Validity of numerical trajectories in the synchronization transition of complex systems

Viana, Ricardo L.; Grebogi, Celso; Pinto, Sandro Ely de Souza; Lopes, S. R.; Batista, Antônio M.; Kurths, Jürgen

doi:10.1103/physreve.68.067204

Cited by 28 publications

(19 citation statements)

References 25 publications

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“…Another related case of interest is to consider different types of local dynamics and different types of ranges for coupling [19][20][21][22][23][24][25][26][27][28] in investigating the variations in patterns of collective behaviors as well as the SMLE. Considering the non-extensivity of the long-range coupling case, nonextensive statistical mechanics may be applied for a deeper understanding of numerical results [29].…”

Section: Discussionmentioning

confidence: 99%

Collective behavior of coupled map lattices with different scales of local coupling

Shi

2011

Chin. Sci. Bull.

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In this paper, I present a numerical study on the collective behavior of one-dimensional coupled map lattices with the nearest coupling to different scales for the whole system. Using the maximum Lyapunov exponent as a tool for subsystem and return mapping, I observed several basic patterns of collective behavior and investigated the contrasts between the different scales. To study the mechanism, the system under entirely random perturbations was investigated using the Monte Carlo method and the contrast with the deterministic approach is given. The results show that the response to a random input is complicated and involves the correlation of different signals and taking into consideration the dynamic properties of the system itself. subsystem, maximum Lyapunov exponent, different scale, return map, random perturbation Citation:Shi W J. Collective behavior of coupled map lattices with different scales of local coupling.Space-time chaos has been studied for many years and is still a major topic of research. Using coupled map lattices (CMLs) as a basic model for studying space-time chaos has received increased attention for some time [1]. Previous research on this model has mainly been applied to global dynamics, while little consideration of the dynamics has been given to different scales of the system as the lattices were assumed to be coarse grained. However, if the system has some collective behavior, the dynamics for different scales of the system should be considered. On the other hand, the relationship between few and many bodied systems continues to attract great interest. A similar pertinent question is whether I can also treat the dynamics of the CML with a deterministic approach and add random perturbations? An interesting paper [2] gives a positive answer to this for a special case, although random noise is artificially added. From the research presented in this paper, it is clear that the question is a difficult one. Furthermore, recent works related to nontrivial collective behavior has attracted wide attention. In particular, using Lyapunov modes gives a very new and profound comprehension of collective spatiotemporal chaotic behaviors [3]. In this paper, I aim to carry out a numerical study of collective behaviors for various local scales of the whole system (the scale here is the length of a subsystem). Common and extensively studied models (CML) are employed.

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Section: Discussionmentioning

confidence: 99%

Collective behavior of coupled map lattices with different scales of local coupling

Shi

2011

Chin. Sci. Bull.

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“…Although the intense work of Grebogi and collaborators (there are many references, see for example [9,15,3,11,16,17,13,14,18,10]) has been of fundamental importance in showing a large number of systems with UDV, we assert that this is not a universal property of non-hyperbolic high dimensional systems. In fact, the aim of this work is the construction of a high-dimensional non-hyperbolic system without UDV.…”

Section: Introductionmentioning

confidence: 86%

Analysis of high dimensional non-hyperbolic coupled systems through finite-time Lyapunov exponents

Disconzi

Brunnet

2008

Physica A: Statistical Mechanics and its Applications

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“…only when N is finite and below some threshold. Paper [35] shows that given N one can obtain the synchronization increasing the coupling range. In [37] we demonstrate that the number of elements that can be synchronized depends on the largest Lyapunov exponent of the partial element: the higher is the exponent, the smaller is N. Figure 1 shows the time dependence of ρ(nT ) above the desynchronization threshold.…”

Section: The Distance To the Synchronization Manifoldmentioning

confidence: 98%

Violation of hyperbolicity via unstable dimension variability in a chain with local hyperbolic chaotic attractors

Kuptsov¹

2013

J. Phys. A: Math. Theor.

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Abstract. We consider a chain of oscillators with hyperbolic chaos coupled via diffusion. When the coupling is strong the chain is synchronized and demonstrates hyperbolic chaos so that there is one positive Lyapunov exponent. With the decay of the coupling the second and the third Lyapunov exponents approach zero simultaneously. The second one becomes positive, while the third one remains close to zero. Its finite-time numerical approximation fluctuates changing the sign within a wide range of the coupling parameter. These fluctuations arise due to the unstable dimension variability which is known to be the source for non-hyperbolicity. We provide a detailed study of this transition using the methods of Lyapunov analysis.

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Validity of numerical trajectories in the synchronization transition of complex systems

Cited by 28 publications

References 25 publications

Collective behavior of coupled map lattices with different scales of local coupling

Collective behavior of coupled map lattices with different scales of local coupling

Analysis of high dimensional non-hyperbolic coupled systems through finite-time Lyapunov exponents

Violation of hyperbolicity via unstable dimension variability in a chain with local hyperbolic chaotic attractors

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