On the bifurcation structure of the mean-field fluctuation in the globally coupled tent map systems

Chawanya, Tsuyoshi; Morita, Satoru

doi:10.1016/s0167-2789(97)00254-6

Cited by 22 publications

(30 citation statements)

References 23 publications

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“…To address the question, we note that in certain coupled dynamical systems a macroscopic variable shows seemingly low-dimensional motions, while microscopic variables keep high dimensional chaos. Such a phenomenon has been extensively studied as a collective motion in coupled map lattice [4], globally coupled oscillators [5], and globally coupled map [6][7][8][9][10][11][12][13][14]. In the present Letter, we focus on the effect of noise on the collective motion of globally coupled map (GCM).…”

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confidence: 99%

Noiseless Collective Motion out of Noisy Chaos

1999

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We consider the effect of microscopic external noise on the collective motion of a globally coupled map in fully desynchronized states. Without the external noise a macroscopic variable shows high-dimensional chaos distinguishable from random motion. With the increase of external noise intensity, the collective motion is successively simplified. The number of effective degrees of freedom in the collective motion is found to decrease as $-\log{\sigma^2}$ with the external noise variance $\sigma^2$. It is shown how the microscopic noise can suppress the number of degrees of freedom at a macroscopic level.Comment: 9 pages RevTex file and 4 postscript figure

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confidence: 99%

Noiseless Collective Motion out of Noisy Chaos

1999

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“…[19][20][21] In this case, each component keeps on moving apparently independent from each other, and the behavior of the system seems stochastic. The emergence of collective motion by an arbitrarily weak coupling indicates that the destruction of the mutual/temporal correlation by chaos cannot be simply remodeled with that by random noise, even in the seemingly highly stochastic states.…”

Section: Globally Coupled Tent-map Systemsmentioning

confidence: 99%

“…The dynamical stability of the stationary distribution ͑corresponding to the SRB measure of a tent map mentioned above͒ has been extensively analyzed 21,[25][26][27][28] and it is shown that the stationary distribution becomes an attractor if ␣ and satisfy the following conditions, 21,25 i.e.,…”

Section: B Idealized Model For the Limit Of Näؕmentioning

confidence: 99%

Quasistable states in globally coupled tent map systems

Chawanya

2003

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The characteristics of long lasting but not perpetual chaotic states appear in a wide parameter region in a globally coupled overcritical tent map system are exhibited. The lifetime of the transient state has essential relevance with the system size. In some parameter region, the lifetime saturates at a certain level, while in another region it seems to diverge as the size of the system grows. In order to uncover the dynamical structures in large system size limit, the dynamics of one-body distribution is investigated as an idealized model for the infinitely large coupled map system. Obtained numerical results indicate the correspondence between the characteristics of long transient behavior in finite size system and that of the attractor or the ruin of attractor in the idealized model.

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“…It should be noticed that similar turbulent phases were reported by Kaneko in globally coupled tent maps. 17) However, recent studies 23) - 25) have shown that this fully turbulent state is unstable; i.e., small amplitude collective motion always occurs in globally coupled tent or piecewise linear maps. Because of this smallness, the nontrivial collective dynamics was not detected numerically in the globally coupled tent maps.…”

Section: Full Turbulencementioning

confidence: 99%

Synchronization and Collective Behavior in Globally Coupled Logarithmic Maps

Cosenza

González

1998

Progress of Theoretical Physics

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The collective phenomena arising in a system of globally coupled chaotic logarithmic maps are investigated by considering the properties of the mean field of the network. Several collective states are found in the phase diagram of the system: synchronized, collective periodic, collective chaotic, and fully turbulent states. In contrast with previously studied globally coupled systems, no splitting of the elements into different groups nor quasiperiodic collective states occur in this model. The organization of the observed nontrivial collective states is related to the presence of unstable periodic orbits in the local dynamics. The role that the properties of the local dynamics play in the emergence and characteristics of nontrivial collective behavior in globally coupled systems is discussed. §1. IntroductionCoupled map lattices (CML) are discrete space, discrete time dynamical systems of interacting elements whose states vary continuously according to specific functions. Globally coupled maps constitute a class of CML where the coupling interaction is a function of all the elements. 1) Though CML models are idealized systems, they have proved capable of capturing much of the phenomenology observed in a variety of complex spatiotemporal processes, with the advantage of being computationally efficient and, in many cases, mathematically tractable. 2) There has been recent interest in the use of CML models in the investigation of cooperative phenomena, such as synchronization or nontrivial collective behavior, which appear in many extended chaotic dynamical systems. 3) -7) Synchronization consists of the complete coincidence in time of the states of the elements in a system, while nontrivial collective behavior is characterized by a well-defined temporal evolution of statistical quantities emerging out of local chaos.An important category of systems with many degrees of freedom which can exhibit these collective effects is globally coupled nonlinear oscillators. Such systems arise naturally in the description of Josephson junctions arrays, charge density waves, multimode lasers, neural dynamics, and ecological and evolution models. 8) -12) Globally coupled maps represent a useful approach to the study of many processes on this kind of systems, in particular to the search for the conditions leading to the occurrence of collective dynamics.Studies in globally coupled chaotic maps have revealed interesting features such as: a) formation of clusters, i.e., differentiated subsets of synchronized elements within the network; 13) b) non-statistical properties in the fluctuations of the mean field of the ensemble; 13), 14) c) global quasiperiodic motion; 15),16) d) different collecDownloaded from https://academic.oup.

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On the bifurcation structure of the mean-field fluctuation in the globally coupled tent map systems

Cited by 22 publications

References 23 publications

Noiseless Collective Motion out of Noisy Chaos

Noiseless Collective Motion out of Noisy Chaos

Quasistable states in globally coupled tent map systems

Synchronization and Collective Behavior in Globally Coupled Logarithmic Maps

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