1
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Nicolaou ZG, Bramburger JJ. Complex localization mechanisms in networks of coupled oscillators: Two case studies. CHAOS (WOODBURY, N.Y.) 2024; 34:013131. [PMID: 38252783 DOI: 10.1063/5.0174550] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/31/2023] [Accepted: 12/14/2023] [Indexed: 01/24/2024]
Abstract
Localized phenomena abound in nature and throughout the physical sciences. Some universal mechanisms for localization have been characterized, such as in the snaking bifurcations of localized steady states in pattern-forming partial differential equations. While much of this understanding has been targeted at steady states, recent studies have noted complex dynamical localization phenomena in systems of coupled oscillators. These localized states can come in the form of symmetry-breaking chimera patterns that exhibit coexistence of coherence and incoherence in symmetric networks of coupled oscillators and gap solitons emerging in the bandgap of parametrically driven networks of oscillators. Here, we report detailed numerical continuations of localized time-periodic states in systems of coupled oscillators, while also documenting the numerous bifurcations they give way to. We find novel routes to localization involving bifurcations of heteroclinic cycles in networks of Janus oscillators and strange bifurcation diagrams resembling chaotic tangles in a parametrically driven array of coupled pendula. We highlight the important role of discrete symmetries and the symmetric branch points that emerge in symmetric models.
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Affiliation(s)
- Zachary G Nicolaou
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-3925, USA
| | - Jason J Bramburger
- Department of Mathematics and Statistics, Concordia University, Montréal, Quebec H3G 1M8, Canada
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2
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Zhu J, Yamakou ME. Self-induced-stochastic-resonance breathing chimeras. Phys Rev E 2023; 108:L022204. [PMID: 37723731 DOI: 10.1103/physreve.108.l022204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2023] [Accepted: 07/22/2023] [Indexed: 09/20/2023]
Abstract
The study by Semenova et al. [Phys. Rev. Lett. 117, 014102 (2016)0031-900710.1103/PhysRevLett.117.014102] discovered a type of chimera state known as coherence-resonance chimera (CRC), which combines the effects of coherence resonance (CR) and the spatial property of classical chimeras. In this Letter, we present yet another form of chimera, which we refer to as self-induced-stochastic-resonance breathing chimera (SISR-BC), which differs fundamentally from the CRC in that it combines the mechanism and effects of self-induced stochastic resonance (SISR, previously shown by DeVille et al. [Phys. Rev. E 72, 031105 (2005)1539-375510.1103/PhysRevE.72.031105] to be intrinsically different from CR), the symmetry breaking in the rotational coupling between the slow and fast subsystems of the coupled oscillators, and the property of breathing chimera-a form of chimera state characterized by nonstationary periodic dynamics of coherent-incoherent patterns with a periodically oscillating global order parameter. Unlike other types of chimeras, including CRC, SISR-BC demonstrates remarkable resilience to a relatively wide range of stochastic perturbations and persists even when the purely excitable system is significantly distant from the Hopf bifurcation threshold-thanks to the mechanism of SISR-and globally attracts random distributions of initial conditions. Considering its potential impact on information processing in neuronal networks, SISR-BC could have special significance and applications.
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Affiliation(s)
- Jinjie Zhu
- State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
| | - Marius E Yamakou
- Department of Data Science, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany
- Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22, 04103 Leipzig, Germany
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3
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Kong LW, Lai YC. Short-lived chimera states. CHAOS (WOODBURY, N.Y.) 2023; 33:2894496. [PMID: 37276573 DOI: 10.1063/5.0145573] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/06/2023] [Accepted: 04/21/2023] [Indexed: 06/07/2023]
Abstract
In the classic Kuramoto system of coupled two-dimensional rotators, chimera states characterized by the coexistence of synchronous and asynchronous groups of oscillators are long-lived because the average lifetime of these states increases exponentially with the system size. Recently, it was discovered that, when the rotators in the Kuramoto model are three-dimensional, the chimera states become short-lived in the sense that their lifetime scales with only the logarithm of the dimension-augmenting perturbation. We introduce transverse-stability analysis to understand the short-lived chimera states. In particular, on the unit sphere representing three-dimensional (3D) rotations, the long-lived chimera states in the classic Kuramoto system occur on the equator, to which latitudinal perturbations that make the rotations 3D are transverse. We demonstrate that the largest transverse Lyapunov exponent calculated with respect to these long-lived chimera states is typically positive, making them short-lived. The transverse-stability analysis turns the previous numerical scaling law of the transient lifetime into an exact formula: the "free" proportional constant in the original scaling law can now be precisely determined in terms of the largest transverse Lyapunov exponent. Our analysis reinforces the speculation that in physical systems, chimera states can be short-lived as they are vulnerable to any perturbations that have a component transverse to the invariant subspace in which they live.
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Affiliation(s)
- Ling-Wei Kong
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
- Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
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4
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Smirnov L, Pikovsky A. Travelling chimeras in oscillator lattices with advective-diffusive coupling. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2023; 381:20220076. [PMID: 36842987 DOI: 10.1098/rsta.2022.0076] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/31/2022] [Accepted: 12/19/2022] [Indexed: 06/18/2023]
Abstract
We consider a one-dimensional array of phase oscillators coupled via an auxiliary complex field. While in the seminal chimera studies by Kumamoto and Battogtokh only diffusion of the field was considered, we include advection which makes the coupling left-right asymmetric. Chimera starts to move and we demonstrate that a weakly turbulent moving pattern appears. It possesses a relatively large synchronous domain where the phases are nearly equal, and a more disordered domain where the local driving field is small. For a dense system with a large number of oscillators, there are strong local correlations in the disordered domain, which at most places looks like a smooth phase profile. We find also exact regular travelling wave chimera-like solutions of different complexity, but only some of them are stable. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.
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Affiliation(s)
- L Smirnov
- Department of Control Theory, Research and Education Mathematical Center 'Mathematics for Future Technologies', Nizhny Novgorod State University, Gagarin Avenue 23, Nizhny Novgorod 603022, Russia
- Institute of Applied Physics of the Russian Academy of Sciences, Ul'yanov Street 46, Nizhny Novgorod 603950, Russia
| | - A Pikovsky
- Institute of Physics and Astronomy, Potsdam University, Potsdam-Golm 14476, Germany
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Ragavan A, Manoranjani M, Senthilkumar DV, Chandrasekar VK. Multistable chimera states in a smallest population of three coupled oscillators. Phys Rev E 2023; 107:044209. [PMID: 37198793 DOI: 10.1103/physreve.107.044209] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2022] [Accepted: 03/28/2023] [Indexed: 05/19/2023]
Abstract
We uncover the emergence of distinct sets of multistable chimera states in addition to chimera death and synchronized states in a smallest population of three globally coupled oscillators with mean-field diffusive coupling. Sequence of torus bifurcations result in the manifestation of distinct periodic orbits as a function of the coupling strength, which in turn result in the genesis of distinct chimera states constituted by two synchronized oscillators coexisting with an asynchronous oscillator. Two subsequent Hopf bifurcations result in homogeneous and inhomogeneous steady states resulting in desynchronized steady states and chimera death state among the coupled oscillators. The periodic orbits and the steady states lose their stability via a sequence of saddle-loop and saddle-node bifurcations finally resulting in a stable synchronized state. We have generalized these results to N coupled oscillators and also deduced the variational equations corresponding to the perturbation transverse to the synchronization manifold and corroborated the synchronized state in the two-parameter phase diagrams using its largest eigenvalue. Chimera states in three coupled oscillators emerge as a solitary state in N coupled oscillator ensemble.
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Affiliation(s)
- A Ragavan
- Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur-613 401, Tamil Nadu, India
| | - M Manoranjani
- 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
| | - 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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6
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Rybalova E, Muni S, Strelkova G. Transition from chimera/solitary states to traveling waves. CHAOS (WOODBURY, N.Y.) 2023; 33:033104. [PMID: 37003811 DOI: 10.1063/5.0138207] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/09/2022] [Accepted: 02/08/2023] [Indexed: 06/19/2023]
Abstract
We study numerically the spatiotemporal dynamics in a ring network of nonlocally coupled nonlinear oscillators, each represented by a two-dimensional discrete-time model of the classical van der Pol oscillator. It is shown that the discretized oscillator exhibits richer behavior, combining the peculiarities of both the original system and its own dynamics. Moreover, a large variety of spatiotemporal structures is observed in the network of discrete van der Pol oscillators when the discretization parameter and the coupling strength are varied. Regimes, such as the coexistence of a multichimera state/a traveling wave and a solitary state are revealed for the first time and are studied in detail. It is established that the majority of the observed chimera/solitary states, including the newly found ones, are transient toward a purely traveling wave mode. The peculiarities of the transition process and the lifetime (transient duration) of the chimera structures and the solitary state are analyzed depending on the system parameters, the observation time, initial conditions, and the influence of external noise.
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Affiliation(s)
- E Rybalova
- Institute of Physics, Saratov State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
| | - S Muni
- Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Campus Road, Mohanpur, West Bengal 741246, India
| | - G Strelkova
- Institute of Physics, Saratov State University, 83 Astrakhanskaya Street, Saratov 410012, Russia
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7
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Laing CR. Chimeras on a ring of oscillator populations. CHAOS (WOODBURY, N.Y.) 2023; 33:013121. [PMID: 36725662 DOI: 10.1063/5.0127306] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/20/2022] [Accepted: 12/16/2022] [Indexed: 06/18/2023]
Abstract
Chimeras occur in networks of coupled oscillators and are characterized by coexisting groups of synchronous oscillators and asynchronous oscillators. We consider a network formed from N equal-sized populations at equally spaced points around a ring. We use the Ott/Antonsen ansatz to derive coupled ordinary differential equations governing the level of synchrony within each population and describe chimeras using a self-consistency argument. For N=2 and 3, our results are compared with previously known ones. We obtain new results for the cases of 4,5,…,12 populations and a numerically based conjecture resulting from the behavior of larger numbers of populations. We find macroscopic chaos when more than five populations are considered, but conjecture that this behavior vanishes as the number of populations is increased.
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Affiliation(s)
- Carlo R Laing
- School of Mathematical and Computational Sciences, Massey University, Private Bag 102-904, North Shore Mail Centre, Auckland, New Zealand
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8
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Lee S, Krischer K. Nontrivial twisted states in nonlocally coupled Stuart-Landau oscillators. Phys Rev E 2022; 106:044210. [PMID: 36397498 DOI: 10.1103/physreve.106.044210] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2022] [Accepted: 09/22/2022] [Indexed: 06/16/2023]
Abstract
A twisted state is an important yet simple form of collective dynamics in an oscillatory medium. Here we describe a nontrivial type of twisted state in a system of nonlocally coupled Stuart-Landau oscillators. The nontrivial twisted state (NTS) is a coherent traveling wave characterized by inhomogeneous profiles of amplitudes and phase gradients, which can be assigned a winding number. To further investigate its properties, several methods are employed. We perform a linear stability analysis in the continuum limit and compare the results with Lyapunov exponents obtained in a finite-size system. The determination of covariant Lyapunov vectors allows us to identify collective modes. Furthermore, we show that the NTS is robust to small heterogeneities in the natural frequencies and present a bifurcation analysis revealing that NTSs are born or annihilated in a saddle-node bifurcation and change their stability in Hopf bifurcations. We observe stable NTSs with winding number 1 and 2. The latter can lose stability in a supercritical Hopf bifurcation, leading to a modulated 2-NTS.
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Affiliation(s)
- Seungjae Lee
- Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
| | - Katharina Krischer
- Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
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9
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Bi H, Fukai T. Amplitude-mediated chimera states in nonlocally coupled Stuart-Landau oscillators. CHAOS (WOODBURY, N.Y.) 2022; 32:083125. [PMID: 36049944 DOI: 10.1063/5.0096284] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2022] [Accepted: 07/20/2022] [Indexed: 06/15/2023]
Abstract
Chimera states achieve the coexistence of coherent and incoherent subgroups through symmetry breaking and emerge in physical, chemical, and biological systems. We show the presence of amplitude-mediated multicluster chimera states in nonlocally coupled Stuart-Landau oscillators. We clarify the prerequisites for having different types of chimera states by analytically and numerically studying how phase transitions occur between these states. Our results demonstrate how the oscillation amplitudes interact with the phase degrees of freedom in chimera states and significantly advance our understanding of the generation mechanisms of such states in coupled oscillator systems.
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Affiliation(s)
- Hongjie Bi
- Okinawa Institute of Science and Technology, Onna-son, Okinawa 904-0495, Japan
| | - Tomoki Fukai
- Okinawa Institute of Science and Technology, Onna-son, Okinawa 904-0495, Japan
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10
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Botha AE, Ansariara M, Emadi S, Kolahchi MR. Chimera Patterns of Synchrony in a Frustrated Array of Hebb Synapses. Front Comput Neurosci 2022; 16:888019. [PMID: 35814347 PMCID: PMC9260432 DOI: 10.3389/fncom.2022.888019] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2022] [Accepted: 05/30/2022] [Indexed: 11/13/2022] Open
Abstract
The union of the Kuramoto–Sakaguchi model and the Hebb dynamics reproduces the Lisman switch through a bistability in synchronized states. Here, we show that, within certain ranges of the frustration parameter, the chimera pattern can emerge, causing a different, time-evolving, distribution in the Hebbian synaptic strengths. We study the stability range of the chimera as a function of the frustration (phase-lag) parameter. Depending on the range of the frustration, two different types of chimeras can appear spontaneously, i.e., from randomized initial conditions. In the first type, the oscillators in the coherent region rotate, on average, slower than those in the incoherent region; while in the second type, the average rotational frequencies of the two regions are reversed, i.e., the coherent region runs, on average, faster than the incoherent region. We also show that non-stationary behavior at finite N can be controlled by adjusting the natural frequency of a single pacemaker oscillator. By slowly cycling the frequency of the pacemaker, we observe hysteresis in the system. Finally, we discuss how we can have a model for learning and memory.
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Affiliation(s)
- A. E. Botha
- Department of Physics, Science Campus, University of South Africa, Private Bag X6, Johannesburg, South Africa
| | - M. Ansariara
- Department of Physics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
| | - S. Emadi
- Department of Biological Sciences, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
| | - M. R. Kolahchi
- Department of Physics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
- *Correspondence: M. R. Kolahchi
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11
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Majhi S, Rakshit S, Ghosh D. Oscillation suppression and chimera states in time-varying networks. CHAOS (WOODBURY, N.Y.) 2022; 32:042101. [PMID: 35489845 DOI: 10.1063/5.0087291] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2022] [Accepted: 03/11/2022] [Indexed: 06/14/2023]
Abstract
Complex network theory has offered a powerful platform for the study of several natural dynamic scenarios, based on the synergy between the interaction topology and the dynamics of its constituents. With research in network theory being developed so fast, it has become extremely necessary to move from simple network topologies to more sophisticated and realistic descriptions of the connectivity patterns. In this context, there is a significant amount of recent works that have emerged with enormous evidence establishing the time-varying nature of the connections among the constituents in a large number of physical, biological, and social systems. The recent review article by Ghosh et al. [Phys. Rep. 949, 1-63 (2022)] demonstrates the significance of the analysis of collective dynamics arising in temporal networks. Specifically, the authors put forward a detailed excerpt of results on the origin and stability of synchronization in time-varying networked systems. However, among the complex collective dynamical behaviors, the study of the phenomenon of oscillation suppression and that of other diverse aspects of synchronization are also considered to be central to our perception of the dynamical processes over networks. Through this review, we discuss the principal findings from the research studies dedicated to the exploration of the two collective states, namely, oscillation suppression and chimera on top of time-varying networks of both static and mobile nodes. We delineate how temporality in interactions can suppress oscillation and induce chimeric patterns in networked dynamical systems, from effective analytical approaches to computational aspects, which is described while addressing these two phenomena. We further sketch promising directions for future research on these emerging collective behaviors in time-varying networks.
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Affiliation(s)
- Soumen Majhi
- Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel
| | - Sarbendu Rakshit
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
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12
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Jaros P, Levchenko R, Kapitaniak T, Maistrenko Y. Chimera states for directed networks. CHAOS (WOODBURY, N.Y.) 2021; 31:103111. [PMID: 34717326 DOI: 10.1063/5.0059765] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2021] [Accepted: 09/28/2021] [Indexed: 06/13/2023]
Abstract
We demonstrate that chimera behavior can be observed in ensembles of phase oscillators with unidirectional coupling. For a small network consisting of only three identical oscillators (cyclic triple), tiny chimera islands arise in the parameter space. They are surrounded by developed chaotic switching behavior caused by a collision of rotating waves propagating in opposite directions. For larger networks, as we show for a hundred oscillators (cyclic century), the islands merge into a single chimera continent, which incorporates the world of chimeras of different configurations. The phenomenon inherits from networks with intermediate ranges of the unidirectional coupling and it diminishes as the coupling range decreases.
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Affiliation(s)
- Patrycja Jaros
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | | | - Tomasz Kapitaniak
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Yuri Maistrenko
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
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13
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Bataille-Gonzalez M, Clerc MG, Omel'chenko OE. Moving spiral wave chimeras. Phys Rev E 2021; 104:L022203. [PMID: 34525661 DOI: 10.1103/physreve.104.l022203] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/29/2021] [Accepted: 08/04/2021] [Indexed: 01/20/2023]
Abstract
We consider a two-dimensional array of heterogeneous nonlocally coupled phase oscillators on a flat torus and study the bound states of two counter-rotating spiral chimeras, shortly two-core spiral chimeras, observed in this system. In contrast to other known spiral chimeras with motionless incoherent cores, the two-core spiral chimeras typically show a drift motion. Due to this drift, their incoherent cores become spatially modulated and develop specific fingerprint patterns of varying synchrony levels. In the continuum limit of infinitely many oscillators, the two-core spiral chimeras can be studied using the Ott-Antonsen equation. Numerical analysis of this equation allows us to reveal the stability region of different spiral chimeras, which we group into three main classes-symmetric, asymmetric, and meandering spiral chimeras.
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Affiliation(s)
- Martin Bataille-Gonzalez
- Departamento de Física and Millenium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - Marcel G Clerc
- Departamento de Física and Millenium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - Oleh E Omel'chenko
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam, Germany
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14
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Simo GR, Njougouo T, Aristides RP, Louodop P, Tchitnga R, Cerdeira HA. Chimera states in a neuronal network under the action of an electric field. Phys Rev E 2021; 103:062304. [PMID: 34271625 DOI: 10.1103/physreve.103.062304] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2020] [Accepted: 05/25/2021] [Indexed: 11/07/2022]
Abstract
The phenomenon of the chimera state symbolizes the coexistence of coherent and incoherent sections of a given population. This phenomenon identified in several physical and biological systems presents several variants, including the multichimera states and the traveling chimera state. Here, we numerically study the influence of a weak external electric field on the dynamics of a network of Hindmarsh-Rose (HR) neurons coupled locally by an electrical interaction and nonlocally by a chemical one. We first focus on the phenomena of traveling chimera states and multicluster oscillating breathers that appear in the electric field's absence. Then in the field's presence, we highlight the presence of a chimera state, a multichimera state, an alternating chimera state, and a multicluster traveling chimera.
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Affiliation(s)
- Gaël R Simo
- Research Unit Condensed Matter, Electronics and Signal Processing, Department of Physics, Faculty of Sciences, University of Dschang, P.O. Box 67, Dschang, Cameroon
| | - Thierry Njougouo
- Research Unit Condensed Matter, Electronics and Signal Processing, Department of Physics, Faculty of Sciences, University of Dschang, P.O. Box 67, Dschang, Cameroon
| | - R P Aristides
- São Paulo State University (UNESP), Instituto de Física Teórica, Rua Doutor Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, 01140-070 São Paulo, Brazil
| | - Patrick Louodop
- Research Unit Condensed Matter, Electronics and Signal Processing, Department of Physics, Faculty of Sciences, University of Dschang, P.O. Box 67, Dschang, Cameroon
| | - Robert Tchitnga
- Research Unit Condensed Matter, Electronics and Signal Processing, Department of Physics, Faculty of Sciences, University of Dschang, P.O. Box 67, Dschang, Cameroon.,Institute of Surface Chemistry and Catalysis, University of Ulm, Albert-Einstein-Allee 47, 89081 Ulm, Germany
| | - Hilda A Cerdeira
- São Paulo State University (UNESP), Instituto de Física Teórica, Rua Doutor Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, 01140-070 São Paulo, Brazil.,Epistemic, Gomez & Gomez Ltda. ME, Avenida Professor Lineu Prestes 2242, Cietec, Sala 244, 05508-000 São Paulo, Brazil
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15
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Dudkowski D, Czołczyński K, Kapitaniak T. Multi-headed loop chimera states in coupled oscillators. CHAOS (WOODBURY, N.Y.) 2021; 31:013135. [PMID: 33754776 DOI: 10.1063/5.0033519] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2020] [Accepted: 01/05/2021] [Indexed: 06/12/2023]
Abstract
In this paper, we introduce a novel type of chimera state, characterized by the geometrical distortion of the coherent ring topology of coupled oscillators. The multi-headed loop chimeras are examined for a simple network of locally coupled pendulum clocks, suspended on the vertical platform. We determine the regions of the occurrence of the observed patterns, their structure, and possible co-existence. The representative examples of behaviors are shown, exhibiting the variety of configurations that can be observed. The statistical analysis of the solutions indicates the geometrical regions of the system with the highest probability of the chimeras' occurrence. We investigate the mechanism of the creation of the observed states, showing that the manipulation of the initial positions of chosen pendula may induce the desired patterns. Apart from the study of the isolated network, we also discuss the scenario of the movable platform, showing a possible influence of the global coupling structure on the stability of the observed states. The stability of loop chimeras is examined for varying both the amplitude and the frequency of the oscillations of the platform. We indicate the excitation parameters for which the solutions can survive as well as be destroyed. The bifurcation analysis included in the paper allows us to discuss the transitions between possible behaviors. The appearance of multi-headed loop chimeras is generalized into large networks of oscillators, showing the universal character of the observed patterns. One should expect to observe similar results also in other types of coupled oscillators, especially the mechanical ones.
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Affiliation(s)
- Dawid Dudkowski
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Krzysztof Czołczyński
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Tomasz Kapitaniak
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
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16
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Bolotov MI, Smirnov LA, Osipov GV, Pikovsky A. Locking and regularization of chimeras by periodic forcing. Phys Rev E 2020; 102:042218. [PMID: 33212667 DOI: 10.1103/physreve.102.042218] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2019] [Accepted: 09/28/2020] [Indexed: 11/07/2022]
Abstract
We study how a chimera state in a one-dimensional medium of nonlocally coupled oscillators responds to a homogeneous in space periodic in time external force. On a macroscopic level, where a chimera can be considered as an oscillating object, forcing leads to entrainment of the chimera's basic frequency inside an Arnold tongue. On a mesoscopic level, where a chimera can be viewed as an inhomogeneous, stationary, or nonstationary pattern, strong forcing can lead to regularization of an unstationary chimera. On a microscopic level of the dynamics of individual oscillators, forcing outside of the Arnold tongue leads to a multiplateau state with nontrivial locking properties.
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Affiliation(s)
- Maxim I Bolotov
- Department of Control Theory, Research and Education Mathematical Center "Mathematics for Future Technologies," Nizhny Novgorod State University, Gagarin Av. 23, 603950, Nizhny Novgorod, Russia
| | - Lev A Smirnov
- Department of Control Theory, Research and Education Mathematical Center "Mathematics for Future Technologies," Nizhny Novgorod State University, Gagarin Av. 23, 603950, Nizhny Novgorod, Russia.,Institute of Applied Physics of the Russian Academy of Sciences, Ul'yanov Str. 46, 603950, Nizhny Novgorod, Russia
| | - Grigory V Osipov
- Department of Control Theory, Research and Education Mathematical Center "Mathematics for Future Technologies," Nizhny Novgorod State University, Gagarin Av. 23, 603950, Nizhny Novgorod, Russia
| | - Arkady Pikovsky
- Department of Control Theory, Research and Education Mathematical Center "Mathematics for Future Technologies," Nizhny Novgorod State University, Gagarin Av. 23, 603950, Nizhny Novgorod, Russia.,Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24-25, 14476 Potsdam, Germany.,National Research University Higher School of Economics, 25/12 Bolshaya Pecherskaya Ulitsa, 603155 Nizhny Novgorod, Russia
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Singha J, Gupte N. Chimera states in coupled map lattices: Spatiotemporally intermittent behavior and an equivalent cellular automaton. CHAOS (WOODBURY, N.Y.) 2020; 30:113102. [PMID: 33261350 DOI: 10.1063/5.0016056] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/10/2020] [Accepted: 10/10/2020] [Indexed: 06/12/2023]
Abstract
We construct an equivalent cellular automaton (CA) for a system of globally coupled sine circle maps with two populations and distinct values for intergroup and intragroup coupling. The phase diagram of the system shows that the coupled map lattice can exhibit chimera states with synchronized and spatiotemporally intermittent subgroups after evolution from random initial conditions in some parameter regimes, as well as to other kinds of solutions in other parameter regimes. The CA constructed by us reflects the global nature and the two population structure of the coupled map lattice and is able to reproduce the phase diagram accurately. The CA depends only on the total number of laminar and burst sites and shows a transition from co-existing deterministic and probabilistic behavior in the chimera region to fully probabilistic behavior at the phase boundaries. This identifies the characteristic signature of the transition of a cellular automaton to a chimera state. We also construct an evolution equation for the average number of laminar/burst sites from the CA, analyze its behavior and solutions, and correlate these with the behavior seen for the coupled map lattice. Our CA and methods of analysis can have relevance in wider contexts.
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Affiliation(s)
- Joydeep Singha
- Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
| | - Neelima Gupte
- Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
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18
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Cisternas J, Cartes C, Descalzi O, Albers T, Radons G. Random walks of trains of dissipative solitons. CHAOS (WOODBURY, N.Y.) 2020; 30:073134. [PMID: 32752625 DOI: 10.1063/5.0006091] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2020] [Accepted: 07/04/2020] [Indexed: 06/11/2023]
Abstract
The propagation of light pulses in dual-core nonlinear optical fibers is studied using a model proposed by Sakaguchi and Malomed. The system consists of a supercritical complex Ginzburg-Landau equation coupled to a linear equation. Our analysis includes single standing and walking solitons as well as walking trains of 3, 5, 6, and 12 solitons. For the characterization of the different scenarios, we used ensemble-averaged square displacement of the soliton trajectories and time-averaged power spectrum of the background waves. Power law spectra, indicative of turbulence, were found to be associated with random walks. The number of solitons (or their separations) can trigger anomalous random walks or totally suppress the background waves.
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Affiliation(s)
- Jaime Cisternas
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Monseñor Alvaro del Portillo 12455, Las Condes, Santiago, Chile
| | - Carlos Cartes
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Monseñor Alvaro del Portillo 12455, Las Condes, Santiago, Chile
| | - Orazio Descalzi
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Monseñor Alvaro del Portillo 12455, Las Condes, Santiago, Chile
| | - Tony Albers
- Institute of Physics, Technische Universität Chemnitz, 09107 Chemnitz, Germany
| | - Günter Radons
- Institute of Physics, Technische Universität Chemnitz, 09107 Chemnitz, Germany
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Ganaie MA, Ghosh S, Mendola N, Tanveer M, Jalan S. Identification of chimera using machine learning. CHAOS (WOODBURY, N.Y.) 2020; 30:063128. [PMID: 32611090 DOI: 10.1063/1.5143285] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2019] [Accepted: 05/07/2020] [Indexed: 06/11/2023]
Abstract
Chimera state refers to the coexistence of coherent and non-coherent phases in identically coupled dynamical units found in various complex dynamical systems. Identification of chimera, on one hand, is essential due to its applicability in various areas including neuroscience and, on the other hand, is challenging due to its widely varied appearance in different systems and the peculiar nature of its profile. Therefore, a simple yet universal method for its identification remains an open problem. Here, we present a very distinctive approach using machine learning techniques to characterize different dynamical phases and identify the chimera state from given spatial profiles generated using various different models. The experimental results show that the performance of the classification algorithms varies for different dynamical models. The machine learning algorithms, namely, random forest, oblique random forest based on Tikhonov, axis-parallel split, and null space regularization achieved more than 96% accuracy for the Kuramoto model. For the logistic maps, random forest and Tikhonov regularization based oblique random forest showed more than 90% accuracy, and for the Hénon map model, random forest, null space, and axis-parallel split regularization based oblique random forest achieved more than 80% accuracy. The oblique random forest with null space regularization achieved consistent performance (more than 83% accuracy) across different dynamical models while the auto-encoder based random vector functional link neural network showed relatively lower performance. This work provides a direction for employing machine learning techniques to identify dynamical patterns arising in coupled non-linear units on large-scale and for characterizing complex spatiotemporal patterns in real-world systems for various applications.
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Affiliation(s)
- M A Ganaie
- Discipline of Mathematics, Indian Institute of Technology Indore, Khandwa Road, Simrol, 453552 Indore, India
| | - Saptarshi Ghosh
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, 453552 Indore, India
| | - Naveen Mendola
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, 453552 Indore, India
| | - M Tanveer
- Discipline of Mathematics, Indian Institute of Technology Indore, Khandwa Road, Simrol, 453552 Indore, India
| | - Sarika Jalan
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, 453552 Indore, India
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20
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Suda Y, Okuda K. Emergence of second coherent regions for breathing chimera states. Phys Rev E 2020; 101:062203. [PMID: 32688598 DOI: 10.1103/physreve.101.062203] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2019] [Accepted: 04/30/2020] [Indexed: 06/11/2023]
Abstract
Chimera states in one-dimensional nonlocally coupled phase oscillators are mostly assumed to be stationary, but breathing chimeras can occasionally appear, branching from the stationary chimeras via Hopf bifurcation. In this paper, we demonstrate two types of breathing chimeras: The type I breathing chimera looks the same as the stationary chimera at a glance, while the type II consists of multiple coherent regions with different average frequencies. Moreover, it is shown that the type I changes to the type II by increasing the breathing amplitude. Furthermore, we develop a self-consistent analysis of the local order parameter, which can be applied to breathing chimeras, and numerically demonstrate this analysis in the present system.
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Affiliation(s)
- Yusuke Suda
- Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
- Institute for the Advancement of Higher Education, Hokkaido University, Sapporo 060-0817, Japan
| | - Koji Okuda
- Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
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Blondeau Soh G, Louodop P, Kengne R, Tchitnga R. Chimera dynamics in an array of coupled FitzHugh-Nagumo system with shift of close neighbors. Heliyon 2020; 6:e03739. [PMID: 32280805 PMCID: PMC7139117 DOI: 10.1016/j.heliyon.2020.e03739] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2019] [Revised: 02/20/2020] [Accepted: 03/31/2020] [Indexed: 11/30/2022] Open
Abstract
In this paper, we consider an array of FitzHugh-Nagumo (FHN) systems with R close neighbors. Each element (j) connects to another (m) and its 2R neighbors. Shifting these neighbors produces particular phenomena such as chimera and multi-chimera. Step traveling chimera is observed for a time dependent shift. Results show that, basing oneself on both shift parameter m and close neighbors R, a full control on the chimera dynamics of the network can be ensured.
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Affiliation(s)
- Guy Blondeau Soh
- Laboratory of Electronics, Automation and Signal Processing, Faculty of Science, Department of Physics, University of Dschang, P.O. Box 67 Dschang, Cameroon
| | - Patrick Louodop
- Laboratory of Electronics, Automation and Signal Processing, Faculty of Science, Department of Physics, University of Dschang, P.O. Box 67 Dschang, Cameroon
| | - Romanic Kengne
- Laboratory of Electronics, Automation and Signal Processing, Faculty of Science, Department of Physics, University of Dschang, P.O. Box 67 Dschang, Cameroon
| | - Robert Tchitnga
- Laboratory of Electronics, Automation and Signal Processing, Faculty of Science, Department of Physics, University of Dschang, P.O. Box 67 Dschang, Cameroon
- Institute of Surface Chemistry and Catalysis, University of Ulm, Albert-Einstein-Allee 47, 89081 Ulm, Germany
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22
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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: 9] [Impact Index Per Article: 1.8] [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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Bansal K, Garcia JO, Tompson SH, Verstynen T, Vettel JM, Muldoon SF. Cognitive chimera states in human brain networks. SCIENCE ADVANCES 2019; 5:eaau8535. [PMID: 30949576 PMCID: PMC6447382 DOI: 10.1126/sciadv.aau8535] [Citation(s) in RCA: 52] [Impact Index Per Article: 10.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2018] [Accepted: 02/13/2019] [Indexed: 05/10/2023]
Abstract
The human brain is a complex dynamical system, and how cognition emerges from spatiotemporal patterns of regional brain activity remains an open question. As different regions dynamically interact to perform cognitive tasks, variable patterns of partial synchrony can be observed, forming chimera states. We propose that the spatial patterning of these states plays a fundamental role in the cognitive organization of the brain and present a cognitively informed, chimera-based framework to explore how large-scale brain architecture affects brain dynamics and function. Using personalized brain network models, we systematically study how regional brain stimulation produces different patterns of synchronization across predefined cognitive systems. We analyze these emergent patterns within our framework to understand the impact of subject-specific and region-specific structural variability on brain dynamics. Our results suggest a classification of cognitive systems into four groups with differing levels of subject and regional variability that reflect their different functional roles.
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Affiliation(s)
- Kanika Bansal
- Human Research and Engineering Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA
- Department of Biomedical Engineering, Columbia University, New York, NY 10027, USA
- Mathematics Department, University at Buffalo, SUNY, Buffalo, NY 14260, USA
| | - Javier O. Garcia
- Human Research and Engineering Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA
- Department of Biomedical Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA
| | - Steven H. Tompson
- Human Research and Engineering Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA
- Department of Biomedical Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA
| | - Timothy Verstynen
- Department of Psychology, Carnegie Mellon University, Pittsburgh, PA 15213, USA
| | - Jean M. Vettel
- Human Research and Engineering Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA
- Department of Biomedical Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA
- Department of Psychological and Brain Sciences, University of California, Santa Barbara, Santa Barbara, CA 93106, USA
| | - Sarah F. Muldoon
- Mathematics Department, University at Buffalo, SUNY, Buffalo, NY 14260, USA
- CDSE Program and Neuroscience Program, University at Buffalo, SUNY, Buffalo, NY 14260, USA
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24
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Kundu S, Bera BK, Ghosh D, Lakshmanan M. Chimera patterns in three-dimensional locally coupled systems. Phys Rev E 2019; 99:022204. [PMID: 30934225 DOI: 10.1103/physreve.99.022204] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/05/2018] [Indexed: 06/09/2023]
Abstract
The coexistence of coherent and incoherent domains, namely the appearance of chimera states, has been studied extensively in many contexts of science and technology since the past decade, though the previous studies are mostly built on the framework of one-dimensional and two-dimensional interaction topologies. Recently, the emergence of such fascinating phenomena has been studied in a three-dimensional (3D) grid formation while considering only the nonlocal interaction. Here we study the emergence and existence of chimera patterns in a three-dimensional network of coupled Stuart-Landau limit-cycle oscillators and Hindmarsh-Rose neuronal oscillators with local (nearest-neighbor) interaction topology. The emergence of different types of spatiotemporal chimera patterns is investigated by taking two distinct nonlinear interaction functions. We provide appropriate analytical explanations in the 3D grid of the network formation and the corresponding numerical justifications are given. We extend our analysis on the basis of the Ott-Antonsen reduction approach in the case of Stuart-Landau oscillators containing infinite numbers of oscillators. Particularly, in the Hindmarsh-Rose neuronal network the existence of nonstationary chimera states is characterized by an instantaneous strength of incoherence and an instantaneous local order parameter. Besides, the condition for achieving exact neuronal synchrony is obtained analytically through a linear stability analysis. The different types of collective dynamics together with chimera states are mapped over a wide range of various parameter spaces.
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Affiliation(s)
- Srilena Kundu
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata 700108, India
| | - Bidesh K Bera
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata 700108, India
- Department of Mathematics, Indian Institute of Technology Ropar, Punjab 140001, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata 700108, India
| | - M Lakshmanan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirapalli 620024, India
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25
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Yao N, Huang ZG, Ren HP, Grebogi C, Lai YC. Self-adaptation of chimera states. Phys Rev E 2019; 99:010201. [PMID: 30780345 DOI: 10.1103/physreve.99.010201] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2018] [Indexed: 06/09/2023]
Abstract
Chimera states in spatiotemporal dynamical systems have been investigated in physical, chemical, and biological systems, and have been shown to be robust against random perturbations. How do chimera states achieve their robustness? We uncover a self-adaptation behavior by which, upon a spatially localized perturbation, the coherent component of the chimera state spontaneously drifts to an optimal location as far away from the perturbation as possible, exposing only its incoherent component to the perturbation to minimize the disturbance. A systematic numerical analysis of the evolution of the spatiotemporal pattern of the chimera state towards the optimal stable state reveals an exponential relaxation process independent of the spatial location of the perturbation, implying that its effects can be modeled as restoring and damping forces in a mechanical system and enabling the articulation of a phenomenological model. Not only is the model able to reproduce the numerical results, it can also predict the trajectory of drifting. Our finding is striking as it reveals that, inherently, chimera states possess a kind of "intelligence" in achieving robustness through self-adaptation. The behavior can be exploited for the controlled generation of chimera states with their coherent component placed in any desired spatial region of the system.
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Affiliation(s)
- Nan Yao
- Department of Applied Physics, Xi'an University of Technology, Xi'an 710048, China
| | - Zi-Gang Huang
- The Key Laboratory of Biomedical Information Engineering of Ministry of Education, National Engineering Research Center of Health Care and Medical Devices, The Key Laboratory of Neuro-informatics & Rehabilitation Engineering of Ministry of Civil Affairs, and Institute of Health and Rehabilitation Science, School of Life Science and Technology, Xi'an Jiaotong University, Xi'an 710049, China
| | - Hai-Peng Ren
- Shaanxi Key Laboratory of Complex System Control and Intelligent Information Processing, Xi'an University of Technology, Xi'an 710048, China
| | - Celso Grebogi
- Institute for Complex Systems and Mathematical Biology, King's College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
- Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
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Wei Z, Parastesh F, Azarnoush H, Jafari S, Ghosh D, Perc M, Slavinec M. Nonstationary chimeras in a neuronal network. ACTA ACUST UNITED AC 2018. [DOI: 10.1209/0295-5075/123/48003] [Citation(s) in RCA: 54] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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27
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Suda Y, Okuda K. Breathing multichimera states in nonlocally coupled phase oscillators. Phys Rev E 2018; 97:042212. [PMID: 29758692 DOI: 10.1103/physreve.97.042212] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2017] [Indexed: 06/08/2023]
Abstract
Chimera states for the one-dimensional array of nonlocally coupled phase oscillators in the continuum limit are assumed to be stationary states in most studies, but a few studies report the existence of breathing chimera states. We focus on multichimera states with two coherent and incoherent regions and numerically demonstrate that breathing multichimera states, whose global order parameter oscillates temporally, can appear. Moreover, we show that the system exhibits a Hopf bifurcation from a stationary multichimera to a breathing one by the linear stability analysis for the stationary multichimera.
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Affiliation(s)
- Yusuke Suda
- Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
| | - Koji Okuda
- Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
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28
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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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Kundu S, Majhi S, Bera BK, Ghosh D, Lakshmanan M. Chimera states in two-dimensional networks of locally coupled oscillators. Phys Rev E 2018; 97:022201. [PMID: 29548198 DOI: 10.1103/physreve.97.022201] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/30/2017] [Indexed: 06/08/2023]
Abstract
Chimera state is defined as a mixed type of collective state in which synchronized and desynchronized subpopulations of a network of coupled oscillators coexist and the appearance of such anomalous behavior has strong connection to diverse neuronal developments. Most of the previous studies on chimera states are not extensively done in two-dimensional ensembles of coupled oscillators by taking neuronal systems with nonlinear coupling function into account while such ensembles of oscillators are more realistic from a neurobiological point of view. In this paper, we report the emergence and existence of chimera states by considering locally coupled two-dimensional networks of identical oscillators where each node is interacting through nonlinear coupling function. This is in contrast with the existence of chimera states in two-dimensional nonlocally coupled oscillators with rectangular kernel in the coupling function. We find that the presence of nonlinearity in the coupling function plays a key role to produce chimera states in two-dimensional locally coupled oscillators. We analytically verify explicitly in the case of a network of coupled Stuart-Landau oscillators in two dimensions that the obtained results using Ott-Antonsen approach and our analytical finding very well matches with the numerical results. Next, we consider another type of important nonlinear coupling function which exists in neuronal systems, namely chemical synaptic function, through which the nearest-neighbor (locally coupled) neurons interact with each other. It is shown that such synaptic interacting function promotes the emergence of chimera states in two-dimensional lattices of locally coupled neuronal oscillators. In numerical simulations, we consider two paradigmatic neuronal oscillators, namely Hindmarsh-Rose neuron model and Rulkov map for each node which exhibit bursting dynamics. By associating various spatiotemporal behaviors and snapshots at particular times, we study the chimera states in detail over a large range of coupling parameter. The existence of chimera states is confirmed by instantaneous angular frequency, order parameter and strength of incoherence.
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Affiliation(s)
- Srilena Kundu
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata-700108, India
| | - Soumen Majhi
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata-700108, India
| | - Bidesh K Bera
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata-700108, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata-700108, India
| | - M Lakshmanan
- Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirapalli-620024, India
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30
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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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English LQ, Zampetaki A, Kevrekidis PG, Skowronski K, Fritz CB, Abdoulkary S. Analysis and observation of moving domain fronts in a ring of coupled electronic self-oscillators. CHAOS (WOODBURY, N.Y.) 2017; 27:103125. [PMID: 29092454 DOI: 10.1063/1.5009088] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
In this work, we consider a ring of coupled electronic (Wien-bridge) oscillators from a perspective combining modeling, simulation, and experimental observation. Following up on earlier work characterizing the pairwise interaction of Wien-bridge oscillators by Kuramoto-Sakaguchi phase dynamics, we develop a lattice model for a chain thereof, featuring an exponentially decaying spatial kernel. We find that for certain values of the Sakaguchi parameter α, states of traveling phase-domain fronts involving the coexistence of two clearly separated regions of distinct dynamical behavior, can establish themselves in the ring lattice. Experiments and simulations show that stationary coexistence domains of synchronization only manifest themselves with the introduction of a local impurity; here an incoherent cluster of oscillators can arise reminiscent of the chimera states in a range of systems with homogeneous oscillators and suitable nonlocal interactions between them.
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Affiliation(s)
- L Q English
- Department of Physics and Astronomy, Dickinson College, Carlisle, Pennsylvania 17013, USA
| | - A Zampetaki
- Zentrum für Optische Quantentechnologien, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003, USA
| | - K Skowronski
- Department of Physics and Astronomy, Dickinson College, Carlisle, Pennsylvania 17013, USA
| | - C B Fritz
- Department of Physics and Astronomy, Dickinson College, Carlisle, Pennsylvania 17013, USA
| | - Saidou Abdoulkary
- Département des Sciences Fondamentales, IMIP University of Maroua, P.O. Box 46, Maroua, Cameroon
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Jalan S, Ghosh S, Patra B. Is repulsion good for the health of chimeras? CHAOS (WOODBURY, N.Y.) 2017; 27:101104. [PMID: 29092446 DOI: 10.1063/1.5005576] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Yes! Very much so. A chimera state refers to the coexistence of a coherent-incoherent dynamical evolution of identically coupled oscillators. We investigate the impact of multiplexing of a layer having repulsively coupled oscillators on the occurrence of chimeras in the layer having attractively coupled identical oscillators. We report that there exists an enhancement in the appearance of the chimera state in one layer of the multiplex network in the presence of repulsive coupling in the other layer. Furthermore, we show that a small amount of inhibition or repulsive coupling in one layer is sufficient to yield the chimera state in another layer by destroying its synchronized behavior. These results can be used to obtain insight into dynamical behaviors of those systems where both attractive and repulsive couplings exist among their constituents.
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Affiliation(s)
- Sarika Jalan
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Simrol, Indore 453552, India
| | - Saptarshi Ghosh
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Simrol, Indore 453552, India
| | - Bibhabasu Patra
- Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Simrol, Indore 453552, India
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33
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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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Rakshit S, Bera BK, Perc M, Ghosh D. Basin stability for chimera states. Sci Rep 2017; 7:2412. [PMID: 28546537 PMCID: PMC5445089 DOI: 10.1038/s41598-017-02409-5] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2016] [Accepted: 04/11/2017] [Indexed: 11/09/2022] Open
Abstract
Chimera states, namely complex spatiotemporal patterns that consist of coexisting domains of spatially coherent and incoherent dynamics, are investigated in a network of coupled identical oscillators. These intriguing spatiotemporal patterns were first reported in nonlocally coupled phase oscillators, and it was shown that such mixed type behavior occurs only for specific initial conditions in nonlocally and globally coupled networks. The influence of initial conditions on chimera states has remained a fundamental problem since their discovery. In this report, we investigate the robustness of chimera states together with incoherent and coherent states in dependence on the initial conditions. For this, we use the basin stability method which is related to the volume of the basin of attraction, and we consider nonlocally and globally coupled time-delayed Mackey-Glass oscillators as example. Previously, it was shown that the existence of chimera states can be characterized by mean phase velocity and a statistical measure, such as the strength of incoherence, by using well prepared initial conditions. Here we show further how the coexistence of different dynamical states can be identified and quantified by means of the basin stability measure over a wide range of the parameter space.
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Affiliation(s)
- Sarbendu Rakshit
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata, 700108, India
| | - Bidesh K Bera
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata, 700108, India
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000, Maribor, Slovenia.,CAMTP - Center for Applied Mathematics and Theoretical Physics, University of Maribor, Mladinska 3, SI-2000, Maribor, Slovenia
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata, 700108, India.
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35
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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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36
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Ashwin P, Coombes S, Nicks R. Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2016; 6:2. [PMID: 26739133 PMCID: PMC4703605 DOI: 10.1186/s13408-015-0033-6] [Citation(s) in RCA: 104] [Impact Index Per Article: 13.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2015] [Accepted: 10/30/2015] [Indexed: 05/20/2023]
Abstract
The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear-for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.
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Affiliation(s)
- Peter Ashwin
- Centre for Systems Dynamics and Control, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Harrison Building, Exeter, EX4 4QF, UK.
| | - Stephen Coombes
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
| | - Rachel Nicks
- School of Mathematics, University of Birmingham, Watson Building, Birmingham, B15 2TT, UK.
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37
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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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38
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Botha AE. Characteristic distribution of finite-time Lyapunov exponents for chimera states. Sci Rep 2016; 6:29213. [PMID: 27374473 PMCID: PMC4931592 DOI: 10.1038/srep29213] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/16/2016] [Accepted: 06/13/2016] [Indexed: 11/30/2022] Open
Abstract
Our fascination with chimera states stems partially from the somewhat paradoxical, yet fundamental trait of identical, and identically coupled, oscillators to split into spatially separated, coherently and incoherently oscillating groups. While the list of systems for which various types of chimeras have already been detected continues to grow, there is a corresponding increase in the number of mathematical analyses aimed at elucidating the fundamental reasons for this surprising behaviour. Based on the model systems, there are strong indications that chimera states may generally be ubiquitous in naturally occurring systems containing large numbers of coupled oscillators - certain biological systems and high-Tc superconducting materials, for example. In this work we suggest a new way of detecting and characterising chimera states. Specifically, it is shown that the probability densities of finite-time Lyapunov exponents, corresponding to chimera states, have a definite characteristic shape. Such distributions could be used as signatures of chimera states, particularly in systems for which the phases of all the oscillators cannot be measured directly. For such cases, we suggest that chimera states could perhaps be detected by reconstructing the characteristic distribution via standard embedding techniques, thus making it possible to detect chimera states in systems where they could otherwise exist unnoticed.
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Affiliation(s)
- André E. Botha
- Department of Physics, University of South Africa, Science Campus, Private Bag X6, Florida 1710, South Africa
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39
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Omelchenko I, Omel'chenko OE, Zakharova A, Wolfrum M, Schöll E. Tweezers for Chimeras in Small Networks. PHYSICAL REVIEW LETTERS 2016; 116:114101. [PMID: 27035303 DOI: 10.1103/physrevlett.116.114101] [Citation(s) in RCA: 35] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2015] [Indexed: 05/26/2023]
Abstract
We propose a control scheme which can stabilize and fix the position of chimera states in small networks. Chimeras consist of coexisting domains of spatially coherent and incoherent dynamics in systems of nonlocally coupled identical oscillators. Chimera states are generally difficult to observe in small networks due to their short lifetime and erratic drifting of the spatial position of the incoherent domain. The control scheme, like a tweezer, might be useful in experiments, where usually only small networks can be realized.
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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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40
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Suda Y, Okuda K. Persistent chimera states in nonlocally coupled phase oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:060901. [PMID: 26764621 DOI: 10.1103/physreve.92.060901] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/13/2015] [Indexed: 06/05/2023]
Abstract
Chimera states in the systems of nonlocally coupled phase oscillators are considered stable in the continuous limit of spatially distributed oscillators. However, it is reported that in the numerical simulations without taking such limit, chimera states are chaotic transient and finally collapse into the completely synchronous solution. In this Rapid Communication, we numerically study chimera states by using the coupling function different from the previous studies and obtain the result that chimera states can be stable even without taking the continuous limit, which we call the persistent chimera state.
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Affiliation(s)
- Yusuke Suda
- Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
| | - Koji Okuda
- Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
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41
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Xie J, Knobloch E, Kao HC. Twisted chimera states and multicore spiral chimera states on a two-dimensional torus. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:042921. [PMID: 26565318 DOI: 10.1103/physreve.92.042921] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2015] [Indexed: 06/05/2023]
Abstract
Chimera states consisting of domains of coherently and incoherently oscillating oscillators in a two-dimensional periodic array of nonlocally coupled phase oscillators are studied. In addition to the one-dimensional chimera states familiar from one spatial dimension, two-dimensional structures termed twisted chimera states and spiral wave chimera states are identified in simulations. The properties of many of these states, including stability, are determined using an evolution equation for a complex order parameter and are found to be in agreement with the simulations.
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Affiliation(s)
- Jianbo Xie
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
| | - Edgar Knobloch
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
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Abstract
A remarkable phenomenon in spatiotemporal dynamical systems is chimera state, where the structurally and dynamically identical oscillators in a coupled networked system spontaneously break into two groups, one exhibiting coherent motion and another incoherent. This phenomenon was typically studied in the setting of non-local coupling configurations. We ask what can happen to chimera states under systematic changes to the network structure when links are removed from the network in an orderly fashion but the local coupling topology remains invariant with respect to an index shift. We find the emergence of multicluster chimera states. Remarkably, as a parameter characterizing the amount of link removal is increased, chimera states of distinct numbers of clusters emerge and persist in different parameter regions. We develop a phenomenological theory, based on enhanced or reduced interactions among oscillators in different spatial groups, to explain why chimera states of certain numbers of clusters occur in certain parameter regions. The theoretical prediction agrees well with numerics.
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Omelchenko I, Zakharova A, Hövel P, Siebert J, Schöll E. Nonlinearity of local dynamics promotes multi-chimeras. CHAOS (WOODBURY, N.Y.) 2015; 25:083104. [PMID: 26328555 DOI: 10.1063/1.4927829] [Citation(s) in RCA: 37] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Chimera states are complex spatio-temporal patterns in which domains of synchronous and asynchronous dynamics coexist in coupled systems of oscillators. We examine how the character of the individual elements influences chimera states by studying networks of nonlocally coupled Van der Pol oscillators. Varying the bifurcation parameter of the Van der Pol system, we can interpolate between regular sinusoidal and strongly nonlinear relaxation oscillations and demonstrate that more pronounced nonlinearity induces multi-chimera states with multiple incoherent domains. We show that the stability regimes for multi-chimera states and the mean phase velocity profiles of the oscillators change significantly as the nonlinearity becomes stronger. Furthermore, we reveal the influence of time delay on chimera patterns.
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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
| | - Philipp Hövel
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Julien Siebert
- 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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Böhm F, Zakharova A, Schöll E, Lüdge K. Amplitude-phase coupling drives chimera states in globally coupled laser networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:040901. [PMID: 25974428 DOI: 10.1103/physreve.91.040901] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/28/2014] [Indexed: 06/04/2023]
Abstract
For a globally coupled network of semiconductor lasers with delayed optical feedback, we demonstrate the existence of chimera states. The domains of coherence and incoherence that are typical for chimera states are found to exist for the amplitude, phase, and inversion of the coupled lasers. These chimera states defy several of the previously established existence criteria. While chimera states in phase oscillators generally demand nonlocal coupling, large system sizes, and specially prepared initial conditions, we find chimera states that are stable for global coupling in a network of only four coupled lasers for random initial conditions. The existence is linked to a regime of multistability between the synchronous steady state and asynchronous periodic solutions. We show that amplitude-phase coupling, a concept common in different fields, is necessary for the formation of the chimera states.
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Affiliation(s)
- Fabian Böhm
- 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
| | - Kathy Lüdge
- Institut für Theoretische Physik, Freie Universität Berlin, 14195 Berlin, Germany
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45
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Xie J, Kao HC, Knobloch E. Chimera states in systems of nonlocal nonidentical phase-coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032918. [PMID: 25871183 DOI: 10.1103/physreve.91.032918] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/02/2015] [Indexed: 06/04/2023]
Abstract
Chimera states consisting of domains of coherently and incoherently oscillating nonlocally coupled phase oscillators in systems with spatial inhomogeneity are studied. The inhomogeneity is introduced through the dependence of the oscillator frequency on its location. Two types of spatial inhomogeneity, localized and spatially periodic, are considered and their effects on the existence and properties of multicluster and traveling chimera states are explored. The inhomogeneity is found to break up splay states, to pin the chimera states to specific locations, and to trap traveling chimeras. Many of these states can be studied by constructing an evolution equation for a complex order parameter. Solutions of this equation are in good agreement with the results of numerical simulations.
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Affiliation(s)
- Jianbo Xie
- Department of Physics, University of California, Berkeley, California 94720, USA
| | | | - Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
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46
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Dudkowski D, Maistrenko Y, Kapitaniak T. Different types of chimera states: an interplay between spatial and dynamical chaos. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032920. [PMID: 25314517 DOI: 10.1103/physreve.90.032920] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2014] [Indexed: 06/04/2023]
Abstract
We discuss the occurrence of chimera states in networks of nonlocally coupled bistable oscillators, in which individual subsystems are characterized by the coexistence of regular (a fixed point or a limit cycle) and chaotic attractors. By analyzing the dependence of the network dynamics on the range and strength of coupling, we identify parameter regions for various chimera states, which are characterized by different types of chaotic behavior at the incoherent interval. Besides previously observed chimeras with space-temporal and spatial chaos in the incoherent intervals we observe another type of chimera state in which the incoherent interval is characterized by a central interval with standard space-temporal chaos and two narrow side intervals with spatial chaos. Our findings for the maps as well as for time-continuous van der Pol-Duffing's oscillators reveal that this type of chimera states represents characteristic spatiotemporal patterns at the transition from coherence to incoherence.
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Affiliation(s)
- Dawid Dudkowski
- Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Yuri Maistrenko
- Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland and Institute of Mathematics and Centre for Medical and Biotechnical Research, National Academy of Sciences of Ukraine, Tereshchenkivska Street 3, 01030, Kyiv, Ukraine
| | - Tomasz Kapitaniak
- Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland
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