Macroscopic chaos in globally coupled maps

Cencini, Massimo; Falcioni, M.; Vergni, Davide; Vulpiani, Angelo

doi:10.1016/s0167-2789(99)00015-9

Cited by 31 publications

(56 citation statements)

References 23 publications

(66 reference statements)

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“…modes, which are delocalized over the collection of the dynamical units and thus exert relevant perturbations to macroscopic variables, without the need for finite-amplitude perturbations as opposed to some earlier claims [20,21]. The CLVs allow us to detect such collective modes with the delocalization criterion, Y (j) 2 ∼ 1/N, and to examine directly their role in the collective dynamics, as demonstrated for the regime of nontrivial collective chaos in globally-coupled limit-cycle oscillators (Section 2).…”

Section: Discussion and Concluding Remarksmentioning

confidence: 99%

Collective Lyapunov modes

Takeuchi¹,

Chaté²

2013

J. Phys. A: Math. Theor.

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Abstract. We show, using covariant Lyapunov vectors in addition to standard Lyapunov analysis, that there exists a set of collective Lyapunov modes in large chaotic systems exhibiting collective dynamics. Associated with delocalized Lyapunov vectors, they act collectively on the trajectory and hence characterize the instability of its collective dynamics. We further develop, for globally-coupled systems, a connection between these collective modes and the Lyapunov modes in the corresponding PerronFrobenius equation. We thereby address the fundamental question of the effective dimension of collective dynamics and discuss the extensivity of chaos in presence of collective dynamics.

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Section: Discussion and Concluding Remarksmentioning

confidence: 99%

Collective Lyapunov modes

Takeuchi¹,

Chaté²

2013

J. Phys. A: Math. Theor.

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“…The velocity (12) does not depend on the chosen threshold values [2,4,10]. Since the dynamics of the difference field (11) is not confined in the tangent space, non linearities can play a crucial role in the information propagation.…”

Section: Information Spreading In Spatially Distributed Systemsmentioning

confidence: 99%

“…Indeed, finite disturbances, which are not confined in the tangent space, but are governed by the complete nonlinear dynamics, play a fundamental role in the erratic behaviors observed in some high dimensional system [8][9][10][11][12]. A rather intriguing phenomenon, termed stable chaos, has been reported in [9]: the authors observed that even a linearly stable system (i.e.…”

Section: Introductionmentioning

confidence: 99%

Linear and nonlinear information flow in spatially extended systems

Cencini

Torcini

2001

Phys. Rev. E

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Infinitesimal and finite amplitude error propagation in spatially extended systems are numerically and theoretically investigated. The information transport in these systems can be characterized in terms of the propagation velocity of perturbations Vp. A linear stability analysis is sufficient to capture all the relevant aspects associated to propagation of infinitesimal disturbances. In particular, this analysis gives the propagation velocity VL of infinitesimal errors. If linear mechanisms prevail on the nonlinear ones Vp = VL. On the contrary, if nonlinear effects are predominant finite amplitude disturbances can eventually propagate faster than infinitesimal ones (i.e. Vp > VL). The finite size Lyapunov exponent can be successfully employed to discriminate the linear or nonlinear origin of information flow. A generalization of finite size Lyapunov exponent to a comoving reference frame allows to state a marginal stability criterion able to provide Vp both in the linear and in the nonlinear case. Strong analogies are found between information spreading and propagation of fronts connecting steady states in reaction-diffusion systems. The analysis of the common characteristics of these two phenomena leads to a better understanding of the role played by linear and nonlinear mechanisms for the flow of information in spatially extended systems.

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“…In this paper, we will mainly consider two quantities: the Finite-Size Lyapunov Exponent (FSLE) and -entropy. The former was introduced in the context of developed turbulence [Aurell et al, 1997] and has proven to be more suited for a broad range of systems where the dynamics exhibit low-dimensional chaotic behavior only on large-scale [Shibata & Kaneko, 1998;Cencini et al, 1999;Gao et al, 2006]. Roughly, if we want to quantify the sensitivity to initial conditions on large scales, it is necessary to consider perturbations which are not infinitesimal.…”

Section: Finite-size Lyapunov and -Entropymentioning

confidence: 99%

“…In an attempt to reconcile these observations, we rely on recently introduced generalization of classical nonlinear tools. In particular, Finite-Size Lyapunov Exponent [Aurell et al, 1997;Shibata & Kaneko, 1998;Cencini et al, 1999;Gao et al, 2006] and the -entropy [Cencini et al, 2000] have proven to be valuable measures to probe the system which displays different behaviors at different scales. In the present context, where low-dimensional dynamics can take place on top of a highly irregular neuronal behavior, those measures appear as the most natural.…”

Section: How To Reconcile These Data?mentioning

confidence: 99%

Brain Dynamics at Multiple Scales: Can One Reconcile the Apparent Low-Dimensional Chaos of Macroscopic Variables With the Seemingly Stochastic Behavior of Single Neurons?

Boustani

Destexhe

2010

Int. J. Bifurcation Chaos

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Nonlinear time series analyses have suggested that the human electroencephalogram (EEG) may share statistical and dynamical properties with chaotic systems. During slow-wave sleep or pathological states like epilepsy, correlation dimension measurements display low values, while in awake and attentive subjects, there is no such low dimensionality, and the EEG is more similar to a stochastic variable. We briefly review these results and contrast them with recordings in cat cerebral cortex, as well as with theoretical models. In awake or sleeping cats, recordings with microelectrodes inserted in cortex show that global variables such as local field potentials (local EEG) are similar to the human EEG. However, neuronal discharges are highly irregular and exponentially distributed, similar to Poisson stochastic processes. To reconcile these results, we investigate models of randomly-connected networks of integrate-and-fire neurons, and also contrast global (averaged) variables, with neuronal activity. The network displays different states, such as "synchronous regular" (SR) or "asynchronous irregular" (AI) states. In SR states, the global variables display coherent behavior with low dimensionality, while in AI states, the global activity is high-dimensionally chaotic with exponentially distributed neuronal discharges, similar to awake cats. Scale-dependent Lyapunov exponents and -entropies show that the seemingly stochastic nature at small scales (neurons) can coexist with more coherent behavior at larger scales (averages). Thus, we suggest that brain activity obeys a similar scheme, with seemingly stochastic dynamics at small scales (neurons), while large scales (EEG) display more coherent behavior or high-dimensional chaos.

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Macroscopic chaos in globally coupled maps

Cited by 31 publications

References 23 publications

Collective Lyapunov modes

Collective Lyapunov modes

Linear and nonlinear information flow in spatially extended systems

Brain Dynamics at Multiple Scales: Can One Reconcile the Apparent Low-Dimensional Chaos of Macroscopic Variables With the Seemingly Stochastic Behavior of Single Neurons?

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