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Liu S, Zhan M. Clustering versus non-clustering phase synchronizations. CHAOS (WOODBURY, N.Y.) 2014; 24:013104. [PMID: 24697366 DOI: 10.1063/1.4861685] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
Clustering phase synchronization (CPS) is a common scenario to the global phase synchronization of coupled dynamical systems. In this work, a novel scenario, the non-clustering phase synchronization (NPS), is reported. It is found that coupled systems do not transit to the global synchronization until a certain sufficiently large coupling is attained, and there is no clustering prior to the global synchronization. To reveal the relationship between CPS and NPS, we further analyze the noise effect on coupled phase oscillators and find that the coupled oscillator system can change from CPS to NPS with the increase of noise intensity or system disorder. These findings are expected to shed light on the mechanism of various intriguing self-organized behaviors in coupled systems.
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Affiliation(s)
- 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 430071, 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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Suresh R, Senthilkumar DV, Lakshmanan M, Kurths J. Global phase synchronization in an array of time-delay systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:016215. [PMID: 20866715 DOI: 10.1103/physreve.82.016215] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2010] [Indexed: 05/29/2023]
Abstract
We report the identification of global phase synchronization (GPS) in a linear array of unidirectionally coupled Mackey-Glass time-delay systems exhibiting highly non-phase-coherent chaotic attractors with complex topological structure. In particular, we show that the dynamical organization of all the coupled time-delay systems in the array to form GPS is achieved by sequential synchronization as a function of the coupling strength. Further, the asynchronous ones in the array with respect to the main sequentially synchronized cluster organize themselves to form clusters before they achieve synchronization with the main cluster. We have confirmed these results by estimating instantaneous phases including phase difference, average phase, average frequency, frequency ratio, and their differences from suitably transformed phase coherent attractors after using a nonlinear transformation of the original non-phase-coherent attractors. The results are further corroborated using two other independent approaches based on recurrence analysis and the concept of localized sets from the original non-phase-coherent attractors directly without explicitly introducing the measure of phase.
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Affiliation(s)
- R Suresh
- Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University, Tiruchirapalli, India.
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Senthilkumar DV, Lakshmanan M, Kurths J. Transition from phase to generalized synchronization in time-delay systems. CHAOS (WOODBURY, N.Y.) 2008; 18:023118. [PMID: 18601485 DOI: 10.1063/1.2911541] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
The notion of phase synchronization in time-delay systems, exhibiting highly non-phase-coherent attractors, has not been realized yet even though it has been well studied in chaotic dynamical systems without delay. We report the identification of phase synchronization in coupled nonidentical piecewise linear and in coupled Mackey-Glass time-delay systems with highly non-phase-coherent regimes. We show that there is a transition from nonsynchronized behavior to phase and then to generalized synchronization as a function of coupling strength. We have introduced a transformation to capture the phase of the non-phase-coherent attractors, which works equally well for both the time-delay systems. The instantaneous phases of the above coupled systems calculated from the transformed attractors satisfy both the phase and mean frequency locking conditions. These transitions are also characterized in terms of recurrence-based indices, namely generalized autocorrelation function P(t), correlation of probability of recurrence, joint probability of recurrence, and similarity of probability of recurrence. We have quantified the different synchronization regimes in terms of these indices. The existence of phase synchronization is also characterized by typical transitions in the Lyapunov exponents of the coupled time-delay systems.
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Affiliation(s)
- D V Senthilkumar
- Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University, Tiruchirapalli-620024, India.
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Senthilkumar DV, Lakshmanan M, Kurths J. Phase synchronization in time-delay systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:035205. [PMID: 17025694 DOI: 10.1103/physreve.74.035205] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/23/2006] [Indexed: 05/12/2023]
Abstract
Though the notion of phase synchronization has been well studied in chaotic dynamical systems without delay, it has not been realized yet in chaotic time-delay systems exhibiting non-phase-coherent hyperchaotic attractors. In this paper we report identification of phase synchronization in coupled time-delay systems exhibiting hyperchaotic attractor. We show that there is a transition from nonsynchronized behavior to phase and then to generalized synchronization as a function of coupling strength. These transitions are characterized by recurrence quantification analysis, by phase differences based on a transformation of the attractors, and also by the changes in the Lyapunov exponents. We have found these transitions in coupled piecewise linear and in Mackey-Glass time-delay systems.
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Affiliation(s)
- D V Senthilkumar
- Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University, Tiruchirapalli, 620 024, India
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Zhang X, Fu M, Xiao J, Hu G. Self-organization of chaos synchronization and pattern formation in coupled chaotic oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:015202. [PMID: 16907144 DOI: 10.1103/physreve.74.015202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2006] [Indexed: 05/11/2023]
Abstract
Pattern formation in spatiotemporal chaotic systems is investigated. Temporally chaotic and spatially ordered patterns are observed by varying the coupling strength. Spatial orderings emerge spontaneously due to self-organization of partial and nonlocal chaos synchronization, governed by various types of spatial symmetries. The first and secondary bifurcations from spatially disordered chaos to chaos with different levels of spatial orderings are observed and the scaling behaviors associated with these bifurcations are statistically analyzed.
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Affiliation(s)
- Xiaoming Zhang
- Department of Physics, Beijing Normal University, Beijing 100875, China
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Guan S, Lai CH, Wei GW. Phase synchronization between two essentially different chaotic systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:016205. [PMID: 16090064 DOI: 10.1103/physreve.72.016205] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2004] [Indexed: 05/03/2023]
Abstract
In this paper, we numerically investigate phase synchronization between two coupled essentially different chaotic oscillators in drive-response configuration. It is shown that phase synchronization can be observed between two coupled systems despite the difference and the large frequency detuning between them. Moreover, the relation between phase synchronization and generalized synchronization is compared with that in coupled parametrically different systems. In the systems studied, it is found that phase synchronization occurs after generalized synchronization in coupled essentially different chaotic systems.
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Affiliation(s)
- Shuguang Guan
- Temasek Laboratories, National University of Singapore, 5 Sports Drive 2, Singapore, 117508
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Maistrenko Y, Popovych O, Burylko O, Tass PA. Mechanism of desynchronization in the finite-dimensional Kuramoto model. PHYSICAL REVIEW LETTERS 2004; 93:084102. [PMID: 15447191 DOI: 10.1103/physrevlett.93.084102] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/18/2003] [Indexed: 05/24/2023]
Abstract
We study how a decrease of the coupling strength causes a desynchronization in the Kuramoto model of N globally coupled phase oscillators. We show that, if the natural frequencies are distributed uniformly or close to that, the synchronized state can robustly split into any number of phase clusters with different average frequencies, even culminating in complete desynchronization. In the simplest case of N=3 phase oscillators, the course of the splitting is controlled by a Cherry flow. The general N-dimensional desynchronization mechanism is numerically illustrated for N=5.
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Affiliation(s)
- Yu Maistrenko
- Institute of Medicine, Research Center Jülich, 52425 Jülich, Germany
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Osipov GV, Kurths J. Regular and chaotic phase synchronization of coupled circle maps. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:016216. [PMID: 11800777 DOI: 10.1103/physreve.65.016216] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2001] [Revised: 09/27/2001] [Indexed: 05/23/2023]
Abstract
We study the effects of regular and chaotic phase synchronization in ensembles of coupled nonidentical circle maps (CMs) and find phase-locking regions for both types of synchronization. We show that synchronization of chaotic CMs is crucially influenced by the three quantities: (i) rotation number difference, (ii) variance of the phase evolution, and (iii)relative duration of intervals of phase increase respect decrease. In the case of regular CMs, only variance and rotation number difference are important. It is demonstrated that with increase of noncoherence of phase evolutionsin the regular and chaotic regime, the regions of the main (1:1) synchronization are usually decreased. We present a chaotic synchronization in the systems of coupled nonidentical circle maps where phase entrainment occurs and it is not accompanied by bifurcations of the chaotic set. For ensembles (chains) of coupled CMs with linear and random distributions of the individual frequencies soft and hard transitions to global synchronization are found.
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Affiliation(s)
- Grigory V Osipov
- Institute of Physics, University Potsdam 10, Am Neuen Palais, D-14415, Potsdam, Germany
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Zheng Z, Hu G, Hu B. Enhancement of phase synchronization through asymmetric couplings. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:7501-7504. [PMID: 11102118 DOI: 10.1103/physreve.62.7501] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/19/2000] [Indexed: 05/23/2023]
Abstract
Phase synchronization in lattices of coupled chaotic oscillators is studied. It is found that phase synchronization can be greatly improved by asymmetric biased coupling. The mechanism responsible for this effect is the transition from a localized wave to synchronized flow and nonlocal phase synchronization.
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Affiliation(s)
- Z Zheng
- Department of Physics, Beijing Normal University, Beijing 100875, China and Department of Physics and Center for Nonlinear Studies, Hong Kong Baptist University, Hong Kong, China
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