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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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2
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Kong LW, Weng Y, Glaz B, Haile M, Lai YC. Reservoir computing as digital twins for nonlinear dynamical systems. CHAOS (WOODBURY, N.Y.) 2023; 33:033111. [PMID: 37003826 DOI: 10.1063/5.0138661] [Citation(s) in RCA: 12] [Impact Index Per Article: 12.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/13/2022] [Accepted: 02/13/2023] [Indexed: 06/19/2023]
Abstract
We articulate the design imperatives for machine learning based digital twins for nonlinear dynamical systems, which can be used to monitor the "health" of the system and anticipate future collapse. The fundamental requirement for digital twins of nonlinear dynamical systems is dynamical evolution: the digital twin must be able to evolve its dynamical state at the present time to the next time step without further state input-a requirement that reservoir computing naturally meets. We conduct extensive tests using prototypical systems from optics, ecology, and climate, where the respective specific examples are a chaotic CO2 laser system, a model of phytoplankton subject to seasonality, and the Lorenz-96 climate network. We demonstrate that, with a single or parallel reservoir computer, the digital twins are capable of a variety of challenging forecasting and monitoring tasks. Our digital twin has the following capabilities: (1) extrapolating the dynamics of the target system to predict how it may respond to a changing dynamical environment, e.g., a driving signal that it has never experienced before, (2) making continual forecasting and monitoring with sparse real-time updates under non-stationary external driving, (3) inferring hidden variables in the target system and accurately reproducing/predicting their dynamical evolution, (4) adapting to external driving of different waveform, and (5) extrapolating the global bifurcation behaviors to network systems of different sizes. These features make our digital twins appealing in applications, such as monitoring the health of critical systems and forecasting their potential collapse induced by environmental changes or perturbations. Such systems can be an infrastructure, an ecosystem, or a regional climate system.
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Affiliation(s)
- Ling-Wei Kong
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
| | - Yang Weng
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
| | - Bryan Glaz
- Vehicle Technology Directorate, CCDC Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland 20783-1138, USA
| | - Mulugeta Haile
- Vehicle Technology Directorate, CCDC Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland 20783-1138, USA
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
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3
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Li X, Lei Y, Ghosh D. Taming non-stationary chimera states in locally coupled oscillators. CHAOS (WOODBURY, N.Y.) 2022; 32:093152. [PMID: 36182401 DOI: 10.1063/5.0118034] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/03/2022] [Accepted: 09/01/2022] [Indexed: 06/16/2023]
Abstract
The imperfect traveling chimera (ITC) state is a novel non-stationary chimera pattern in which the incoherent domain of oscillators spreads into the coherent domain. We investigate the ITC state in locally coupled pendulum oscillators with heterogeneous driving forces. We introduce the heterogeneous phase value in the driving forces by two different ways, namely, the random phase from uniform distribution and random phase directions with identical amplitude. We discover two transition mechanisms from ITC to coherent state through traveling chimera-like state by taking the two different phase heterogeneity. The transition phenomena are investigated using cylindrical and polar coordinate phase spaces. In the numerical study, we propose a quantitative measurement named "spatiotemporal consistency" strength for distinguishing the ITC from the traveling one. Our research facilitates the exploration of potential applications of heterogeneous interactions in neuroscience.
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Affiliation(s)
- Xueqi Li
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
| | - Youming Lei
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
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4
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Franović I, Omel'chenko OE, Wolfrum M. Bumps, chimera states, and Turing patterns in systems of coupled active rotators. Phys Rev E 2021; 104:L052201. [PMID: 34942776 DOI: 10.1103/physreve.104.l052201] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2021] [Accepted: 10/20/2021] [Indexed: 11/07/2022]
Abstract
Self-organized coherence-incoherence patterns, called chimera states, have first been reported in systems of Kuramoto oscillators. For coupled excitable units, similar patterns where coherent units are at rest are called bump states. Here, we study bumps in an array of active rotators coupled by nonlocal attraction and global repulsion. We demonstrate how they can emerge in a supercritical scenario from completely coherent Turing patterns: a single incoherent unit appears in a homoclinic bifurcation, undergoing subsequent transitions to quasiperiodic and chaotic behavior, which eventually transforms into extensive chaos with many incoherent units. We present different types of transitions and explain the formation of coherence-incoherence patterns according to the classical paradigm of short-range activation and long-range inhibition.
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Affiliation(s)
- Igor Franović
- Scientific Computing Laboratory, Center for the Study of Complex Systems, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
| | - Oleh E Omel'chenko
- University of Potsdam, Institute of Physics and Astronomy, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
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5
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Khatun AA, Jafri HH, Punetha N. Controlling chimera states in chaotic oscillator ensembles through linear augmentation. Phys Rev E 2021; 103:042202. [PMID: 34005985 DOI: 10.1103/physreve.103.042202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2019] [Accepted: 03/07/2021] [Indexed: 11/07/2022]
Abstract
In this work, we show how "chimera states," namely, the dynamical situation when synchronized and desynchronized domains coexist in an oscillator ensemble, can be controlled through a linear augmentation (LA) technique. Specifically, in the networks of coupled chaotic oscillators, we obtain chimera states through induced multistability and demonstrate how LA can be used to control the size and spatial location of the incoherent and coherent populations in the ensemble. We examine basins of attraction of the system to analyze the effects of LA on its multistable behavior and thus on chimera states. Stability of the synchronized dynamics is analyzed through a master stability function. We find that these results are independent of a system's initial conditions and the strategy is applicable to the networks of globally, locally as well as nonlocally coupled oscillators. Our results suggest that LA control can be an effective method to control chimera states and to realize a desired collective dynamics in such ensembles.
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Affiliation(s)
- Anjuman Ara Khatun
- Department of Physics, Aligarh Muslim University, Aligarh 202 002, India
| | - Haider Hasan Jafri
- Department of Physics, Aligarh Muslim University, Aligarh 202 002, India
| | - Nirmal Punetha
- Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India
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Ruzzene G, Omelchenko I, Sawicki J, Zakharova A, Schöll E, Andrzejak RG. Remote pacemaker control of chimera states in multilayer networks of neurons. Phys Rev E 2020; 102:052216. [PMID: 33327161 DOI: 10.1103/physreve.102.052216] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2020] [Accepted: 10/30/2020] [Indexed: 06/12/2023]
Abstract
Networks of coupled nonlinear oscillators allow for the formation of nontrivial partially synchronized spatiotemporal patterns, such as chimera states, in which there are coexisting coherent (synchronized) and incoherent (desynchronized) domains. These complementary domains form spontaneously, and it is impossible to predict where the synchronized group will be positioned within the network. Therefore, possible ways to control the spatial position of the coherent and incoherent groups forming the chimera states are of high current interest. In this work we investigate how to control chimera patterns in multiplex networks of FitzHugh-Nagumo neurons, and in particular we want to prove that it is possible to remotely control chimera states exploiting the multiplex structure. We introduce a pacemaker oscillator within the network: this is an oscillator that does not receive input from the rest of the network but is sending out information to its neighbors. The pacemakers can be positioned in one or both layers. Their presence breaks the spatial symmetry of the layer in which they are introduced and allows us to control the position of the incoherent domain. We demonstrate how the remote control is possible for both uni- and bidirectional coupling between the layers. Furthermore we show which are the limitations of our control mechanisms when it is generalized from single-layer to multilayer networks.
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Affiliation(s)
- Giulia Ruzzene
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
| | - Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Jakub Sawicki
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
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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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Ruzzene G, Omelchenko I, Schöll E, Zakharova A, Andrzejak RG. Controlling chimera states via minimal coupling modification. CHAOS (WOODBURY, N.Y.) 2019; 29:051103. [PMID: 31154763 DOI: 10.1063/1.5097570] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/26/2019] [Accepted: 04/17/2019] [Indexed: 06/09/2023]
Abstract
We propose a method to control chimera states in a ring-shaped network of nonlocally coupled phase oscillators. This method acts exclusively on the network's connectivity. Using the idea of a pacemaker oscillator, we investigate which is the minimal action needed to control chimeras. We implement the pacemaker choosing one oscillator and making its links unidirectional. Our results show that a pacemaker induces chimeras for parameters and initial conditions for which they do not form spontaneously. Furthermore, the pacemaker attracts the incoherent part of the chimera state, thus controlling its position. Beyond that, we find that these control effects can be achieved with modifications of the network's connectivity that are less invasive than a pacemaker, namely, the minimal action of just modifying the strength of one connection allows one to control chimeras.
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Affiliation(s)
- Giulia Ruzzene
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
| | - Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
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9
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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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Omelchenko I, Zakharova A. Intriguing coexistence of synchrony and asynchrony in the brain: Comment on "Chimera states in neuronal networks: A review" by Soumen Majhi, Bidesh K. Bera, Dibakar Ghosh, Matjaẑ Perc. Phys Life Rev 2019; 28:134-136. [PMID: 30910384 DOI: 10.1016/j.plrev.2019.02.013] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/27/2019] [Accepted: 02/27/2019] [Indexed: 11/17/2022]
Affiliation(s)
- Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany.
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany.
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11
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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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12
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Malchow AK, Omelchenko I, Schöll E, Hövel P. Robustness of chimera states in nonlocally coupled networks of nonidentical logistic maps. Phys Rev E 2018; 98:012217. [PMID: 30110780 DOI: 10.1103/physreve.98.012217] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2018] [Indexed: 06/08/2023]
Abstract
We investigate the dynamics of nonlocally coupled time-discrete maps with emphasis on the occurrence and robustness of chimera states. These peculiar, hybrid states are characterized by a coexistence of coherent and incoherent regions. We consider logistic maps coupled on a one-dimensional ring with finite coupling radius. Domains of chimera existence form different tongues in the parameter space of coupling range and coupling strength. For a sufficiently large coupling strength, each tongue refers to a wave number describing the structure of the spatial profile. We also analyze the period-adding scheme within these tongues and multiplicity of period solutions. Furthermore, we study the robustness of chimeras with respect to parameter inhomogeneities and find that these states persist for different widths of the parameter distribution. Finally, we explore the spatial structure of the chimera using a spatial correlation function.
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Affiliation(s)
- Anne-Kathleen Malchow
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Philipp Hövel
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
- Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität zu Berlin, Philippstraße 13, 10115 Berlin, Germany
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