Cluster Synchronization in Three-Dimensional Lattices of Diffusively Coupled Oscillators

Белых, В. Н.; Belykh, Igor; Hasler, M.; Nevidin, K.

doi:10.1142/s0218127403006923

Cited by 48 publications

(26 citation statements)

References 39 publications

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“…Synchronized states can be created in at least two ways : by architectural and internal 7 symmetries [11,12,6,36] or by diffusion-like couplings [48,20,33,26,2]. Actually, we shall see that both, together or separately, can create flow-invariant subspaces corresponding to concurrently synchronized states.…”

Section: Symmetries Diffusion-like Couplings Flow-invariant Subspacmentioning

confidence: 94%

Stable concurrent synchronization in dynamic system networks

2007

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In a network of dynamical systems, concurrent synchronization is a regime where multiple groups of fully synchronized elements coexist. In the brain, concurrent synchronization may occur at several scales, with multiple "rhythms" interacting and functional assemblies combining neural oscillators of many different types. Mathematically, stable concurrent synchronization corresponds to convergence to a flow-invariant linear subspace of the global state space. We derive a general condition for such convergence to occur globally and exponentially. We also show that, under mild conditions, global convergence to a concurrently synchronized regime is preserved under basic system combinations such as negative feedback or hierarchies, so that stable concurrently synchronized aggregates of arbitrary size can be constructed. Robustnesss of stable concurrent synchronization to variations in individual dynamics is also quantified. Simple applications of these results to classical questions in systems neuroscience and robotics are discussed.

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Section: Symmetries Diffusion-like Couplings Flow-invariant Subspacmentioning

confidence: 94%

Stable concurrent synchronization in dynamic system networks

2007

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“…16 Statement 3: ͑a͒ The system ͑2͒ has a family of cluster synchronization manifolds M (d 1 ,d 2 ) that are an intersection of invariant manifolds M (d 1 ) and M (d 2 ) existing in the case of the 1D system ͑1͒. The corresponding cluster regimes are a topological product of synchronization regimes in the two directions of the 2D lattice.…”

Section: A Existence Of Cluster Synchronization Manifoldsmentioning

confidence: 99%

“…Using the approach developed in the previous papers, 5,14,16 we proceed now with the study of global stability of the cluster synchronization manifold M (1,N)ϭ͕X i, j ϭX j , i, jϭ1, . .…”

Section: Stability Of the Invariant Manifoldsmentioning

confidence: 99%

“…7, and references therein͒. In the subsequent years, new synchronization phenomena were found including the most interesting cases of full [1][2][3][4][5][6][7][8][9] and cluster synchronization, [10][11][12][13][14][15][16][17] generalized, 18 phase, 19 and lag 20 synchronization, riddled basins of attraction, 21 attractor bubbling, 22 on-off intermittency, 23 etc.…”

Section: Introductionmentioning

confidence: 99%

“…Full synchronization in twodimensional ͑2D͒ and 3D lattices of locally coupled limit-cycle 26,27 systems and chaotic oscillators [33][34][35][36]16 has recently been studied analytically.…”

Section: Introductionmentioning

confidence: 99%

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Persistent clusters in lattices of coupled nonidentical chaotic systems

Belykh

Белых

Nevidin

et al. 2003

Chaos: An Interdisciplinary Journal of Nonlinear Science

114

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Two-dimensional ͑2D͒ lattices of diffusively coupled chaotic oscillators are studied. In previous work, it was shown that various cluster synchronization regimes exist when the oscillators are identical. Here, analytical and numerical studies allow us to conclude that these cluster synchronization regimes persist when the chaotic oscillators have slightly different parameters. In the analytical approach, the stability of almost-perfect synchronization regimes is proved via the Lyapunov function method for a wide class of systems, and the synchronization error is estimated. Examples include a 2D lattice of nonidentical Lorenz systems with scalar diffusive coupling. In the numerical study, it is shown that in lattices of Lorenz and Rössler systems the cluster synchronization regimes are stable and robust against up to 10%-15% parameter mismatch and against small noise. © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1514202͔Lattices of coupled chaotic oscillators model many systems of interest in physics, biology, and engineering. In particular, the phenomenon of cluster synchronization, i.e., the synchronization of groups of oscillators, has received much attention. This phenomenon depends heavily on the coupling configuration between the oscillators. While in a network of globally coupled identical oscillators in principle any subset of the oscillators may synchronize, in diffusively coupled oscillators only very few of the decompositions into subsets of synchronized oscillators are possible. They have been characterized in recent papers in detail "see Refs. 14-16…. In that work, the oscillators were assumed to be identical and the symmetries of the resulting system of coupled oscillators were exploited. In more realistic models of physical systems, however, the individual oscillators have slightly different parameters and therefore the perfect symmetries in the coupled systems no longer exist. Similarly, perfect cluster synchronization cannot exist anymore, but approximate synchronization is still possible. The question then arises whether the cluster synchronization regimes that are observed in systems with identical oscillators persist approximatively under small parameter mismatch and the addition of small noise. This paper gives a positive answer and explores the limit of approximate synchronization both analytically and numerically.

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Cluster synchronization control for the complex network with dynamic connection relations

Chen,

Wang,

Peng

et al. 2024

Optim Control Appl Methods

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In this article, the cluster synchronization control for a complex dynamic network (CDN) with mutually coupled nodes subsystem (NS) and connection relations subsystem is investigated. The process of cluster synchronization is segmented into three dynamic continuous stages: first, the free movement, then the task‐oriented clustering, and finally the cluster synchronization. The control scheme, founded on the clustering nodes of CDN, is proposed, such that the cluster synchronization arises when clustering all the nodes into two different sub‐networks (SNs). This approach originates from the flight motion for a group of autonomous aircrafts. At a certain moment, free flight is interrupted, and all the aircrafts split into two SNs according to the given clustering rules. Then the two controlled SNs are required to achieve cluster synchronization, that is, every SN converges to the same in speed. In order to split all the nodes into two different SNs, a clustering algorithm according to the “central nodes” is proposed. Based on Lyapunov stability theory, by designing the suitable inner cluster links, the adaptive controller of NS for each SN is synthesized to guarantee the stability of the error system. It is shown that the eventual inner topology of each SN is displayed in the pre‐given form when the cluster synchronization achieves. Finally, the merits and effectiveness of the proposed control scheme are verified by a simulation example.

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Cluster Synchronization in Three-Dimensional Lattices of Diffusively Coupled Oscillators

Cited by 48 publications

References 39 publications

Stable concurrent synchronization in dynamic system networks

Stable concurrent synchronization in dynamic system networks

Persistent clusters in lattices of coupled nonidentical chaotic systems

Cluster synchronization control for the complex network with dynamic connection relations

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