On Chaotic Synchronization in a Linear Array of Chua's Circuits

Белых, В. Н.; Verichev, N. N.; Kocarev, Lj.; Chua, Lynn

doi:10.1142/9789812798855_0014

Cited by 13 publications

(10 citation statements)

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“…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%

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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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Section: Stability Of the Invariant Manifoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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“…We first need to check the boundedness of the solutions, namely that no trajectories of the system go to infinity. (3) We show that the analysis of the asymptotic behaviour of model (1), (3) for N = 2 can be restricted to a bounded region of the positive orthant of the state space because such a region is an absorbing domain, in the sense that all trajectories enter the domain in finite time and remain in it forever. Two connected patches of the kind (3) are described by: (2) − z (1) ), (2) ),…”

Section: Appendix a Global Synchronization Of The Two-patch Networkmentioning

confidence: 98%

“…Most methods for global synchronization of periodic and chaotic oscillators are based on the eigenvalues of the connectivity matrix and on the dynamics of the single oscillator [3,8,19,43,62,63]. An alternative approach, called connection graph method (CGM) [4], combines the Liapunov function approach with graph theoretical arguments.…”

Section: Global Synchronizationmentioning

confidence: 99%

“…In this appendix, we discuss the global synchronization of the metapopulation (1), (3) in the special case N = 2, i.e., when there are only two coupled patches. For the sake of clarity, we choose the dispersion profile H 0 = diag[1, 1, 1], i.e., we assume that the three populations disperse equally, but the extension of the analysis to the case of generic dispersal profiles is straightforward.…”

Section: Appendix a Global Synchronization Of The Two-patch Networkmentioning

confidence: 99%

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Synchrony in tritrophic food chain metacommunities

Belykh

Piccardi

Rinaldi

2009

Journal of Biological Dynamics

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The synchronous behaviour of interacting communities is studied in this paper. Each community is described by a tritrophic food chain model, and the communities interact through a network with arbitrary topology, composed of patches and migration corridors. The analysis of the local synchronization properties (via the master stability function approach) shows that, if only one species can migrate, the dispersal of the consumer (i.e., the intermediate trophic level) is the most effective mechanism for promoting synchronization. When analysing the effects of the variations of demographic parameters, it is found that factors that stabilize the single community also tend to favour synchronization. Global synchronization is finally analysed by means of the connection graph method, yielding a lower bound on the value of the dispersion rate that guarantees the synchronization of the metacommunity for a given network topology.

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