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Bera BK, Ghosh D, Banerjee T. Imperfect traveling chimera states induced by local synaptic gradient coupling. Phys Rev E 2016; 94:012215. [PMID: 27575131 DOI: 10.1103/physreve.94.012215] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2016] [Indexed: 06/06/2023]
Abstract
In this paper, we report the occurrence of chimera patterns in a network of neuronal oscillators, which are coupled through local, synaptic gradient coupling. We discover a new chimera pattern, namely the imperfect traveling chimera state, where the incoherent traveling domain spreads into the coherent domain of the network. Remarkably, we also find that chimera states arise even for one-way local coupling, which is in contrast to the earlier belief that only nonlocal, global, or nearest-neighbor local coupling can give rise to chimera state; this find further relaxes the essential connectivity requirement of getting a chimera state. We choose a network of identical bursting Hindmarsh-Rose neuronal oscillators, and we show that depending upon the relative strength of the synaptic and gradient coupling, several chimera patterns emerge. We map all the spatiotemporal behaviors in parameter space and identify the transitions among several chimera patterns, an in-phase synchronized state, and a global amplitude death state.
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Affiliation(s)
- Bidesh K Bera
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700 108, India
| | - Dibakar Ghosh
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700 108, India
| | - Tanmoy Banerjee
- Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan, Burdwan 713 104, West Bengal, India
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Ning D, Wu X, Lu JA, Lü J. Driving-based generalized synchronization in two-layer networks via pinning control. CHAOS (WOODBURY, N.Y.) 2015; 25:113104. [PMID: 26627564 DOI: 10.1063/1.4935069] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Synchronization of complex networks has been extensively investigated in various fields. In the real world, one network is usually affected by another one but coexists in harmony with it, which can be regarded as another kind of synchronization--generalized synchronization (GS). In this paper, the GS in two-layer complex networks with unidirectional inter-layer coupling via pinning control is investigated based on the auxiliary-system approach. Specifically, for two-layer networks under study, one is considered as the drive network and the other is the response one. According to the auxiliary-system approach, output from the drive layer is designed as input for the response one, and an identical duplication of the response layer is constructed, which is driven by the same driving signals. A sufficient condition for achieving GS via pinning control is presented. Numerical simulations are further provided to illustrate the correctness of the theoretical results. It is also revealed that the least number of pinned nodes needed for achieving GS decreases with the increasing density of the response layer. In addition, it is found that when the intra-layer coupling strength of the response network is large, nodes with larger degrees should be selected to pin first for the purpose of achieving GS. However, when the coupling strength is small, it is more preferable to pin nodes with smaller degrees. This work provides engineers with a convenient approach to realize harmonious coexistence of various complex systems, which can further facilitate the selection of pinned systems and reduce control cost.
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Affiliation(s)
- Di Ning
- School of Mathematics and Statistics, Wuhan University, Hubei 430072, China
| | - Xiaoqun Wu
- School of Mathematics and Statistics, Wuhan University, Hubei 430072, China
| | - Jun-an Lu
- School of Mathematics and Statistics, Wuhan University, Hubei 430072, China
| | - Jinhu Lü
- Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
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He Z, Wang X, Zhang GY, Zhan M. Control for a synchronization-desynchronization switch. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:012909. [PMID: 25122362 DOI: 10.1103/physreve.90.012909] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2014] [Indexed: 06/03/2023]
Abstract
How to freely enhance or suppress synchronization of networked dynamical systems is of great importance in many disciplines. A unified precise control method for a synchronization-desynchronization switch, called the pull-push control method, is suggested. Namely, synchronization can be achieved when the original systems are desynchronous by pulling (or protecting) one node or a certain subset of nodes, whereas desynchronization can be accomplished when the systems are already synchronous by pushing (or kicking) one node or a certain subset of nodes. With this method, the controlled nodes should be chosen by the generalized eigenvector centrality of the critical synchronization mode of the Laplacian matrix. Compared with existing control methods for synchronization, it displays high efficiency, flexibility, and precision as well.
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Affiliation(s)
- Zhiwei He
- Wuhan Center for Magnetic Resonance, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China and University of the Chinese Academy of Sciences, Beijing 100049, China
| | - Xingang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Guo-Yong Zhang
- Wuhan Center for Magnetic Resonance, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China and College of Computer Science and Technology, Hubei Normal University, Huangshi 435002, China
| | - Meng Zhan
- Wuhan Center for Magnetic Resonance, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
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Zhan M, Liu S, He Z. Matching rules for collective behaviors on complex networks: optimal configurations for vibration frequencies of networked harmonic oscillators. PLoS One 2013; 8:e82161. [PMID: 24386088 PMCID: PMC3873270 DOI: 10.1371/journal.pone.0082161] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/17/2013] [Accepted: 10/22/2013] [Indexed: 11/18/2022] Open
Abstract
The structure-dynamics-function has become one of central problems in modern sciences, and it is a great challenge to unveil the organization rules for different dynamical processes on networks. In this work, we study the vibration spectra of the classical mass spring model with different masses on complex networks, and pay our attention to how the mass spatial configuration influences the second-smallest vibrational frequency (ω2) and the largest one (ωN). For random networks, we find that ω2 becomes maximal and ωN becomes minimal if the node degrees are point-to-point-positively correlated with the masses. In these cases, we call it point-to-point matching. Moreover, ω2 becomes minimal under the condition that the heaviest mass is placed on the lowest-degree vertex, and ωN is maximal as long as the lightest mass is placed on the highest-degree vertex, and in both cases all other masses can be arbitrarily settled. Correspondingly, we call it single-point matching. These findings indicate that the matchings between the node dynamics (parameter) and the node position rule the global systems dynamics, and sometimes only one node is enough to control the collective behaviors of the whole system. Therefore, the matching rules might be the common organization rules for collective behaviors on networks.
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Affiliation(s)
- Meng Zhan
- Wuhan Center for Magnetic Resonance, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, China
| | - Shuai Liu
- Wuhan Center for Magnetic Resonance, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, China
- University of the Chinese Academy of Sciences, Beijing, China
| | - Zhiwei He
- Wuhan Center for Magnetic Resonance, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, China
- University of the Chinese Academy of Sciences, Beijing, China
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Zou W, Yao C, Zhan M. Eliminating delay-induced oscillation death by gradient coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:056203. [PMID: 21230559 DOI: 10.1103/physreve.82.056203] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/17/2010] [Revised: 10/10/2010] [Indexed: 05/30/2023]
Abstract
In this work, we investigate gradient coupling effect on oscillation death in a ring of N delay-coupled oscillators. We find that the gradient coupling monotonically reduces the domain of death island in the parameter space of the diffusive coupling and time delay, and thus the death island can be completely eliminated once the gradient coupling strength exceeds a certain threshold, whose value is found to be a constant if N is sufficiently large. For two special cases, a ring with zero gradient coupling and a one-way ring for identical diffusive and gradient couplings, all previous results in the literature are recovered. In particular, for the one-way ring, a size effect of N is discovered, which indicates that under this situation the death can always be eliminated if N is above a critical N(max). All the described results are proved to hold generally in coupled oscillator systems.
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Affiliation(s)
- Wei Zou
- School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
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Zou W, Zhan M. Complete periodic synchronization in coupled systems. CHAOS (WOODBURY, N.Y.) 2008; 18:043115. [PMID: 19123625 DOI: 10.1063/1.3025253] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
Recently, complete chaotic synchronization in coupled systems has been well studied. In this paper, we study complete synchronization in coupled periodic oscillators with diffusive and gradient couplings. Eight typical types of critical curve for the transverse Lyapunov exponent of standard mode, which give rise to different synchronization-desynchronization patterns, are classified. All possible desynchronous behaviors including steady state, periodic state, quasiperiodic state, low-dimensional chaotic state, and two types of high-dimensional chaotic state are identified, and two classical synchronization-desynchronizaiton bifurcations--the shortest wavelength bifurcation and Hopf bifurcation from synchronous periodic state--are classified.
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Affiliation(s)
- Wei Zou
- Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
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Fradkov AL, Andrievsky B, Evans RJ. Synchronization of nonlinear systems under information constraints. CHAOS (WOODBURY, N.Y.) 2008; 18:037109. [PMID: 19045483 DOI: 10.1063/1.2977459] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
A brief survey of control and synchronization under information constraints (limited information capacity of the coupling channel) is given. Limit possibilities of nonlinear observer-based synchronization systems with first-order coders or full-order coders are considered in more detail. The existing and new theoretical results for multidimensional drive-response Lurie systems (linear part plus nonlinearity depending only on measurable outputs) are presented. It is shown that the upper bound of the limit synchronization error (LSE) is proportional to the upper bound of the transmission error. As a consequence, the upper and lower bounds of LSE are proportional to the maximum coupling signal rate and inversely proportional to the information transmission rate (channel capacity). The analysis is extended to networks having a "chain," "star," or "star-chain" topology. Adaptive chaotic synchronization under information constraints is analyzed. The results are illustrated by example: master-slave synchronization of two chaotic Chua systems coupled via a channel with limited capacity.
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Affiliation(s)
- Alexander L Fradkov
- Institute for Problems of Mechanical Engineering, Russian Academy of Sciences, 61, Bolshoy V.O. Av., 199178 Saint Petersburg, Russia.
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Zaher AA. Design of fast state observers using a backstepping-like approach with application to synchronization of chaotic systems. CHAOS (WOODBURY, N.Y.) 2008; 18:023114. [PMID: 18601481 DOI: 10.1063/1.2907737] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
A simple technique is introduced to build fast state observers for chaotic systems when only a scalar time series of the output is available. This technique relies on using a backstepping-like approach via introducing new virtual states that can be observed using the drive-response synchronization mechanism. The proposed dynamic structure of the virtual states allows for employing control parameters that can adjust the convergence rate of the observed states. In addition, these control parameters can be used to improve the transient performance of the response system to accommodate small and large variations of the initial conditions, thus achieving superior performance to conventional synchronization techniques. Simple Lyapunov functions are used to estimate the range of the control parameters that guarantees stable operation of the proposed technique. Three benchmark chaotic systems are considered for illustration; namely, the Lorenz, Chua, and Rossler systems. The conflict between stability and agility of the states observer is analyzed and a simple tuning mechanism is introduced. Implementation of the proposed technique in both analog and digital forms is also addressed and experimental results are reported ensuring feasibility and real-time applicability. Finally, advantages and limitations are discussed and a comparison with conventional synchronization methods is investigated.
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Affiliation(s)
- Ashraf A Zaher
- Physics Department, Science College, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
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