The origin of non-chaotic behavior in identically driven systems

Gade, Prashant M.; Basu, Chinmay

doi:10.1016/0375-9601(96)00306-4

Cited by 29 publications

(17 citation statements)

References 16 publications

(29 reference statements)

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“…Synchronization with λ < 0 occurs for σ 2 > 0.24. So, in contrast to the examples in [1][2][3][4][5][6][7][8], the sensitivity of this chaotic map can be suppressed by zero-mean noise, so that systems starting from different initial conditions will finally collapse into the same final orbit.The synchronization can be understood from the view point of the convergence region of the map. We perform two calculations: one is the distribution of the state of the system; and the other the distribution of the finite-time Lyapunov exponents defined as[5]…”

mentioning

confidence: 94%

Synchronization of chaotic maps by symmetric common noise

Lai

Zhou

1998

Europhys. Lett.

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Synchronization of identical chaotic systems subjected to common noise has been the subject of recent research. Studies on several chaotic systems have shown that, the synchronization is actually induced by the non-zero mean of the noise, and symmetric noise with zero-mean cannot lead to synchronization. Here it is presented that synchronization can be achieved by zero-mean noise in some chaotic maps with large convergence regions. The effect of common noise on the synchronization of identical chaotic systems has attracted much attention since the work by Maritan and Banavar [1], which claimed that two identical chaotic systems subjected to the same strong enough noise can be synchronized, with the logistic map and the Lorenz system as examples. Some authors have reconsidered their conclusion. Pikovsky [2] pointed out that the largest Lyapunov exponent of the noisy logistic map is positive which is in contradiction to the criterion of negative largest Lyapunov exponent for synchronization [3]. It was also pointed out [2] that the synchronization observed in [1] is an outcome of finite precision in numerical simulations, which was further confirmed by a detailed study by Longa et al [4].Several other authors, on the other hand, reconsidered the problem by examining the properties of the noise added to the systems. For the case of noisy logistic mapwhere the random number ξ is chosen from the interval [−W, W ] with the constraint x n+1 ∈ (0, 1); otherwise, a new random number is chosen. Such a state-dependent noise is no longer symmetric [5], but has a negative mean [6], and it is this nonzero mean that plays an important role in the coalescence of trajectories. A zero-mean noise, although is still state-dependent, cannot lead to synchronization [6]. For the case of the Lorenz system, synchronization was observed by Maritan and Banavar for uniform noise in [0, W ], but not for symmetric noise. In [5], it was shown that the largest Lyapunov exponent of the noisy Lorenz system is the same as that of the system driven constantly by the mean value of the noise, indicating that the bias of the noise plays the central role in synchronization. It was pointed out that the origin of nonchaotic behavior is that the Lorenz system driven by large enough constant perturbations is actually stable at the fixed points [5,7]. Very recently, Sanchez et al [8] analyzed the synchronization of chaotic systems by noise in an experiment with the Chua circuit, again drawing the conclusion that synchronization may be achieved only by biased noise, and not symmetric noise.So it seems that symmetric noise cannot convert a chaotic system into a nonchaotic one, so that synchronization will occur for systems in common noise. In this letter, we are going to present an example that synchronization can actually be achieved by symmetric, zero-mean common noise. In order to avoid the effect of the boundary of a system, such as the logistic map, on the realization of noise, we choose a system that can be driven by noise of any level. The cha...

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confidence: 94%

Synchronization of chaotic maps by symmetric common noise

Lai

Zhou

1998

Europhys. Lett.

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“…Gade and Basu showed that this synchronization phenomena is indeed physical in certain cases, and for synchronization of those system the randomness is not vital [19]. Moreover we have shown that, in this kind of synchronization, the distribution of the random variable is vital and there exists on-off intermittency in the boundary of synchronization region [6].…”

Section: Introductionmentioning

confidence: 87%

“…One should notice that the random variable is multiplied commonly to every particles in the summation of Eq. (19), and this leads to on-off intermittency ofS n .…”

Section: A Uncoupled Map Lattice With Homogeneous Backgroundmentioning

confidence: 99%

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Mechanism of synchronization in a random dynamical system

et al. 2001

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The mechanism of synchronization in the random Zaslavsky map is investigated. From the error dynamics of two particles, the structure of phase space was analyzed, and a transcritical bifurcation between a saddle and a stable fixed point was found. We have verified the structure of on-off intermittency in terms of a biased random walk. Furthermore, for the generalized case of the ensemble of particles, a modified definition of the size of a snapshot attractor was exploited to establish the link with a random walk. As a result, the structure of on-off intermittency in the ensemble of particles was explicitly revealed near the transition.

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“…One of the challenging problems is to understand the formation of structures, localized in both space and time, in turbulent fluid [15][16][17]. Recently lot of attention has been devoted to the role of fluctuations in onset, selection and evolution of such patterns and structures [18][19][20][21][22][23][24][25][26]. An interesting and counterintuitive observation is that the presence of noise can help sustain structures which otherwise would have been absent in an evolving dynamical system.…”

Section: Introductionmentioning

confidence: 99%

Observation of stochastic coherence in coupled map lattices

Roy

Amritkar

1997

Phys. Rev. E

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Chaotic evolution of structures in Coupled map lattice driven by identical noise on each site is studied (a structure is a group of neighbouring lattice-sites for whom values of dynamical variable follow certain predefined pattern). Number of structures is seen to follow a power-law decay with length of the structure for a given noise-strength. An interesting phenomenon, which we call stochastic coherence, is reported in which a rise of bell-shaped type in abundance and lifetimes of these structures is observed for a range of noise-strength values.Coupled map lattice (CML) model has been observed to exhibit diverse spatio-temporal patterns and structures [1], and serves as a simple model for experimental systems such as Rayleigh-Bénard convection, TaylorCouette flow, B-Z reaction etc. [2]. A major interest is in understanding formation of structures, localised in both space and time, in turbulent fluid [3]. Role of fluctuations in onset, selection and evolution of such patterns and structures has been studied in some detail [4,5]. In this communication we report a novel phenomenon observed in the dynamics of structures in a chaotically evolving one dimensional CML driven by identical noise. By a structure we mean a group of neighbouring sites whose variable-values follow certain prespecified spatial pattern. Distribution of these structures during evolution of the lattice shows that their numbers exhibit powerlaw decay with length of the structure for a given noisestrength, with an exponent which is a function of noisestrength. It is observed that average length of these structures shows a bell-shaped curve with a characteristic peak, as a function of noise-strength. Similar behaviour is observed for average lifetime of these structures during their evolution, within same range of noise values. We call this phenomenon stochastic coherence.We consider a one dimensional CML of the formwhere, is value of the variable located at site i at time t, η t is the additive noise, ε is the (nearest neighbour) coupling strength, and L is the size of the lattice. Logistic function F (x) = µx(1 − x) is used as local dynamics governing nonlinear site-evolution with nonlinearity parameter µ. We have used both openboundary conditions, and periodic-boundary conditions x t (L + i) = x t (i), for our system. For noise η t we have used a uniformly distributed random number bounded between −W and +W , where W is defined as noise-strength parameter. We define a structure as a region of space in which the dynamical variables at sites within this region follow a predefined spatial pattern [6]. To study coherence in the system we choose a spatial pattern where the difference in the values of the variables of neighbouring sites within the structure is less than a predefined small positive number say δ, i.e., |x t (i) − x t (i ± 1)| ≤ δ. We call δ the structure parameter. We look for such patterns, or coherent structures, to appear in course of evolution of the model given by Eq. 1.We now present our main observations. Values of µ, ε and L ...

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The origin of non-chaotic behavior in identically driven systems

Cited by 29 publications

References 16 publications

Synchronization of chaotic maps by symmetric common noise

Synchronization of chaotic maps by symmetric common noise

Mechanism of synchronization in a random dynamical system

Observation of stochastic coherence in coupled map lattices

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