Multiplicative stochastic processes in statistical physics

Schenzle, A.; Brand, Helmut R.

doi:10.1103/physreva.20.1628

Cited by 444 publications

(221 citation statements)

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“…In the presence of noise, this elimination is no longer straightforward since the damped modes are continuously excited by noise. This 1 This inadequacy of the linear stability analysis in the presence of multiplicative noise has been also shown for more complex model equations. 2 Thresholds corresponding to different stability criteria have been compared in [15].…”

Section: Introductionmentioning

confidence: 99%

“…First, the linear stability analysis of the x = 0 solution is misleading, as it was shown in [13]: the linear growth rates of the moments become positive for µ n < 0, µ n depending on n, whereas it can be shown that when the nonlinear term is taken into account all the moments x n go to zero for µ < 0, so that the bifurcation occurs for µ = 0 independently of n [1,14]. 1 On the contrary, if the most probable value of x is taken as an order parameter, the bifurcation from zero occurs for µ > 0 and increasing with the noise intensity. 2 As a consequence, the knowledge of the probability density function(PDF) of x as a function of µ is required in order to fully describe the system.…”

Section: Introductionmentioning

confidence: 99%

“…Stabilization, i.e. inhibition of the instability, has been reported in various situations [1,2,6]. Moreover, it has been found that multiplicative noise can modify the bifurcation diagram, generating new solutions that do not exist in the corresponding deterministic system.…”

Section: Introductionmentioning

confidence: 99%

“…(1) in the vicinity of the bifurcation point µ = 0. This reduction is no longer possible in the presence of noise, and the bifurcation from zero is shifted to larger values of µ if x 2 is taken as an order parameter in Eq.…”

Section: Introductionmentioning

confidence: 99%

“…It has been known for a long time that random fluctuations of the control parameters of a dynamical system, or multiplicative noise, may generate surprising effects, such as stabilization by noise [1,2], different types of noise induced transitions [3,4], stochastic resonance (see for example [5]), etc. Although these effects have been predicted for a great variety of systems, only a few quantitative experiments have been * Corresponding author.…”

Section: Introductionmentioning

confidence: 99%

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Effect of multiplicative noise on parametric instabilities

Berthet¹,

Petrossian

Residori

et al. 2003

Physica D: Nonlinear Phenomena

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We report a study on the effect of external multiplicative noise on parametric instabilities using two different experimental systems: an electronic RLC circuit, parametrically pumped with a voltage-variable capacitor, and surface waves generated by vertically vibrating a layer of fluid (the Faraday instability). Both systems are forced by the superposition of a sinusoidal and a noisy component. We study the statistical properties of the response of both systems to noisy parametric forcing and compare them with theoretical predictions. When the detuning from parametric resonance is such that the bifurcation in the absence of noise is supercritical, both systems behave in the same way under the influence of noise. We find that the effect of noise is twofold: on one hand, it triggers the instability before its deterministic onset under the form of oscillatory bursts; on the other hand, it inhibits the nonlinearly saturated oscillatory response above the deterministic onset. When the detuning is such that the bifurcation is subcritical, we find that the two systems behave differently. In the case of the electronic oscillator, noise mostly triggers random transitions between the two states of the bistable region that exists in the absence of noise, whereas in the surface wave experiment new states are created by noise and the bistable region is strongly enlarged.

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Section: Introductionmentioning

confidence: 99%