular. We study the synchronization transition in a phase oscillator system with two nonvanishing Fourier-modes in the interaction function, hence going beyond the Kuramoto paradigm. We show that the transition scenarios crucially depend on the interplay of the two coupling modes. We describe the multistability induced by the presence of a second coupling mode. By extending the collective coordinate approach, we describe the emergence of various states observed in the transition from incoherence to coherence. Remarkably, our analysis suggests that, in essence, the two-mode coupling gives rise to states characterized by two independent but interacting groups of oscillators. We believe that these findings will stimulate future research on dynamical systems, including complex interaction functions beyond the Kuramoto-type.
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Affiliation(s)
- Rico Berner
- Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
| | - Annie Lu
- Department of Mathematics, Washington State University, Pullman, Washington 99164-3113, USA
| | - Igor M Sokolov
- Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
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4
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Kachhvah AD, Jalan S. First-order route to antiphase clustering in adaptive simplicial complexes. Phys Rev E 2022; 105:L062203. [PMID: 35854537 DOI: 10.1103/physreve.105.l062203] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2022] [Accepted: 06/17/2022] [Indexed: 06/15/2023]
Abstract
This Letter investigates the transition to synchronization of oscillator ensembles encoded by simplicial complexes in which pairwise and higher-order coupling weights alter with time through a rate-based adaptive mechanism inspired by the Hebbian learning rule. These simultaneously evolving disparate adaptive coupling weights lead to a phenomenon in that the in-phase synchronization is completely obliterated; instead, the antiphase synchronization is originated. In addition, the onsets of antiphase synchronization and desynchronization are manageable through both dyadic and triadic learning rates. The theoretical validation of these numerical assessments is delineated thoroughly by employing Ott-Antonsen dimensionality reduction. The framework and results of the Letter would help understand the underlying synchronization behavior of a range of real-world systems, such as the brain functions and social systems where interactions evolve with time.
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Affiliation(s)
- Ajay Deep Kachhvah
- Complex Systems Laboratory, Department of Physics, Indian Institute of Technology Indore - Simrol, Indore 453552, India
| | - Sarika Jalan
- Complex Systems Laboratory, Department of Physics, Indian Institute of Technology Indore - Simrol, Indore 453552, India
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5
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Wang X, Zheng Z, Xu C. Collective dynamics of phase oscillator populations with three-body interactions. Phys Rev E 2021; 104:054208. [PMID: 34942717 DOI: 10.1103/physreve.104.054208] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2021] [Accepted: 10/26/2021] [Indexed: 11/07/2022]
Abstract
Many-body interactions between dynamical agents have caught particular attention in recent works that found wide applications in physics, neuroscience, and sociology. In this paper we investigate such higher order (nonadditive) interactions on collective dynamics in a system of globally coupled heterogeneous phase oscillators. We show that the three-body interactions encoded microscopically in nonlinear couplings give rise to added dynamic phenomena occurring beyond the pairwise interactions. The system in general displays an abrupt desynchronization transition characterized by irreversible explosive synchronization via an infinite hysteresis loop. More importantly, we give a mathematical argument that such an abrupt dynamic pattern is a universally expected effect. Furthermore, the origin of this abrupt transition is uncovered by performing a rigorous stability analysis of the equilibrium states, as well as by providing a detailed description of the spectrum structure of linearization around the steady states. Our work reveals a self-organized phenomenon that is responsible for the rapid switching to synchronization in diverse complex systems exhibiting critical transitions with nonpairwise interactions.
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Affiliation(s)
- Xuan Wang
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
| | - Zhigang Zheng
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China.,School of Mathematical Science, Huaqiao University, Quanzhou 362021, China
| | - Can Xu
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China.,School of Mathematical Science, Huaqiao University, Quanzhou 362021, China
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6
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Rosenblum M, Pikovsky A. Nonlinear phase coupling functions: a numerical study. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20190093. [PMID: 31656143 PMCID: PMC6833997 DOI: 10.1098/rsta.2019.0093] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/04/2023]
Abstract
Phase reduction is a general tool widely used to describe forced and interacting self-sustained oscillators. Here, we explore the phase coupling functions beyond the usual first-order approximation in the strength of the force. Taking the periodically forced Stuart-Landau oscillator as the paradigmatic model, we determine and numerically analyse the coupling functions up to the fourth order in the force strength. We show that the found nonlinear phase coupling functions can be used for predicting synchronization regions of the forced oscillator. 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 Rosenblum
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
- Control Theory Department, Institute of Information Technologies, Mathematics and Mechanics, Lobachevsky University Nizhny Novgorod, Nizhny Novgorod, Russia
- e-mail:
| | - Arkady Pikovsky
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
- Control Theory Department, Institute of Information Technologies, Mathematics and Mechanics, Lobachevsky University Nizhny Novgorod, Nizhny Novgorod, Russia
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7
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Stankovski T, Pereira T, McClintock PVE, Stefanovska A. Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20190039. [PMID: 31656134 PMCID: PMC6834002 DOI: 10.1098/rsta.2019.0039] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Accepted: 08/13/2019] [Indexed: 06/10/2023]
Abstract
Dynamical systems are widespread, with examples in physics, chemistry, biology, population dynamics, communications, climatology and social science. They are rarely isolated but generally interact with each other. These interactions can be characterized by coupling functions-which contain detailed information about the functional mechanisms underlying the interactions and prescribe the physical rule specifying how each interaction occurs. Coupling functions can be used, not only to understand, but also to control and predict the outcome of the interactions. This theme issue assembles ground-breaking work on coupling functions by leading scientists. After overviewing the field and describing recent advances in the theory, it discusses novel methods for the detection and reconstruction of coupling functions from measured data. It then presents applications in chemistry, neuroscience, cardio-respiratory physiology, climate, electrical engineering and social science. Taken together, the collection summarizes earlier work on coupling functions, reviews recent developments, presents the state of the art, and looks forward to guide the future evolution of the field. 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)
- Tomislav Stankovski
- Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
- Faculty of Medicine, Ss Cyril and Methodius University, Skopje 1000, Macedonia
| | - Tiago Pereira
- Department of Mathematics, Imperial College London, London SW7 2AZ, UK
- Institute of Mathematical and Computer Sciences, University of Sao Paulo, Sao Carlos 13566-590, Brazil
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8
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Delabays R. Dynamical equivalence between Kuramoto models with first- and higher-order coupling. CHAOS (WOODBURY, N.Y.) 2019; 29:113129. [PMID: 31779348 DOI: 10.1063/1.5118941] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/06/2019] [Accepted: 11/11/2019] [Indexed: 06/10/2023]
Abstract
The Kuramoto model with high-order coupling has recently attracted some attention in the field of coupled oscillators in order, for instance, to describe clustering phenomena in sets of coupled agents. Instead of considering interactions given directly by the sine of oscillators' angle differences, the interaction is given by the sum of sines of integer multiples of these angle differences. This can be interpreted as a Fourier decomposition of a general 2π-periodic interaction function. We show that in the case where only one multiple of the angle differences is considered, which we refer to as the "Kuramoto model with simple qth-order coupling," the system is dynamically equivalent to the original Kuramoto model. In other words, any property of the Kuramoto model with simple higher-order coupling can be recovered from the standard Kuramoto model.
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Affiliation(s)
- Robin Delabays
- School of Engineering, University of Applied Sciences of Western Switzerland, CH-1950 Sion, Switzerland
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9
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Senthilkumar DV, Chandrasekar VK. Local and global chimera states in a four-oscillator system. Phys Rev E 2019; 100:032211. [PMID: 31639927 DOI: 10.1103/physreve.100.032211] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2019] [Indexed: 11/07/2022]
Abstract
Emergence of distinct collective dynamical states in the two smallest populations of nonlinear oscillators with global coupling among them is delineated. In particular, considering four oscillators constituting two populations of two oscillators each, we show the emergence of local chimera, global chimera, and global amplitude chimera states by breaking the permutation symmetry within the populations, in addition to the local synchronized and global synchronized states. Further, the symmetry breaking facilitates the onset of local chimera accompanied by the anomalous synchronization even with an identical nonisochronicity parameter in contrast to the existing literature. Furthermore, the spread of local synchronized and local chimera states is found to increase upon increasing the frequency difference among the populations, while the spread of local chimera, global chimera, and global amplitude chimera states increases for larger values of the nonisochronicity parameter. In addition, we have also deduced the stability condition for the global synchronized state using both the amplitude and its phase model, and find that the global synchronized state attains stability even for smaller values of the coupling strength for the amplitude model when compared to its phase reduced model.
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Affiliation(s)
- D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram-695 551, India
| | - V K Chandrasekar
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur-613 401, Tamil Nadu, India
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10
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Skardal PS, Arenas A. Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplexes. PHYSICAL REVIEW LETTERS 2019; 122:248301. [PMID: 31322386 DOI: 10.1103/physrevlett.122.248301] [Citation(s) in RCA: 74] [Impact Index Per Article: 14.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/28/2019] [Revised: 05/28/2019] [Indexed: 05/20/2023]
Abstract
Collective behavior in large ensembles of dynamical units with nonpairwise interactions may play an important role in several systems ranging from brain function to social networks. Despite recent work pointing to simplicial structure, i.e., higher-order interactions between three or more units at a time, their dynamical characteristics remain poorly understood. Here we present an analysis of the collective dynamics of such a simplicial system, namely coupled phase oscillators with three-way interactions. The simplicial structure gives rise to a number of novel phenomena, most notably a continuum of abrupt desynchronization transitions with no abrupt synchronization transition counterpart, as well as extensive multistability whereby infinitely many stable partially synchronized states exist. Our analysis sheds light on the complexity that can arise in physical systems with simplicial interactions like the human brain and the role that simplicial interactions play in storing information.
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Affiliation(s)
| | - Alex Arenas
- Departament d'Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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11
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Peter F, Pikovsky A. Transition to collective oscillations in finite Kuramoto ensembles. Phys Rev E 2018; 97:032310. [PMID: 29776135 DOI: 10.1103/physreve.97.032310] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2017] [Indexed: 01/01/2023]
Abstract
We present an alternative approach to finite-size effects around the synchronization transition in the standard Kuramoto model. Our main focus lies on the conditions under which a collective oscillatory mode is well defined. For this purpose, the minimal value of the amplitude of the complex Kuramoto order parameter appears as a proper indicator. The dependence of this minimum on coupling strength varies due to sampling variations and correlates with the sample kurtosis of the natural frequency distribution. The skewness of the frequency sample determines the frequency of the resulting collective mode. The effects of kurtosis and skewness hold in the thermodynamic limit of infinite ensembles. We prove this by integrating a self-consistency equation for the complex Kuramoto order parameter for two families of distributions with controlled kurtosis and skewness, respectively.
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Affiliation(s)
- Franziska Peter
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24-25, 14476 Potsdam, Germany
| | - Arkady Pikovsky
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24-25, 14476 Potsdam, Germany.,Research Institute for Supercomputing, Nizhny Novgorod State University, Gagarin Av. 23, 606950, Nizhny Novgorod, Russia
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12
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Skardal PS, Taylor D, Sun J, Arenas A. Erosion of synchronization: Coupling heterogeneity and network structure. PHYSICA D. NONLINEAR PHENOMENA 2016. [PMID: 27909350 DOI: 10.1016/j.physd.2015.10.015,] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Grants] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with well-known spectral properties, which allows us to derive further analytical results.
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Affiliation(s)
- Per Sebastian Skardal
- Department of Mathematics, Trinity College, Hartford, CT 06106, USA; Departament d'Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
| | - Dane Taylor
- Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA
| | - Jie Sun
- Department of Mathematics, Clarkson University, Potsdam, NY 13699, USA
| | - Alex Arenas
- Departament d'Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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13
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Skardal PS, Taylor D, Sun J, Arenas A. Erosion of synchronization: Coupling heterogeneity and network structure. PHYSICA D. NONLINEAR PHENOMENA 2016; 323-324:40-48. [PMID: 27909350 PMCID: PMC5125783 DOI: 10.1016/j.physd.2015.10.015] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with well-known spectral properties, which allows us to derive further analytical results.
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Affiliation(s)
- Per Sebastian Skardal
- Department of Mathematics, Trinity College, Hartford, CT 06106, USA
- Departament d’Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
| | - Dane Taylor
- Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA
| | - Jie Sun
- Department of Mathematics, Clarkson University, Potsdam, NY 13699, USA
| | - Alex Arenas
- Departament d’Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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14
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Kryukov AK, Petrov VS, Osipov GV, Kurths J. Multistability of synchronous regimes in rotator ensembles. CHAOS (WOODBURY, N.Y.) 2015; 25:123121. [PMID: 26723160 DOI: 10.1063/1.4938181] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We study collective dynamics in rotator ensembles and focus on the multistability of synchronous regimes in a chain of coupled rotators. We provide a detailed analysis of the number of coexisting regimes and estimate in particular, the synchronization boundary for different types of individual frequency distribution. The number of wave-based regimes coexisting for the same parameters and its dependence on the chain length are estimated. We give an analytical estimation for the synchronization frequency of the in-phase regime for a uniform individual frequency distribution.
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Affiliation(s)
- A K Kryukov
- Department of Control Theory, Nizhny Novgorod State University, 23, Gagarin Avenue, 603950 Nizhny Novgorod, Russia
| | - V S Petrov
- Department of Control Theory, Nizhny Novgorod State University, 23, Gagarin Avenue, 603950 Nizhny Novgorod, Russia
| | - G V Osipov
- Department of Control Theory, Nizhny Novgorod State University, 23, Gagarin Avenue, 603950 Nizhny Novgorod, Russia
| | - J Kurths
- Institute of Applied Physics, Russian Academy of Sciences, 603950 Nizhny Novgorod, Russia
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15
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Pikovsky A, Rosenblum M. Dynamics of globally coupled oscillators: Progress and perspectives. CHAOS (WOODBURY, N.Y.) 2015; 25:097616. [PMID: 26428569 DOI: 10.1063/1.4922971] [Citation(s) in RCA: 94] [Impact Index Per Article: 10.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/20/2023]
Abstract
In this paper, we discuss recent progress in research of ensembles of mean field coupled oscillators. Without an ambition to present a comprehensive review, we outline most interesting from our viewpoint results and surprises, as well as interrelations between different approaches.
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Affiliation(s)
- Arkady Pikovsky
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany
| | - Michael Rosenblum
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany
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16
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Komarov M, Pikovsky A. Intercommunity resonances in multifrequency ensembles of coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012906. [PMID: 26274246 DOI: 10.1103/physreve.92.012906] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2015] [Indexed: 06/04/2023]
Abstract
We generalize the Kuramoto model of globally coupled oscillators to multifrequency communities. A situation when mean frequencies of two subpopulations are close to the resonance 2:1 is considered in detail. We construct uniformly rotating solutions describing synchronization inside communities and between them. Remarkably, cross coupling across the frequencies can promote synchrony even when ensembles are separately asynchronous. We also show that the transition to synchrony due to the cross coupling is accompanied by a huge multiplicity of distinct synchronous solutions, which is directly related to a multibranch entrainment. On the other hand, for synchronous populations, the cross-frequency coupling can destroy phase locking and lead to chaos of mean fields.
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Affiliation(s)
- Maxim Komarov
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str 24, D-14476, Potsdam, Germany
- Department of Cell Biology and Neuroscience, University of California Riverside, 900 University Ave. Riverside, California 92521, USA
- Department of Control Theory, Nizhni Novgorod State University, Gagarin Av. 23, 606950, Nizhni Novgorod, Russia
| | - Arkady Pikovsky
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str 24, D-14476, Potsdam, Germany
- Department of Control Theory, Nizhni Novgorod State University, Gagarin Av. 23, 606950, Nizhni Novgorod, Russia
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17
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Skardal PS, Taylor D, Sun J, Arenas A. Erosion of synchronization in networks of coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:010802. [PMID: 25679557 DOI: 10.1103/physreve.91.010802] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/28/2014] [Indexed: 05/16/2023]
Abstract
We report erosion of synchronization in networks of coupled phase oscillators, a phenomenon where perfect phase synchronization is unattainable in steady state, even in the limit of infinite coupling. An analysis reveals that the total erosion is separable into the product of terms characterizing coupling frustration and structural heterogeneity, both of which amplify erosion. The latter, however, can differ significantly from degree heterogeneity. Finally, we show that erosion is marked by the reorganization of oscillators according to their node degrees rather than their natural frequencies.
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Affiliation(s)
- Per Sebastian Skardal
- Departament d'Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
| | - Dane Taylor
- Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, North Carolina 27709, USA and Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA
| | - Jie Sun
- Department of Mathematics, Clarkson University, Potsdam, New York, 13699, USA
| | - Alex Arenas
- Departament d'Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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18
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Iatsenko D, McClintock P, Stefanovska A. Glassy states and super-relaxation in populations of coupled phase oscillators. Nat Commun 2014; 5:4118. [PMID: 24947553 PMCID: PMC4083435 DOI: 10.1038/ncomms5118] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2013] [Accepted: 05/14/2014] [Indexed: 11/15/2022] Open
Abstract
Large networks of coupled oscillators appear in many branches of science, so that the kinds of phenomena they exhibit are not only of intrinsic interest but also of very wide importance. In 1975, Kuramoto proposed an analytically tractable model to describe these systems, which has since been successfully applied in many contexts and remains a subject of intensive research. Some related problems, however, remain unclarified for decades, such as the existence and properties of the oscillator glass state. Here we present a detailed analysis of a very general form of the Kuramoto model. In particular, we find the conditions when it can exhibit glassy behaviour, which represents a kind of synchronous disorder in the present case. Furthermore, we discover a new and intriguing phenomenon that we refer to as super-relaxation where the oscillators feel no interaction at all while relaxing to incoherence. Our findings offer the possibility of creating glassy states and observing super-relaxation in real systems.
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Affiliation(s)
- D. Iatsenko
- Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
| | | | - A. Stefanovska
- Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
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Dodla R, Wilson CJ. Interaction function of oscillating coupled neurons. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:042704. [PMID: 24229210 PMCID: PMC3928969 DOI: 10.1103/physreve.88.042704] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2013] [Revised: 08/28/2013] [Indexed: 06/02/2023]
Abstract
Large scale simulations of electrically coupled neuronal oscillators often employ the phase coupled oscillator paradigm to understand and predict network behavior. We study the nature of the interaction between such coupled oscillators using weakly coupled oscillator theory. By employing piecewise linear approximations for phase response curves and voltage time courses and parametrizing their shapes, we compute the interaction function for all such possible shapes and express it in terms of discrete Fourier modes. We find that reasonably good approximation is achieved with four Fourier modes that comprise of both sine and cosine terms.
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Affiliation(s)
- Ramana Dodla
- Department of Biology, University of Texas at San Antonio, San Antonio, Texas 78249, USA
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20
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Duggento A, Stankovski T, McClintock PVE, Stefanovska A. Dynamical Bayesian inference of time-evolving interactions: from a pair of coupled oscillators to networks of oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:061126. [PMID: 23367912 DOI: 10.1103/physreve.86.061126] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2012] [Revised: 11/25/2012] [Indexed: 06/01/2023]
Abstract
Living systems have time-evolving interactions that, until recently, could not be identified accurately from recorded time series in the presence of noise. Stankovski et al. [Phys. Rev. Lett. 109, 024101 (2012)] introduced a method based on dynamical Bayesian inference that facilitates the simultaneous detection of time-varying synchronization, directionality of influence, and coupling functions. It can distinguish unsynchronized dynamics from noise-induced phase slips. The method is based on phase dynamics, with Bayesian inference of the time-evolving parameters being achieved by shaping the prior densities to incorporate knowledge of previous samples. We now present the method in detail using numerically generated data, data from an analog electronic circuit, and cardiorespiratory data. We also generalize the method to encompass networks of interacting oscillators and thus demonstrate its applicability to small-scale networks.
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Affiliation(s)
- Andrea Duggento
- Medical Physics Section, Faculty of Medicine, Tor Vergata University, Rome, Italy
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21
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Stankovski T, Duggento A, McClintock PVE, Stefanovska A. Inference of time-evolving coupled dynamical systems in the presence of noise. PHYSICAL REVIEW LETTERS 2012; 109:024101. [PMID: 23030162 DOI: 10.1103/physrevlett.109.024101] [Citation(s) in RCA: 44] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/21/2012] [Revised: 05/07/2012] [Indexed: 05/03/2023]
Abstract
A new method is introduced for analysis of interactions between time-dependent coupled oscillators, based on the signals they generate. It distinguishes unsynchronized dynamics from noise-induced phase slips and enables the evolution of the coupling functions and other parameters to be followed. It is based on phase dynamics, with Bayesian inference of the time-evolving parameters achieved by shaping the prior densities to incorporate knowledge of previous samples. The method is tested numerically and applied to reveal and quantify the time-varying nature of cardiorespiratory interactions.
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Affiliation(s)
- Tomislav Stankovski
- Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom
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22
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Ota K, Omori T, Watanabe S, Miyakawa H, Okada M, Aonishi T. Measurement of infinitesimal phase response curves from noisy real neurons. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:041902. [PMID: 22181170 DOI: 10.1103/physreve.84.041902] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/02/2011] [Revised: 03/27/2011] [Indexed: 05/31/2023]
Abstract
We sought to measure infinitesimal phase response curves (iPRCs) from rat hippocampal CA1 pyramidal neurons. It is difficult to measure iPRCs from noisy neurons because of the dilemma that either the linearity or the signal-to-noise ratio of responses to external perturbations must be sacrificed. To overcome this difficulty, we used an iPRC measurement model formulated as the Langevin phase equation (LPE) to extract iPRCs in the Bayesian scheme. We then simultaneously verified the effectiveness of the measurement model and the reliability of the estimated iPRCs by demonstrating that LPEs with the estimated iPRCs could predict the stochastic behaviors of the same neurons, whose iPRCs had been measured, when they were perturbed by periodic stimulus currents. Our results suggest that the LPE is an effective model for real oscillating neurons and that many theoretical frameworks based on it may be applicable to real nerve systems.
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Affiliation(s)
- Keisuke Ota
- Brain Science Institute, RIKEN, Saitama 351-0198, Japan
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23
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Skardal PS, Ott E, Restrepo JG. Cluster synchrony in systems of coupled phase oscillators with higher-order coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:036208. [PMID: 22060476 DOI: 10.1103/physreve.84.036208] [Citation(s) in RCA: 41] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2011] [Indexed: 05/31/2023]
Abstract
We study the phenomenon of cluster synchrony that occurs in ensembles of coupled phase oscillators when higher-order modes dominate the coupling between oscillators. For the first time, we develop a complete analytic description of the dynamics in the limit of a large number of oscillators and use it to quantify the degree of cluster synchrony, cluster asymmetry, and switching. We use a variation of the recent dimensionality-reduction technique of Ott and Antonsen [Chaos 18, 037113 (2008)] and find an analytic description of the degree of cluster synchrony valid on a globally attracting manifold. Shaped by this manifold, there is an infinite family of steady-state distributions of oscillators, resulting in a high degree of multistability in the cluster asymmetry. We also show how through external forcing the degree of asymmetry can be controlled, and suggest that systems displaying cluster synchrony can be used to encode and store data.
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Affiliation(s)
- Per Sebastian Skardal
- Department of Applied Mathematics, University of Colorado at Boulder, Colorado 80309, USA.
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24
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Tanaka T, Aoyagi T. Multistable attractors in a network of phase oscillators with three-body interactions. PHYSICAL REVIEW LETTERS 2011; 106:224101. [PMID: 21702602 DOI: 10.1103/physrevlett.106.224101] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/13/2010] [Indexed: 05/20/2023]
Abstract
Three-body interactions have been found in physics, biology, and sociology. To investigate their effect on dynamical systems, as a first step, we study numerically and theoretically a system of phase oscillators with a three-body interaction. As a result, an infinite number of multistable synchronized states appear above a critical coupling strength, while a stable incoherent state always exists for any coupling strength. Owing to the infinite multistability, the degree of synchrony in an asymptotic state can vary continuously within some range depending on the initial phase pattern.
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Affiliation(s)
- Takuma Tanaka
- Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, Japan.
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25
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Sen A, Dodla R, Johnston GL, Sethia GC. Amplitude Death, Synchrony, and Chimera States in Delay Coupled Limit Cycle Oscillators. UNDERSTANDING COMPLEX SYSTEMS 2009. [DOI: 10.1007/978-3-642-02329-3_1] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/11/2023]
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26
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Indic P, Schwartz WJ, Paydarfar D. Design principles for phase-splitting behaviour of coupled cellular oscillators: clues from hamsters with 'split' circadian rhythms. J R Soc Interface 2008; 5:873-83. [PMID: 18077247 PMCID: PMC2607461 DOI: 10.1098/rsif.2007.1248] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Nonlinear interactions among coupled cellular oscillators are likely to underlie a variety of complex rhythmic behaviours. Here we consider the case of one such behaviour, a doubling of rhythm frequency caused by the spontaneous splitting of a population of synchronized oscillators into two subgroups each oscillating in anti-phase (phase-splitting). An example of biological phase-splitting is the frequency doubling of the circadian locomotor rhythm in hamsters housed in constant light, in which the pacemaker in the suprachiasmatic nucleus (SCN) is reconfigured with its left and right halves oscillating in anti-phase. We apply the theory of coupled phase oscillators to show that stable phase-splitting requires the presence of negative coupling terms, through delayed and/or inhibitory interactions. We also find that the inclusion of real biological constraints (that the SCN contains a finite number of non-identical noisy oscillators) implies the existence of an underlying non-uniform network architecture, in which the population of oscillators must interact through at least two types of connections. We propose that a key design principle for the frequency doubling of a population of biological oscillators is inhomogeneity of oscillator coupling.
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Affiliation(s)
- Premananda Indic
- Department of Neurology, University of Massachusetts Medical School, Worcester, MA 01655, USA
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27
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Niedermayer T, Eckhardt B, Lenz P. Synchronization, phase locking, and metachronal wave formation in ciliary chains. CHAOS (WOODBURY, N.Y.) 2008; 18:037128. [PMID: 19045502 DOI: 10.1063/1.2956984] [Citation(s) in RCA: 45] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
Synchronization and wave formation in one-dimensional ciliary arrays are studied analytically and numerically. We develop a simple model for ciliary motion that is complex enough to describe well the behavior of beating cilia but simple enough to study collective effects analytically. Beating cilia are described as phase oscillators moving on circular trajectories with a variable radius. This radial degree of freedom turns out to be essential for the occurrence of hydrodynamically induced synchronization of ciliary beating between neighboring cilia. The transitions to the synchronized and phase-locked state of two cilia and the formation of metachronal waves in ciliary chains with different boundary conditions are discussed.
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Affiliation(s)
- Thomas Niedermayer
- Fachbereich Physik, Philipps-Universität Marburg, 35032 Marburg, Germany
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28
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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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29
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Osterhage H, Mormann F, Wagner T, Lehnertz K. Detecting directional coupling in the human epileptic brain: limitations and potential pitfalls. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:011914. [PMID: 18351883 DOI: 10.1103/physreve.77.011914] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2007] [Revised: 06/04/2007] [Indexed: 05/26/2023]
Abstract
We study directional relationships-in the driver-responder sense-in networks of coupled nonlinear oscillators using a phase modeling approach. Specifically, we focus on the identification of drivers in clusters with varying levels of synchrony, mimicking dynamical interactions between the seizure generating region (epileptic focus) and other brain structures. We demonstrate numerically that such an identification is not always possible in a reliable manner. Using the same analysis techniques as in model systems, we study multichannel electroencephalographic recordings from two patients suffering from focal epilepsy. Our findings demonstrate that--depending on the degree of intracluster synchrony--certain subsystems can spuriously appear to be driving others, which should be taken into account when analyzing field data with unknown underlying dynamics.
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Affiliation(s)
- Hannes Osterhage
- Department of Epileptology, Neurophysics Group, University of Bonn, Sigmund-Freud-Strasse 25, 53105 Bonn, Germany.
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30
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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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31
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Toth R, Taylor AF, Tinsley MR. Collective Behavior of a Population of Chemically Coupled Oscillators. J Phys Chem B 2006; 110:10170-6. [PMID: 16706479 DOI: 10.1021/jp060732z] [Citation(s) in RCA: 75] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023]
Abstract
Experiments are performed in which a large number (approximately 10(4)) of relaxation oscillators are globally coupled through the concentration of chemicals in the surrounding solution. Each oscillator consists of a microscopic catalyst-loaded particle that displays oscillations in the concentrations of chemical species when suspended in catalyst-free Belousov-Zhabotinsky (BZ) reaction solution. In the absence of stirring, the uncoupled particles display a range of oscillatory frequencies. In the well-stirred system, oscillations appear in the surrounding solution for greater than a critical number density of particles (n(crit)). There is a growth in the amplitude of oscillations with increasing n, accompanied by a slight increase or no change in frequency. A model is proposed to account for the behavior, in which the transfer of activator and inhibitor to and from the bulk medium is considered for each particle. We demonstrate that the appearance and subsequent growth in the amplitude of oscillations may be associated with partial synchronization of the oscillators.
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Affiliation(s)
- Rita Toth
- Department of Chemistry, University of Leeds, UK
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32
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Abrams DM, Strogatz SH. Chimera states for coupled oscillators. PHYSICAL REVIEW LETTERS 2004; 93:174102. [PMID: 15525081 DOI: 10.1103/physrevlett.93.174102] [Citation(s) in RCA: 583] [Impact Index Per Article: 29.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2004] [Indexed: 05/07/2023]
Abstract
Arrays of identical oscillators can display a remarkable spatiotemporal pattern in which phase-locked oscillators coexist with drifting ones. Discovered two years ago, such "chimera states" are believed to be impossible for locally or globally coupled systems; they are peculiar to the intermediate case of nonlocal coupling. Here we present an exact solution for this state, for a ring of phase oscillators coupled by a cosine kernel. We show that the stable chimera state bifurcates from a spatially modulated drift state, and dies in a saddle-node bifurcation with an unstable chimera state.
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Affiliation(s)
- Daniel M Abrams
- Department of Theoretical and Applied Mechanics, Cornell University, 212 Kimball Hall, Ithaca, New York 14853-1503, USA.
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33
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Aonishi T, Kurata K, Okada M. Acceleration effect of coupled oscillator systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:046223. [PMID: 12005992 DOI: 10.1103/physreve.65.046223] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/12/2001] [Indexed: 05/23/2023]
Abstract
We have developed a curved isochron clock (CIC) by modifying the radial isochron clock to provide a clean example of the acceleration (deceleration) effect. By analyzing a two-body system of coupled CICs, we determined that an unbalanced mutual interaction caused by curved isochron sets is the minimum mechanism needed for generating the acceleration (deceleration) effect in coupled oscillator systems. From this we can see that the Sakaguchi and Kuramoto (SK) model, which is a class of nonfrustrated mean field model, has an acceleration (deceleration) effect mechanism. To study frustrated coupled oscillator systems, we extended the SK model to two oscillator associative memory models, one with symmetric and the other with asymmetric dilution of coupling, which also have the minimum mechanism of the acceleration (deceleration) effect. We theoretically found that the Onsager reaction term (ORT), which is unique to frustrated systems, plays an important role in the acceleration (deceleration) effect. These two models are ideal for evaluating the effect of the ORT because, with the exception of the ORT, they have the same order parameter equations. We found that the two models have identical macroscopic properties, except for the acceleration effect caused by the ORT. By comparing the results of the two models, we can extract the effect of the ORT from only the rotation speeds of the oscillators.
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Affiliation(s)
- Toru Aonishi
- Brain Science Institute, RIKEN, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan
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34
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Aonishi T, Okada M. Multibranch entrainment and slow evolution among branches in coupled oscillators. PHYSICAL REVIEW LETTERS 2002; 88:024102. [PMID: 11801017 DOI: 10.1103/physrevlett.88.024102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2001] [Indexed: 05/23/2023]
Abstract
In globally coupled oscillators, it is believed that strong higher harmonics of coupling functions are essential for multibranch entrainment (MBE), in which there exist many stable states, whose number scales as approximately O(expN) (where N is the system size). The existence of MBE implies the nonergodicity of the system. Then, because this apparent breaking of ergodicity is caused by microscopic energy barriers, this seems to be in conflict with a basic principle of statistical physics. Using macroscopic dynamical theories, we demonstrate that there is no such ergodicity breaking, and such a system slowly evolves among branch states, jumping over microscopic energy barriers due to the influence of thermal noise. This phenomenon can be regarded as an example of slow dynamics driven by a perturbation along a neutrally stable manifold consisting of an infinite number of branch states.
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Affiliation(s)
- Toru Aonishi
- Brain Science Institute, RIKEN, 2-1 Hirosawa, Wako-shi, Saitama, 351-0198 Japan
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