1
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Kar R, Yadav A, Chandrasekar VK, Senthilkumar DV. Effect of higher-order interactions on chimera states in two populations of Kuramoto oscillators. CHAOS (WOODBURY, N.Y.) 2024; 34:023110. [PMID: 38363957 DOI: 10.1063/5.0181279] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2023] [Accepted: 01/23/2024] [Indexed: 02/18/2024]
Abstract
We investigate the effect of the fraction of pairwise and higher-order interactions on the emergent dynamics of the two populations of globally coupled Kuramoto oscillators with phase-lag parameters. We find that the stable chimera exists between saddle-node and Hopf bifurcations, while the breathing chimera lives between Hopf and homoclinic bifurcations in the two-parameter phase diagrams. The higher-order interaction facilitates the onset of the bifurcation transitions at a much lower disparity between the inter- and intra-population coupling strengths. Furthermore, the higher-order interaction facilitates the spread of breathing chimera in a large region of the parameter space while suppressing the spread of the stable chimera. A low degree of heterogeneity among the phase-lag parameters promotes the spread of both stable chimera and breathing chimera to a large region of the parameter space for a large fraction of the higher-order coupling. In contrast, a large degree of heterogeneity is found to decrease the spread of both chimera states for a large fraction of the higher-order coupling. A global synchronized state is observed above a critical value of heterogeneity among the phase-lag parameters. We have deduced the low-dimensional evolution equations for the macroscopic order parameters using the Ott-Antonsen Ansatz. We have also deduced the analytical saddle-node and Hopf bifurcation curves from the evolution equations for the macroscopic order parameters and found them to match with the bifurcation curves obtained using the software XPPAUT and with the simulation results.
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Affiliation(s)
- Rumi Kar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695551, Kerala, India
| | - Akash Yadav
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695551, Kerala, India
| | - V K Chandrasekar
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur 613 401, Tamil Nadu, India
| | - D V Senthilkumar
- School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695551, Kerala, India
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2
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Mishra A, Saha S, Dana SK. Chimeras in globally coupled oscillators: A review. CHAOS (WOODBURY, N.Y.) 2023; 33:092101. [PMID: 37703474 DOI: 10.1063/5.0143872] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2023] [Accepted: 08/21/2023] [Indexed: 09/15/2023]
Abstract
The surprising phenomenon of chimera in an ensemble of identical oscillators is no more strange behavior of network dynamics and reality. By this time, this symmetry breaking self-organized collective dynamics has been established in many networks, a ring of non-locally coupled oscillators, globally coupled networks, a three-dimensional network, and multi-layer networks. A variety of coupling and dynamical models in addition to the phase oscillators has been used for a successful observation of chimera patterns. Experimental verification has also been done using metronomes, pendula, chemical, and opto-electronic systems. The phenomenon has also been shown to appear in small networks, and hence, it is not size-dependent. We present here a brief review of the origin of chimera patterns restricting our discussions to networks of globally coupled identical oscillators only. The history of chimeras in globally coupled oscillators is older than what has been reported in nonlocally coupled phase oscillators much later. We elaborate the story of the origin of chimeras in globally coupled oscillators in a chronological order, within our limitations, and with brief descriptions of the significant contributions, including our personal experiences. We first introduce chimeras in non-locally coupled and other network configurations, in general, and then discuss about globally coupled networks in more detail.
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Affiliation(s)
- Arindam Mishra
- Department of Physics, National University of Singapore, Singapore 117551
| | - Suman Saha
- Cognitive Brain Dynamics Laboratory, National Brain Research Centre, Gurugram 122051, India
| | - Syamal K Dana
- Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India
- Division of Dynamics, Lodz University of Technology, 90-924 Lodz, Poland
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3
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Effects of Synaptic Pruning on Phase Synchronization in Chimera States of Neural Network. APPLIED SCIENCES-BASEL 2022. [DOI: 10.3390/app12041942] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/01/2023]
Abstract
This research explores the effect of synaptic pruning on a ring-shaped neural network of non-locally coupled FitzHugh–Nagumo (FHN) oscillators. The neurons in the pruned region synchronize with each other, and they repel the coherent domain of the chimera states. Furthermore, the width of the pruned region decides the precision and efficiency of the control effect on the position of coherent domains. This phenomenon gives a systematic comprehension of the relation between pruning and synchronization in neural networks from a new aspect that has never been addressed. An explanation of this mechanism is also given.
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4
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Anesiadis K, Provata A. Synchronization in Multiplex Leaky Integrate-and-Fire Networks With Nonlocal Interactions. FRONTIERS IN NETWORK PHYSIOLOGY 2022; 2:910862. [PMID: 36926067 PMCID: PMC10013047 DOI: 10.3389/fnetp.2022.910862] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/01/2022] [Accepted: 05/25/2022] [Indexed: 11/13/2022]
Abstract
We study synchronization phenomena in a multiplex network composed of two rings with identical Leaky Integrate-and-Fire (LIF) oscillators located on the nodes of the rings. Within each ring the LIF oscillators interact nonlocally, while between rings there are one-to-one inter-ring interactions. This structure is motivated by the observed connectivity between the two hemispheres of the brain: within each hemisphere the various brain regions interact with neighboring regions, while across hemispheres each region interacts, primarily, with the functionally homologous region. We consider both positive (excitatory) and negative (inhibitory) linking. We identify numerically various parameter regimes where the multiplex network develops coexistence of active and subthreshold domains, chimera states, solitary states, full coherence or incoherence. In particular, for weak inter-ring coupling (weak multiplexing) different synchronization patterns on the two rings are supported. These are stable and are obtained when the intra-ring coupling values are near the critical points separating qualitatively distinct synchronization regimes, e.g., between the travelling fronts regime and the chimera state one.
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Affiliation(s)
- K Anesiadis
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos", Athens, Greece.,School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Athens, Greece
| | - A Provata
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos", Athens, Greece
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5
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Khatun AA, Jafri HH, Punetha N. Controlling chimera states in chaotic oscillator ensembles through linear augmentation. Phys Rev E 2021; 103:042202. [PMID: 34005985 DOI: 10.1103/physreve.103.042202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2019] [Accepted: 03/07/2021] [Indexed: 11/07/2022]
Abstract
In this work, we show how "chimera states," namely, the dynamical situation when synchronized and desynchronized domains coexist in an oscillator ensemble, can be controlled through a linear augmentation (LA) technique. Specifically, in the networks of coupled chaotic oscillators, we obtain chimera states through induced multistability and demonstrate how LA can be used to control the size and spatial location of the incoherent and coherent populations in the ensemble. We examine basins of attraction of the system to analyze the effects of LA on its multistable behavior and thus on chimera states. Stability of the synchronized dynamics is analyzed through a master stability function. We find that these results are independent of a system's initial conditions and the strategy is applicable to the networks of globally, locally as well as nonlocally coupled oscillators. Our results suggest that LA control can be an effective method to control chimera states and to realize a desired collective dynamics in such ensembles.
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Affiliation(s)
- Anjuman Ara Khatun
- Department of Physics, Aligarh Muslim University, Aligarh 202 002, India
| | - Haider Hasan Jafri
- Department of Physics, Aligarh Muslim University, Aligarh 202 002, India
| | - Nirmal Punetha
- Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India
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6
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Ruzzene G, Omelchenko I, Sawicki J, Zakharova A, Schöll E, Andrzejak RG. Remote pacemaker control of chimera states in multilayer networks of neurons. Phys Rev E 2020; 102:052216. [PMID: 33327161 DOI: 10.1103/physreve.102.052216] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2020] [Accepted: 10/30/2020] [Indexed: 06/12/2023]
Abstract
Networks of coupled nonlinear oscillators allow for the formation of nontrivial partially synchronized spatiotemporal patterns, such as chimera states, in which there are coexisting coherent (synchronized) and incoherent (desynchronized) domains. These complementary domains form spontaneously, and it is impossible to predict where the synchronized group will be positioned within the network. Therefore, possible ways to control the spatial position of the coherent and incoherent groups forming the chimera states are of high current interest. In this work we investigate how to control chimera patterns in multiplex networks of FitzHugh-Nagumo neurons, and in particular we want to prove that it is possible to remotely control chimera states exploiting the multiplex structure. We introduce a pacemaker oscillator within the network: this is an oscillator that does not receive input from the rest of the network but is sending out information to its neighbors. The pacemakers can be positioned in one or both layers. Their presence breaks the spatial symmetry of the layer in which they are introduced and allows us to control the position of the incoherent domain. We demonstrate how the remote control is possible for both uni- and bidirectional coupling between the layers. Furthermore we show which are the limitations of our control mechanisms when it is generalized from single-layer to multilayer networks.
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Affiliation(s)
- Giulia Ruzzene
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
| | - Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Jakub Sawicki
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
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7
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Lilienkamp T, Parlitz U. Susceptibility of transient chimera states. Phys Rev E 2020; 102:032219. [PMID: 33075925 DOI: 10.1103/physreve.102.032219] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2019] [Accepted: 08/26/2020] [Indexed: 11/07/2022]
Abstract
Chaotic dynamics of a dynamical system is not necessarily persistent. If there is (without any active intervention from outside) a transition towards a (possibly nonchaotic) attractor, this phenomenon is called transient chaos, which can be observed in a variety of systems, e.g., in chemical reactions, population dynamics, neuronal activity, or cardiac dynamics. Also, chimera states, which show coherent and incoherent dynamics in spatially distinct regions of the system, are often chaotic transients. In many practical cases, the control of the chaotic dynamics (either the termination or the preservation of the chaotic dynamics) is desired. Although the self-termination typically occurs quite abruptly and can so far in general not be properly predicted, previous studies showed that in many systems a 'terminal transient phase" (TTP) prior to the self-termination existed, where the system was less susceptible against small but finite perturbations in different directions in state space. In this study, we show that, in the specific case of chimera states, these susceptible directions can be related to the structure of the chimera, which we divide into the coherent part, the incoherent part and the boundary in between. That means, in practice, if self-termination is close we can identify the direction of perturbation which is likely to maintain the chaotic dynamics (the chimera state). This finding improves the general understanding of the state space structure during the TTP, and could contribute also to practical applications like future control strategies of epileptic seizures which have been recently related to the collapse of chimera states.
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Affiliation(s)
- Thomas Lilienkamp
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany.,German Center for Cardiovascular Research (DZHK), Partner Site Göttingen, Robert-Koch-Straße 42a, 37075 Göttingen, Germany
| | - Ulrich Parlitz
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany.,German Center for Cardiovascular Research (DZHK), Partner Site Göttingen, Robert-Koch-Straße 42a, 37075 Göttingen, Germany.,Institut für Dynamik komplexer Systeme, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
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8
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Kang L, Tian C, Huo S, Liu Z. A two-layered brain network model and its chimera state. Sci Rep 2019; 9:14389. [PMID: 31591418 PMCID: PMC6779761 DOI: 10.1038/s41598-019-50969-5] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/07/2019] [Accepted: 09/23/2019] [Indexed: 01/14/2023] Open
Abstract
Based on the data of cerebral cortex, we present a two-layered brain network model of coupled neurons where the two layers represent the left and right hemispheres of cerebral cortex, respectively, and the links between the two layers represent the inter-couplings through the corpus callosum. By this model we show that abundant patterns of synchronization can be observed, especially the chimera state, depending on the parameters of system such as the coupling strengths and coupling phase. Further, we extend the model to a more general two-layered network to better understand the mechanism of the observed patterns, where each hemisphere of cerebral cortex is replaced by a highly clustered subnetwork. We find that the number of inter-couplings is another key parameter for the emergence of chimera states. Thus, the chimera states come from a matching between the structure parameters such as the number of inter-couplings and clustering coefficient etc and the dynamics parameters such as the intra-, inter-coupling strengths and coupling phase etc. A brief theoretical analysis is provided to explain the borderline of synchronization. These findings may provide helpful clues to understand the mechanism of brain functions.
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Affiliation(s)
- Ling Kang
- Department of Physics, East China Normal University, Shanghai, 200062, P.R. China
| | - Changhai Tian
- Department of Physics, East China Normal University, Shanghai, 200062, P.R. China
- School of Data Science, Tongren University, Tongren, 554300, P.R. China
| | - Siyu Huo
- Department of Physics, East China Normal University, Shanghai, 200062, P.R. China
| | - Zonghua Liu
- Department of Physics, East China Normal University, Shanghai, 200062, P.R. China.
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9
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Ghosh S, Schülen L, Deep Kachhvah A, Zakharova A, Jalan S. Taming chimeras in networks through multiplexing delays. ACTA ACUST UNITED AC 2019. [DOI: 10.1209/0295-5075/127/30002] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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10
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Ruzzene G, Omelchenko I, Schöll E, Zakharova A, Andrzejak RG. Controlling chimera states via minimal coupling modification. CHAOS (WOODBURY, N.Y.) 2019; 29:051103. [PMID: 31154763 DOI: 10.1063/1.5097570] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/26/2019] [Accepted: 04/17/2019] [Indexed: 06/09/2023]
Abstract
We propose a method to control chimera states in a ring-shaped network of nonlocally coupled phase oscillators. This method acts exclusively on the network's connectivity. Using the idea of a pacemaker oscillator, we investigate which is the minimal action needed to control chimeras. We implement the pacemaker choosing one oscillator and making its links unidirectional. Our results show that a pacemaker induces chimeras for parameters and initial conditions for which they do not form spontaneously. Furthermore, the pacemaker attracts the incoherent part of the chimera state, thus controlling its position. Beyond that, we find that these control effects can be achieved with modifications of the network's connectivity that are less invasive than a pacemaker, namely, the minimal action of just modifying the strength of one connection allows one to control chimeras.
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Affiliation(s)
- Giulia Ruzzene
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
| | - Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
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11
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Kasatkin DV, Klinshov VV, Nekorkin VI. Itinerant chimeras in an adaptive network of pulse-coupled oscillators. Phys Rev E 2019; 99:022203. [PMID: 30934254 DOI: 10.1103/physreve.99.022203] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/13/2018] [Indexed: 11/07/2022]
Abstract
In a network of pulse-coupled oscillators with adaptive coupling, we discover a dynamical regime which we call an "itinerant chimera." Similarly as in classical chimera states, the network splits into two domains, the coherent and the incoherent. The drastic difference is that the composition of the domains is volatile, i.e., the oscillators demonstrate spontaneous switching between the domains. This process can be seen as traveling of the oscillators from one domain to another or as traveling of the chimera core across the network. We explore the basic features of the itinerant chimeras, such as the mean and the variance of the core size, and the oscillators lifetime within the core. We also study the scaling behavior of the system and show that the observed regime is not a finite-size effect but a key feature of the collective dynamics which persists even in large networks.
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Affiliation(s)
- Dmitry V Kasatkin
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950, Nizhny Novgorod, Russia
| | - Vladimir V Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950, Nizhny Novgorod, Russia
| | - Vladimir I Nekorkin
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950, Nizhny Novgorod, Russia
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12
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Argyropoulos G, Kasimatis T, Provata A. Chimera patterns and subthreshold oscillations in two-dimensional networks of fractally coupled leaky integrate-and-fire neurons. Phys Rev E 2019; 99:022208. [PMID: 30934230 DOI: 10.1103/physreve.99.022208] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/15/2018] [Indexed: 11/07/2022]
Abstract
We discuss the effects that fractal coupling induces on chimera states in a network of leaky integrate-and-fire (LIF) oscillators arranged in a two-dimensional toroidal geometry. We provide evidence that the introduction of a hierarchical coupling topology in the form of a Sierpinski carpet gives rise to complex spatial structures such as multiple spots, stripe-and-grid chimeras, as well as traveling waves and subthreshold oscillations. Unlike in the case of typical nonlocal connectivity, when tuning the coupling strength to small positive values a spot chimera is formed with internal structure reminiscent of the fractal connectivity scheme. This is in line with previous results for one-dimensional networks, where hierarchical connectivity also induces chimeras with stratified spatial arrangements. In the case of negative coupling, cooperative effects produce subthreshold oscillating regions with traveling active islands crossing through them. Subthreshold oscillations and traveling waves are frequently reported in biological neural network experiments.
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Affiliation(s)
- G Argyropoulos
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos," GR-15341 Athens, Greece.,School of Electrical and Computer Engineering, National Technical University of Athens, GR-15780 Athens, Greece
| | - T Kasimatis
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos," GR-15341 Athens, Greece.,School of Applied Mathematical and Physical Sciences, National Technical University of Athens, GR-15780 Athens, Greece
| | - A Provata
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos," GR-15341 Athens, Greece
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13
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Omelchenko I, Zakharova A. Intriguing coexistence of synchrony and asynchrony in the brain: Comment on "Chimera states in neuronal networks: A review" by Soumen Majhi, Bidesh K. Bera, Dibakar Ghosh, Matjaẑ Perc. Phys Life Rev 2019; 28:134-136. [PMID: 30910384 DOI: 10.1016/j.plrev.2019.02.013] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/27/2019] [Accepted: 02/27/2019] [Indexed: 11/17/2022]
Affiliation(s)
- Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany.
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany.
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14
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Awal NM, Bullara D, Epstein IR. The smallest chimera: Periodicity and chaos in a pair of coupled chemical oscillators. CHAOS (WOODBURY, N.Y.) 2019; 29:013131. [PMID: 30709119 DOI: 10.1063/1.5060959] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2018] [Accepted: 12/28/2018] [Indexed: 06/09/2023]
Abstract
Symmetrically coupled identical oscillators were once believed to support only totally synchronous or totally asynchronous states. More recently, chimera states, in which a subset of oscillators behaves coherently while the other subset exhibits disorder, have been found in large arrays of oscillators, coupled either locally or globally. We demonstrate for the first time the existence of a chimera state with only two diffusively coupled identical oscillators, one behaving nearly periodically (coherently) and the other chaotically (incoherently). We attribute this behavior to a "master-slave" interaction, which arises via a symmetry-breaking canard explosion.
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Affiliation(s)
- Naziru M Awal
- Department of Chemistry, Brandeis University, Waltham, Massachusetts 02453, USA
| | - Domenico Bullara
- Department of Chemistry, Brandeis University, Waltham, Massachusetts 02453, USA
| | - Irving R Epstein
- Department of Chemistry, Brandeis University, Waltham, Massachusetts 02453, USA
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15
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Banerjee T, Biswas D, Ghosh D, Schöll E, Zakharova A. Networks of coupled oscillators: From phase to amplitude chimeras. CHAOS (WOODBURY, N.Y.) 2018; 28:113124. [PMID: 30501215 DOI: 10.1063/1.5054181] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/30/2018] [Accepted: 10/31/2018] [Indexed: 06/09/2023]
Abstract
We show that amplitude-mediated phase chimeras and amplitude chimeras can occur in the same network of nonlocally coupled identical oscillators. These are two different partial synchronization patterns, where spatially coherent domains coexist with incoherent domains and coherence/incoherence referring to both amplitude and phase or only the amplitude of the oscillators, respectively. By changing the coupling strength, the two types of chimera patterns can be induced. We find numerically that the amplitude chimeras are not short-living transients but can have a long lifetime. Also, we observe variants of the amplitude chimeras with quasiperiodic temporal oscillations. We provide a qualitative explanation of the observed phenomena in the light of symmetry breaking bifurcation scenarios. We believe that this study will shed light on the connection between two disparate chimera states having different symmetry-breaking properties.
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Affiliation(s)
- Tanmoy Banerjee
- Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan, Burdwan, 713 104 West Bengal, India
| | - Debabrata Biswas
- Department of Physics, Rampurhat College, Birbhum, 731 224 West Bengal, India
| | - Debarati Ghosh
- Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan, Burdwan, 713 104 West Bengal, India
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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16
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Clerc MG, Coulibaly S, Ferré MA, Rojas RG. Chimera states in a Duffing oscillators chain coupled to nearest neighbors. CHAOS (WOODBURY, N.Y.) 2018; 28:083126. [PMID: 30180634 DOI: 10.1063/1.5025038] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/06/2018] [Accepted: 07/31/2018] [Indexed: 06/08/2023]
Abstract
Coupled nonlinear oscillators can present complex spatiotemporal behaviors. Here, we report the coexistence of coherent and incoherent domains, called chimera states, in an array of identical Duffing oscillators coupled to their nearest neighbors. The chimera states show a significant variation of amplitude in the desynchronized domain. These intriguing states are observed in the bistability region between a homogeneous state and a spatiotemporal chaotic one. These dynamical behaviors are characterized by their Lyapunov spectra and their global phase coherence order parameter. The local coupling between oscillators prevents one domain from invading the other one. Depending on initial conditions, a family of chimera states appear, organized in a snaking-like diagram.
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Affiliation(s)
- M G Clerc
- Departamento de Física and Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla, 487-3 Santiago, Chile
| | - S Coulibaly
- Université de Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
| | - M A Ferré
- Departamento de Física and Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla, 487-3 Santiago, Chile
| | - R G Rojas
- Intituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla, 4059 Valparaíso, Chile
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17
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Ghosh S, Jalan S. Engineering chimera patterns in networks using heterogeneous delays. CHAOS (WOODBURY, N.Y.) 2018; 28:071103. [PMID: 30070528 DOI: 10.1063/1.5042133] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2018] [Accepted: 06/26/2018] [Indexed: 06/08/2023]
Abstract
Symmetry breaking spatial patterns, referred to as chimera states, have recently been catapulted into the limelight due to their coexisting coherent and incoherent hybrid dynamics. Here, we present a method to engineer a chimera state by using an appropriate distribution of heterogeneous time delays on the edges of a network. The time delays in interactions, intrinsic to natural or artificial complex systems, are known to induce various modifications in spatiotemporal behaviors of the coupled dynamics on networks. Using a coupled chaotic map with the identical coupling environment, we demonstrate that control over the spatial location of the incoherent region of a chimera state in a network can be achieved by appropriately introducing time delays. This method allows for the engineering of tailor-made one cluster or multi-cluster chimera patterns. Furthermore, borrowing a measure of eigenvector localization from the spectral graph theory, we introduce a spatial inverse participation ratio, which provides a robust way for the identification of the chimera state. This report highlights the necessity to consider the heterogeneous time delays to develop applications for the chimera states in particular and understand coupled dynamical systems in general.
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Affiliation(s)
- Saptarshi Ghosh
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
| | - Sarika Jalan
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India
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18
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Malchow AK, Omelchenko I, Schöll E, Hövel P. Robustness of chimera states in nonlocally coupled networks of nonidentical logistic maps. Phys Rev E 2018; 98:012217. [PMID: 30110780 DOI: 10.1103/physreve.98.012217] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2018] [Indexed: 06/08/2023]
Abstract
We investigate the dynamics of nonlocally coupled time-discrete maps with emphasis on the occurrence and robustness of chimera states. These peculiar, hybrid states are characterized by a coexistence of coherent and incoherent regions. We consider logistic maps coupled on a one-dimensional ring with finite coupling radius. Domains of chimera existence form different tongues in the parameter space of coupling range and coupling strength. For a sufficiently large coupling strength, each tongue refers to a wave number describing the structure of the spatial profile. We also analyze the period-adding scheme within these tongues and multiplicity of period solutions. Furthermore, we study the robustness of chimeras with respect to parameter inhomogeneities and find that these states persist for different widths of the parameter distribution. Finally, we explore the spatial structure of the chimera using a spatial correlation function.
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Affiliation(s)
- Anne-Kathleen Malchow
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Philipp Hövel
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
- Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität zu Berlin, Philippstraße 13, 10115 Berlin, Germany
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19
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Chouzouris T, Omelchenko I, Zakharova A, Hlinka J, Jiruska P, Schöll E. Chimera states in brain networks: Empirical neural vs. modular fractal connectivity. CHAOS (WOODBURY, N.Y.) 2018; 28:045112. [PMID: 31906648 DOI: 10.1063/1.5009812] [Citation(s) in RCA: 46] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
Abstract
Complex spatiotemporal patterns, called chimera states, consist of coexisting coherent and incoherent domains and can be observed in networks of coupled oscillators. The interplay of synchrony and asynchrony in complex brain networks is an important aspect in studies of both the brain function and disease. We analyse the collective dynamics of FitzHugh-Nagumo neurons in complex networks motivated by its potential application to epileptology and epilepsy surgery. We compare two topologies: an empirical structural neural connectivity derived from diffusion-weighted magnetic resonance imaging and a mathematically constructed network with modular fractal connectivity. We analyse the properties of chimeras and partially synchronized states and obtain regions of their stability in the parameter planes. Furthermore, we qualitatively simulate the dynamics of epileptic seizures and study the influence of the removal of nodes on the network synchronizability, which can be useful for applications to epileptic surgery.
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Affiliation(s)
- Teresa Chouzouris
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Jaroslav Hlinka
- Institute of Computer Science, Czech Academy of Sciences, Pod Vodarenskou vezi 2, 18207 Prague, Czech Republic
| | - Premysl Jiruska
- Institute of Physiology, Czech Academy of Sciences, Videnska 1083, 14220 Prague, Czech Republic
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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20
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Omelchenko I, Omel'chenko OE, Zakharova A, Schöll E. Optimal design of tweezer control for chimera states. Phys Rev E 2018; 97:012216. [PMID: 29448470 DOI: 10.1103/physreve.97.012216] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2017] [Indexed: 06/08/2023]
Abstract
Chimera states are complex spatio-temporal patterns which consist of coexisting domains of spatially coherent and incoherent dynamics in systems of coupled oscillators. In small networks, chimera states usually exhibit short lifetimes and erratic drifting of the spatial position of the incoherent domain. A tweezer feedback control scheme can stabilize and fix the position of chimera states. We analyze the action of the tweezer control in small nonlocally coupled networks of Van der Pol and FitzHugh-Nagumo oscillators, and determine the ranges of optimal control parameters. We demonstrate that the tweezer control scheme allows for stabilization of chimera states with different shapes, and can be used as an instrument for controlling the coherent domains size, as well as the maximum average frequency difference of the oscillators.
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Affiliation(s)
- Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | | | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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21
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Choe CU, Kim RS, Ri JS. Chimera and modulated drift states in a ring of nonlocally coupled oscillators with heterogeneous phase lags. Phys Rev E 2017; 96:032224. [PMID: 29346960 DOI: 10.1103/physreve.96.032224] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2017] [Indexed: 06/07/2023]
Abstract
We consider a ring of phase oscillators with nonlocal coupling strength and heterogeneous phase lags. We analyze the effects of heterogeneity in the phase lags on the existence and stability of a variety of steady states. A nonlocal coupling with heterogeneous phase lags that allows the system to be solved analytically is suggested and the stability of solutions along the Ott-Antonsen invariant manifold is explored. We present a complete bifurcation diagram for stationary patterns including the uniform drift and modulated drift states as well as chimera state, which reveals that the stable modulated drift state and a continuum of metastable drift states could occur due to the heterogeneity of the phase lags. We verify our theoretical results using the direct numerical simulations of the model system.
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Affiliation(s)
- Chol-Ung Choe
- Center for Nonlinear Science, University of Science, Unjong-District, Pyongyang, Democratic People's Republic of Korea
| | - Ryong-Son Kim
- Center for Nonlinear Science, University of Science, Unjong-District, Pyongyang, Democratic People's Republic of Korea
| | - Ji-Song Ri
- Center for Nonlinear Science, University of Science, Unjong-District, Pyongyang, Democratic People's Republic of Korea
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22
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Ponomarenko VI, Kulminskiy DD, Prokhorov MD. Chimeralike states in networks of bistable time-delayed feedback oscillators coupled via the mean field. Phys Rev E 2017; 96:022209. [PMID: 28950647 DOI: 10.1103/physreve.96.022209] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2017] [Indexed: 06/07/2023]
Abstract
We study the collective dynamics of oscillators in a network of identical bistable time-delayed feedback systems globally coupled via the mean field. The influence of delay and inertial properties of the mean field on the collective behavior of globally coupled oscillators is investigated. A variety of oscillation regimes in the network results from the presence of bistable states with substantially different frequencies in coupled oscillators. In the physical experiment and numerical simulation we demonstrate the existence of chimeralike states, in which some of the oscillators in the network exhibit synchronous oscillations, while all other oscillators remain asynchronous.
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Affiliation(s)
- V I Ponomarenko
- Saratov Branch of Kotel'nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov 410019, Russia
- Department of Nano- and Biomedical Technologies, Saratov State University, Astrakhanskaya Street, 83, Saratov, 410012, Russia
| | - D D Kulminskiy
- Saratov Branch of Kotel'nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov 410019, Russia
- Department of Nano- and Biomedical Technologies, Saratov State University, Astrakhanskaya Street, 83, Saratov, 410012, Russia
| | - M D Prokhorov
- Saratov Branch of Kotel'nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov 410019, Russia
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Andrzejak RG, Ruzzene G, Malvestio I. Generalized synchronization between chimera states. CHAOS (WOODBURY, N.Y.) 2017; 27:053114. [PMID: 28576111 DOI: 10.1063/1.4983841] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Networks of coupled oscillators in chimera states are characterized by an intriguing interplay of synchronous and asynchronous motion. While chimera states were initially discovered in mathematical model systems, there is growing experimental and conceptual evidence that they manifest themselves also in natural and man-made networks. In real-world systems, however, synchronization and desynchronization are not only important within individual networks but also across different interacting networks. It is therefore essential to investigate if chimera states can be synchronized across networks. To address this open problem, we use the classical setting of ring networks of non-locally coupled identical phase oscillators. We apply diffusive drive-response couplings between pairs of such networks that individually show chimera states when there is no coupling between them. The drive and response networks are either identical or they differ by a variable mismatch in their phase lag parameters. In both cases, already for weak couplings, the coherent domain of the response network aligns its position to the one of the driver networks. For identical networks, a sufficiently strong coupling leads to identical synchronization between the drive and response. For non-identical networks, we use the auxiliary system approach to demonstrate that generalized synchronization is established instead. In this case, the response network continues to show a chimera dynamics which however remains distinct from the one of the driver. Hence, segregated synchronized and desynchronized domains in individual networks congregate in generalized synchronization across networks.
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Affiliation(s)
- Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Catalonia, Spain
| | - Giulia Ruzzene
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Catalonia, Spain
| | - Irene Malvestio
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Catalonia, Spain
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24
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Gjurchinovski A, Schöll E, Zakharova A. Control of amplitude chimeras by time delay in oscillator networks. Phys Rev E 2017; 95:042218. [PMID: 28505829 DOI: 10.1103/physreve.95.042218] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2017] [Indexed: 06/07/2023]
Abstract
We investigate the influence of time-delayed coupling in a ring network of nonlocally coupled Stuart-Landau oscillators upon chimera states, i.e., space-time patterns with coexisting partially coherent and partially incoherent domains. We focus on amplitude chimeras, which exhibit incoherent behavior with respect to the amplitude rather than the phase and are transient patterns, and we show that their lifetime can be significantly enhanced by coupling delay. To characterize their transition to phase-lag synchronization (coherent traveling waves) and other coherent structures, we generalize the Kuramoto order parameter. Contrasting the results for instantaneous coupling with those for constant coupling delay, for time-varying delay, and for distributed-delay coupling, we demonstrate that the lifetime of amplitude chimera states and related partially incoherent states can be controlled, i.e., deliberately reduced or increased, depending upon the type of coupling delay.
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Affiliation(s)
- Aleksandar Gjurchinovski
- Institute of Physics, Faculty of Natural Sciences and Mathematics, Sts. Cyril and Methodius University, P.O. Box 162, 1000 Skopje, Macedonia
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
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25
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Krishnagopal S, Lehnert J, Poel W, Zakharova A, Schöll E. Synchronization patterns: from network motifs to hierarchical networks. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2017; 375:20160216. [PMID: 28115613 PMCID: PMC5311436 DOI: 10.1098/rsta.2016.0216] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 11/04/2016] [Indexed: 05/12/2023]
Abstract
We investigate complex synchronization patterns such as cluster synchronization and partial amplitude death in networks of coupled Stuart-Landau oscillators with fractal connectivities. The study of fractal or self-similar topology is motivated by the network of neurons in the brain. This fractal property is well represented in hierarchical networks, for which we present three different models. In addition, we introduce an analytical eigensolution method and provide a comprehensive picture of the interplay of network topology and the corresponding network dynamics, thus allowing us to predict the dynamics of arbitrarily large hierarchical networks simply by analysing small network motifs. We also show that oscillation death can be induced in these networks, even if the coupling is symmetric, contrary to previous understanding of oscillation death. Our results show that there is a direct correlation between topology and dynamics: hierarchical networks exhibit the corresponding hierarchical dynamics. This helps bridge the gap between mesoscale motifs and macroscopic networks.This article is part of the themed issue 'Horizons of cybernetical physics'.
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Affiliation(s)
- Sanjukta Krishnagopal
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
- Department of Physics, Birla Institute for Technology and Science Pilani, Pilani, Goa 403726, India
| | - Judith Lehnert
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
| | - Winnie Poel
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
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26
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Schmidt A, Kasimatis T, Hizanidis J, Provata A, Hövel P. Chimera patterns in two-dimensional networks of coupled neurons. Phys Rev E 2017; 95:032224. [PMID: 28415206 DOI: 10.1103/physreve.95.032224] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2016] [Indexed: 06/07/2023]
Abstract
We discuss synchronization patterns in networks of FitzHugh-Nagumo and leaky integrate-and-fire oscillators coupled in a two-dimensional toroidal geometry. A common feature between the two models is the presence of fast and slow dynamics, a typical characteristic of neurons. Earlier studies have demonstrated that both models when coupled nonlocally in one-dimensional ring networks produce chimera states for a large range of parameter values. In this study, we give evidence of a plethora of two-dimensional chimera patterns of various shapes, including spots, rings, stripes, and grids, observed in both models, as well as additional patterns found mainly in the FitzHugh-Nagumo system. Both systems exhibit multistability: For the same parameter values, different initial conditions give rise to different dynamical states. Transitions occur between various patterns when the parameters (coupling range, coupling strength, refractory period, and coupling phase) are varied. Many patterns observed in the two models follow similar rules. For example, the diameter of the rings grows linearly with the coupling radius.
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Affiliation(s)
- Alexander Schmidt
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Theodoros Kasimatis
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos", 15310 Athens, Greece
- School of Applied Mathematical and Physical Sciences, National Technical University of Athens, 15780 Athens, Greece
| | - Johanne Hizanidis
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos", 15310 Athens, Greece
- Crete Center for Quantum Complexity and Nanotechnology, Department of Physics, University of Crete, 71003 Heraklion, Greece
| | - Astero Provata
- Institute of Nanoscience and Nanotechnology, National Center for Scientific Research "Demokritos", 15310 Athens, Greece
| | - Philipp Hövel
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
- Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität zu Berlin, Philippstraße 13, 10115 Berlin, Germany
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27
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Kalle P, Sawicki J, Zakharova A, Schöll E. Chimera states and the interplay between initial conditions and non-local coupling. CHAOS (WOODBURY, N.Y.) 2017; 27:033110. [PMID: 28364760 DOI: 10.1063/1.4977866] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Chimera states are complex spatio-temporal patterns that consist of coexisting domains of coherent and incoherent dynamics. We study chimera states in a network of non-locally coupled Stuart-Landau oscillators. We investigate the impact of initial conditions in combination with non-local coupling. Based on an analytical argument, we show how the coupling phase and the coupling strength are linked to the occurrence of chimera states, flipped profiles of the mean phase velocity, and the transition from a phase- to an amplitude-mediated chimera state.
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Affiliation(s)
- Peter Kalle
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Jakub Sawicki
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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28
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Shena J, Hizanidis J, Kovanis V, Tsironis GP. Turbulent chimeras in large semiconductor laser arrays. Sci Rep 2017; 7:42116. [PMID: 28165053 PMCID: PMC5292712 DOI: 10.1038/srep42116] [Citation(s) in RCA: 36] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2016] [Accepted: 01/04/2017] [Indexed: 11/08/2022] Open
Abstract
Semiconductor laser arrays have been investigated experimentally and theoretically from the viewpoint of temporal and spatial coherence for the past forty years. In this work, we are focusing on a rather novel complex collective behavior, namely chimera states, where synchronized clusters of emitters coexist with unsynchronized ones. For the first time, we find such states exist in large diode arrays based on quantum well gain media with nearest-neighbor interactions. The crucial parameters are the evanescent coupling strength and the relative optical frequency detuning between the emitters of the array. By employing a recently proposed figure of merit for classifying chimera states, we provide quantitative and qualitative evidence for the observed dynamics. The corresponding chimeras are identified as turbulent according to the irregular temporal behavior of the classification measure.
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Affiliation(s)
- J. Shena
- Crete Center for Quantum Complexity and Nanotechnology, Department of Physics, University of Crete, 71003 Heraklion, Greece
- Department of Physics, School of Science and Technology, Nazarbayev University, 53 Kabanbay Batyr Ave, Astana, Republic of Kazakhstan
- National University of Science and Technology MISiS, Leninsky prosp. 4, Moscow, 119049, Russia
| | - J. Hizanidis
- Crete Center for Quantum Complexity and Nanotechnology, Department of Physics, University of Crete, 71003 Heraklion, Greece
| | - V. Kovanis
- Department of Physics, School of Science and Technology, Nazarbayev University, 53 Kabanbay Batyr Ave, Astana, Republic of Kazakhstan
| | - G. P. Tsironis
- Crete Center for Quantum Complexity and Nanotechnology, Department of Physics, University of Crete, 71003 Heraklion, Greece
- Institute of Electronic Structure and Laser, Foundation for Research and Technology–Hellas, P.O. Box 1527, 71110 Heraklion, Greece
- National University of Science and Technology MISiS, Leninsky prosp. 4, Moscow, 119049, Russia
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29
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Kouvaris NE, Requejo RJ, Hizanidis J, Díaz-Guilera A. Chimera states in a network-organized public goods game with destructive agents. CHAOS (WOODBURY, N.Y.) 2016; 26:123108. [PMID: 28039967 DOI: 10.1063/1.4971974] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We found that a network-organized metapopulation of cooperators, defectors, and destructive agents playing the public goods game with mutations can collectively reach global synchronization or chimera states. Global synchronization is accompanied by a collective periodic burst of cooperation, whereas chimera states reflect the tendency of the networked metapopulation to be fragmented in clusters of synchronous and incoherent bursts of cooperation. Numerical simulations have shown that the system's dynamics switches between these two steady states through a first order transition. Depending on the parameters determining the dynamical and topological properties, chimera states with different numbers of coherent and incoherent clusters are observed. Our results present the first systematic study of chimera states and their characterization in the context of evolutionary game theory. This provides a valuable insight into the details of their occurrence, extending the relevance of such states to natural and social systems.
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Affiliation(s)
- Nikos E Kouvaris
- Departament de Física de la Matèria Condensada, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
| | - Rubén J Requejo
- Departament de Física de la Matèria Condensada, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
| | - Johanne Hizanidis
- Crete Center for Quantum Complexity and Nanotechnology, Physics Department, University of Crete, 71003 Heraklion, Greece
| | - Albert Díaz-Guilera
- Departament de Física de la Matèria Condensada, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
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30
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Banerjee T, Dutta PS, Zakharova A, Schöll E. Chimera patterns induced by distance-dependent power-law coupling in ecological networks. Phys Rev E 2016; 94:032206. [PMID: 27739698 DOI: 10.1103/physreve.94.032206] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2016] [Indexed: 06/06/2023]
Abstract
This paper reports the occurrence of several chimera patterns and the associated transitions among them in a network of coupled oscillators, which are connected by a long-range interaction that obeys a distance-dependent power law. This type of interaction is common in physics and biology and constitutes a general form of coupling scheme, where by tuning the power-law exponent of the long-range interaction the coupling topology can be varied from local via nonlocal to global coupling. To explore the effect of the power-law coupling on collective dynamics, we consider a network consisting of a realistic ecological model of oscillating populations, namely the Rosenzweig-MacArthur model, and show that the variation of the power-law exponent mediates transitions between spatial synchrony and various chimera patterns. We map the possible spatiotemporal states and their scenarios that arise due to the interplay between the coupling strength and the power-law exponent.
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Affiliation(s)
- Tanmoy Banerjee
- Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan, Burdwan 713 104, West Bengal, India
| | - Partha Sharathi Dutta
- Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140 001, Punjab, India
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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31
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Ulonska S, Omelchenko I, Zakharova A, Schöll E. Chimera states in networks of Van der Pol oscillators with hierarchical connectivities. CHAOS (WOODBURY, N.Y.) 2016; 26:094825. [PMID: 27781460 DOI: 10.1063/1.4962913] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
Chimera states are complex spatio-temporal patterns that consist of coexisting domains of coherent and incoherent dynamics. We analyse chimera states in ring networks of Van der Pol oscillators with hierarchical coupling topology. We investigate the stepwise transition from a nonlocal to a hierarchical topology and propose the network clustering coefficient as a measure to establish a link between the existence of chimera states and the compactness of the initial base pattern of a hierarchical topology; we show that a large clustering coefficient promotes the occurrence of chimeras. Depending on the level of hierarchy and base pattern, we obtain chimera states with different numbers of incoherent domains. We investigate the chimera regimes as a function of coupling strength and nonlinearity parameter of the individual oscillators. The analysis of a network with larger base pattern resulting in larger clustering coefficient reveals two different types of chimera states and highlights the increasing role of amplitude dynamics.
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Affiliation(s)
- Stefan Ulonska
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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32
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Martens EA, Bick C, Panaggio MJ. Chimera states in two populations with heterogeneous phase-lag. CHAOS (WOODBURY, N.Y.) 2016; 26:094819. [PMID: 27781471 DOI: 10.1063/1.4958930] [Citation(s) in RCA: 30] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/20/2023]
Abstract
The simplest network of coupled phase-oscillators exhibiting chimera states is given by two populations with disparate intra- and inter-population coupling strengths. We explore the effects of heterogeneous coupling phase-lags between the two populations. Such heterogeneity arises naturally in various settings, for example, as an approximation to transmission delays, excitatory-inhibitory interactions, or as amplitude and phase responses of oscillators with electrical or mechanical coupling. We find that breaking the phase-lag symmetry results in a variety of states with uniform and non-uniform synchronization, including in-phase and anti-phase synchrony, full incoherence (splay state), chimera states with phase separation of 0 or π between populations, and states where both populations remain desynchronized. These desynchronized states exhibit stable, oscillatory, and even chaotic dynamics. Moreover, we identify the bifurcations through which chimeras emerge. Stable chimera states and desynchronized solutions, which do not arise for homogeneous phase-lag parameters, emerge as a result of competition between synchronized in-phase, anti-phase equilibria, and fully incoherent states when the phase-lags are near ±π2 (cosine coupling). These findings elucidate previous experimental results involving a network of mechanical oscillators and provide further insight into the breakdown of synchrony in biological systems.
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Affiliation(s)
- Erik A Martens
- Department of Biomedical Sciences, University of Copenhagen, Blegdamsvej 3, 2200 Copenhagen, Denmark
| | - Christian Bick
- Department of Mathematics, University of Exeter, Exeter, United Kingdom
| | - Mark J Panaggio
- Mathematics Department, Rose-Hulman Institute of Technology, Terre Haute, Indiana 47803, USA
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Hizanidis J, Lazarides N, Tsironis GP. Robust chimera states in SQUID metamaterials with local interactions. Phys Rev E 2016; 94:032219. [PMID: 27739822 DOI: 10.1103/physreve.94.032219] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/11/2016] [Indexed: 06/06/2023]
Abstract
We report on the emergence of robust multiclustered chimera states in a dissipative-driven system of symmetrically and locally coupled identical superconducting quantum interference device (SQUID) oscillators. The "snakelike" resonance curve of the single SQUID is the key to the formation of the chimera states and is responsible for the extreme multistability exhibited by the coupled system that leads to attractor crowding at the geometrical resonance (inductive-capacitive) frequency. Until now, chimera states were mostly believed to exist for nonlocal coupling. Our findings provide theoretical evidence that nearest-neighbor interactions are indeed capable of supporting such states in a wide parameter range. SQUID metamaterials are the subject of intense experimental investigations, and we are highly confident that the complex dynamics demonstrated in this paper can be confirmed in the laboratory.
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Affiliation(s)
- J Hizanidis
- Department of Physics, Crete Center for Quantum Complexity and Nanotechnology, University of Crete, P.O. Box 2208, 71003 Heraklion, Greece; Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, P.O. Box 1527, 71110 Heraklion, Greece; and National University of Science and Technology MISiS, Leninsky Prospekt 4, Moscow 119049, Russia
| | - N Lazarides
- Department of Physics, Crete Center for Quantum Complexity and Nanotechnology, University of Crete, P.O. Box 2208, 71003 Heraklion, Greece; Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, P.O. Box 1527, 71110 Heraklion, Greece; and National University of Science and Technology MISiS, Leninsky Prospekt 4, Moscow 119049, Russia
| | - G P Tsironis
- Department of Physics, Crete Center for Quantum Complexity and Nanotechnology, University of Crete, P.O. Box 2208, 71003 Heraklion, Greece; Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, P.O. Box 1527, 71110 Heraklion, Greece; and National University of Science and Technology MISiS, Leninsky Prospekt 4, Moscow 119049, Russia
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Abstract
The position of the coherent and incoherent domain of a chimera state in a ring of nonlocally coupled oscillators is strongly influenced by the initial conditions, making nontrivial the problem of confining them in a specific region of the structure. In this paper we propose the use of spatial pinning to induce a chimera state where the nodes belonging to one domain, either the coherent or the incoherent, are fixed by the control action. We design two different techniques according to the dynamics to be forced in the region of pinned nodes, and validate them on FitzHugh-Nagumo and Kuramoto oscillators. Furthermore, we introduce a suitable strategy to deal with the effects of finite size in small structures.
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Affiliation(s)
| | - Mattia Frasca
- DIEEI, Università degli Studi di Catania, Catania, Italy
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Andrzejak RG, Rummel C, Mormann F, Schindler K. All together now: Analogies between chimera state collapses and epileptic seizures. Sci Rep 2016; 6:23000. [PMID: 26957324 PMCID: PMC4783711 DOI: 10.1038/srep23000] [Citation(s) in RCA: 112] [Impact Index Per Article: 14.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/25/2015] [Accepted: 02/26/2016] [Indexed: 01/22/2023] Open
Abstract
Conceptually and structurally simple mathematical models of coupled oscillator networks can show a rich variety of complex dynamics, providing fundamental insights into many real-world phenomena. A recent and not yet fully understood example is the collapse of coexisting synchronous and asynchronous oscillations into a globally synchronous motion found in networks of identical oscillators. Here we show that this sudden collapse is promoted by a further decrease of synchronization, rather than by critically high synchronization. This strikingly counterintuitive mechanism can be found also in nature, as we demonstrate on epileptic seizures in humans. Analyzing spatiotemporal correlation profiles derived from intracranial electroencephalographic recordings (EEG) of seizures in epilepsy patients, we found a pronounced decrease of correlation at the seizure onsets. Applying our findings in a closed-loop control scheme to models of coupled oscillators in chimera states, we succeed in both provoking and preventing outbreaks of global synchronization. Our findings not only advance the understanding of networks of coupled dynamics but can open new ways to control them, thus offering a vast range of potential new applications.
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Affiliation(s)
- Ralph G. Andrzejak
- Universitat Pompeu Fabra, Department of Information and Communication Technologies, Barcelona, Spain
| | - Christian Rummel
- Support Center for Advanced Neuroimaging, University Institute for Diagnostic and Interventional Neuroradiology, Inselspital, Bern University Hospital, University of Bern, Bern, Switzerland
| | - Florian Mormann
- Department of Epileptology, University of Bonn, Bonn, Germany
| | - Kaspar Schindler
- Schlaf-Wach-Epilepsie-Zentrum (SWEZ), Department of Neurology, Inselspital, Bern University Hospital, University of Bern, Bern, Switzerland
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