Coupled map lattices as models of deterministic and stochastic differential delay equations

Losson, Jérôme; Mackey, Michael C.

doi:10.1103/physreve.52.115

Cited by 21 publications

(10 citation statements)

References 30 publications

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“…Thus the second eigenvalue of the product of the SPTOs reflects the contribution of the past activity to the present density as shown in Eq. (11). The modulus of the second eigenvalue shows an overall decreasing trend with increasing number of impulses and its angle alternates between 0 and π in the 2:1 stochastic phase-locking region.…”

Section: Numerical Resultsmentioning

confidence: 88%

“…Markov operators have a wide range of applications; for example, they can be applied to analyses of coupled map lattices [11,12], nonlinear oscillators driven by stochastic inputs [5,6,13,14], neuronal dynamics [15][16][17][18][19][20][21], and neural networks [22]. * yamanobe@med.hokudai.ac.jp Markov operators can be constructed using stochastic kernels, which are transition densities corresponding to the given stochastic differential equations.…”

Section: Introductionmentioning

confidence: 99%

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Stochastic phase transition operator

Yamanobe

2011

Phys. Rev. E

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In this study a Markov operator is introduced that represents the density evolution of an impulse-driven stochastic biological oscillator. The operator's stochastic kernel is constructed using the asymptotic expansion of stochastic processes instead of solving the Fokker-Planck equation. The Markov operator is shown to successfully approximate the density evolution of the biological oscillator considered. The response of the oscillator to both periodic and time-varying impulses can be analyzed using the operator's transient and stationary properties. Furthermore, an unreported stochastic dynamic bifurcation for the biological oscillator is obtained by using the eigenvalues of the product of the Markov operators.

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Section: Numerical Resultsmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Stochastic phase transition operator

Yamanobe

2011

Phys. Rev. E

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“…From a modeling perspective, the interaction of differential delay with noise leads one to study stochastic effects on infinite dimensional systems. In approximating the phase space, finite dimensional formulations based on coupled map lattices were use to initially examine approximations to steady state densities [16]. Here the authors employ a high dimensional Perron-Frobenius formulation to extract the density in the thermodynamic limit.…”

Section: Introductionmentioning

confidence: 99%

Noise Induced Switching in Delayed Systems

Schwartz¹,

Carr²,

Billings³

et al. 2012

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“…To properly understand the significance of the two quantities that we intend to calculate (information capacity and locking time), and understand how these two quantities can compete with each other, it is instructive to review the properties of the isolated tent map S(x), and, by simple a generalization, of the CML with = 0 (more information can be found in [5][6][7]). More specifically, we have that the dynamics of the tent map alone are characterized by a Lyapunov exponent equal to ln a.…”

mentioning

confidence: 99%

“…Thus, for 0 < a < 1, all orbits of the map converge to the unique fixed point x * = 0 independent of the value of the initial conditions, while for a = 1, all initial points of the map are fixed points. When 1 < a ≤ 2, the dynamics of the tent map switches abruptly to a chaotic regime (see Behind the chaotic behaviour of the temporal trajectory, the underlying regularity of the dynamics apparent in the banded structure of the bifurcation diagram is the signature of a property called statistical periodicity which can be observed in the density distribution of an ensemble of tent maps [2,6,7,9]. Specifically, for 2 1/2 1/(n+1) < a ≤ 2 1/2 1/n , the densities of the tent map have periodicity with period T = (n + 1) for n = 1, 2, · · · .…”

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confidence: 99%

Information capacity and pattern formation in a tent map network featuring statistical periodicity

2003

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We provide quantitative support to the observation that lattices of coupled maps are "efficient" information coding devices. It has been suggested recently that lattices of coupled maps may provide a model of information coding in the nervous system because of their ability to create structured and stimulus-dependent activity patterns which have the potential to be used for storing information. In this paper, we give an upper bound to the effective number of patterns that can be used to store information in the lattice by evaluating numerically its information capacity or information rate as a function of the coupling strength between the maps. We also estimate the time taken by the lattice to establish a limiting activity pattern.

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Coupled map lattices as models of deterministic and stochastic differential delay equations

Cited by 21 publications

References 30 publications

Stochastic phase transition operator

Stochastic phase transition operator

Noise Induced Switching in Delayed Systems

Information capacity and pattern formation in a tent map network featuring statistical periodicity

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