Analytical and numerical studies of noise-induced synchronization of chaotic systems

Toral, Raúl; Mirasso, Claudio R.; Hernández-Garcı́a, Emilio; Piro, Oreste

doi:10.1063/1.1386397

Cited by 157 publications

(106 citation statements)

References 82 publications

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“…It may lead to new phenomenon like coherence resonance in optical systems [26], changes in the bifurcation values [27], synchronisation of chaos [9] etc. These are besides the extreme sensitivity of chaotic systems to starting conditions which could also be affected by noise.…”

Section: Building Rnn Models From Short Chaotic and Noisy Observed Datamentioning

confidence: 99%

“…The first may be called open-loop or feed-forward system where the behaviour of the system is altered by applying a properly chosen excitation or perturbation or external action at some chosen time-interval using very little extra energy. This has been successfully utilized in many cases often by simply adding a deliberate noise [9]. The second method is based on stabilisation of an unstable periodic orbit of the chaotic system using information from the Poincaré map of the system to arrive at magnitude of the perturbation of a control parameter needed to make a nearby unstable periodic orbit to jump to a fixed point on the unstable periodic orbit.…”

Section: Introductionmentioning

confidence: 99%

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Modelling and control of chaotic processes through their Bifurcation Diagrams generated with the help of Recurrent Neural Network models: Part 1—simulation studies

Krishnaiah¹,

Kumar²,

Faruqi³

2006

Journal of Process Control

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Many real-world processes tend to be chaotic and also do not lead to satisfactory analytical modelling. It has been shown here that for such chaotic processes represented through short chaotic noisy time-series, a multi-input and multi-output recurrent neural networks model can be built which is capable of capturing the process trends and predicting the future values from any given starting condition. It is further shown that this capability can be achieved by the Recurrent Neural Network model when it is trained to very low value of mean squared error. Such a model can then be used for constructing the Bifurcation Diagram of the process leading to determination of desirable operating conditions. Further, this multi-input and multi-output model makes the process accessible for control using open-loop/closed-loop approaches or bifurcation control etc. All these studies have been carried out using a low dimensional discrete chaotic system of Hénon Map as a representative of some real-world processes.

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Section: Building Rnn Models From Short Chaotic and Noisy Observed Datamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Modelling and control of chaotic processes through their Bifurcation Diagrams generated with the help of Recurrent Neural Network models: Part 1—simulation studies

Krishnaiah¹,

Kumar²,

Faruqi³

2006

Journal of Process Control

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“…(For other examples of excitable maps, see [7].) While the origin is the unique global attractor, map (2) possesses an unstable fixed point, excitation across which triggers a chaotic transient.…”

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confidence: 99%

Noise-Induced Macroscopic Bifurcations in Globally Coupled Chaotic Units

et al. 2004

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Large populations of globally-coupled identical maps subjected to independent additive noise are shown to undergo qualitative changes as the features of the stochastic process are varied. We show that for strong coupling, the collective dynamics can be described in terms of a few effective macroscopic degrees of freedom, whose deterministic equations of motion are systematically derived through an order parameter expansion.

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“…First the phenomenon was considered an artifact arising from a finite precision in numerical simulations ͑Pik-ovsky, 1994͒. Later it was attributed to a nonzero mean value of the noisy signal ͑Herzel and Freund, 1995;Malescio, 1996;Sánchez et al, 1997͒. More recently it has been demonstrated for zero-mean, additive Gaussian white noises of large enough intensity in certain chaotic maps ͑Lai and Zou, 1998; Toral et al, 2001͒ ͓the case with parametric noise was considered by Minai and Anand ͑1998͔͒. The generalization to the weaker level of phase synchronization was considered in the case of two nonidentical chaotic systems under common noise by Zhou and Kurths ͑2002a͒, and experimentally observed in noisy-neuronal oscillators by Neiman and Russell ͑2002͒.…”

Section: B Stochastic Synchronization Of Chaotic Oscillatorsmentioning

confidence: 99%

Spatiotemporal order out of noise

2007

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Natural systems are undeniably subject to random fluctuations, arising from either environmental variability or thermal effects. The consideration of those fluctuations supposes to deal with noisy quantities whose variance might at times be a sizable fraction of their mean levels. It is known that, under these conditions, noisy fluctuations can interact with the system's nonlinearities to render counterintuitive behavior, in which an increase in the noise level produces a more regular behavior. In systems with spatial degrees of freedom, this regularity takes the form of spatiotemporal order. An overview is presented of the mechanisms through which noise induces, enhances, and sustains ordered behavior in passive and active nonlinear media, and different spatiotemporal phenomena are described resulting from these effects. The general theoretical framework used in the analysis of these effects is reviewed, encompassing the theory of stochastic partial differential equations and coupled sets of ordinary stochastic differential equations. Experimental observations of self-organized behavior arising out of noise are also described, and details on the numerical algorithms needed in the computer simulation of these problems are given.

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Analytical and numerical studies of noise-induced synchronization of chaotic systems

Cited by 157 publications

References 82 publications

Modelling and control of chaotic processes through their Bifurcation Diagrams generated with the help of Recurrent Neural Network models: Part 1—simulation studies

Modelling and control of chaotic processes through their Bifurcation Diagrams generated with the help of Recurrent Neural Network models: Part 1—simulation studies

Noise-Induced Macroscopic Bifurcations in Globally Coupled Chaotic Units

Spatiotemporal order out of noise

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