White Noise Perturbation Research Articles

Motion of a Rigid Body under Random Perturbation

We use stochastic analysis to study the random motion of a rigid body under a white noise perturbation. We obtain a formula for the angular velocity in an average sense and discuss the stability near a principle axis.

Large eddy simulations of spatially growing subsonic and supersonic turbulent round jets

This paper presents an application of large eddy simulations (LESs) using the filtered structure function model to spatially developing compressible round jets issuing from a perturbed upstream velocity profile close to a top hat. For centreline Mach number M = 0.9 and Reynolds number Re = 3600, the numerical solution compares satisfactorily against a forced-jet direct numerical simulation (DNS) and experimental data, both previously reported. High Reynolds number (Re = 36 000) ‘free’ jets at Mach 0.7 (case 1) and 1.4 (case 2) are studied. Here, an isotropic random white-noise perturbation is superposed on the upstream velocity. The Mach 0.7 jet has a convective Mach number of 0.35, and is weakly affected by compressibility. In this case, axisymmetric vortex rings are first shed from the nozzle and undergo alternate pairing further downstream. Then turbulence develops. The centreline velocity decay and some other statistical quantities are, in the self-similarity region, in very good agreement with previous incompressible experiments. At Mach 1.4, an impressive upstream reduction of the jet spreading rate is observed, due to an important delay of Kelvin–Helmholtz instability due to compressibility effects. Alternate pairing occurs immediately, and vortices are much more elongated in the flow direction. Further downstream, the jet becomes subsonic, develops into turbulence and spreads out again at a rate comparable with its subsonic counterpart. The potential-core length is increased by 27% from the subsonic to the supersonic case. This is in agreement with several laboratory experiments. Finally, the effects of Mach number increase upon various statistical quantities such as Reynolds stresses and radial lengthscale are studied. Results compare favourably against some experiments and temporal DNSs. From the point of view of Lumley's anisotropy invariant map evaluated on the whole physical domain, the Mach 0.7 jet is dominated by axisymmetric structures and the Mach 1.4 jet by streamwise perturbations.

The predictability of deep convection in cloud‐resolving simulations over land

AbstractCloud‐resolving models (CRMs) can be used to provide subgrid information for use in improving the representation of convection in large‐scale models. However, it is shown here that, for a 24‐hour simulation of convection over land, the total daily rainfall and instant rain rates can often be hard to predict. An ensemble of 2D CRM runs, which differ only in the initial white‐noise temperature perturbations added to the lowest 200 m, show a large spread of results. The size of the spread is shown to be dependent on the nature and strength of the forcing with the most significant differences between ensemble members when only a small number of convective cells exist in the domain. Further sensitivity experiments show that stronger forcing or increased domain size can reduce the spread of results due to an increased number of clouds. However, in all the simulations described in this paper, convection tended to organize into fewer but larger clouds during the day and, as this occurs, the ensemble members began to diverge even with a domain size of 1000 km. © Crown copyright, 2004.

Dynamic control by sinusoidal perturbation and by Gaussian noise of a system of two nonlinear oscillators: computation and experimental results.

In this paper we report numerical and experimental studies of the dynamic control of the inter-anode plasma of a double electrical discharge and of a system of two coupled nonlinear oscillators modeling this plasma. We compare the transition between chaotic dynamics and periodic dynamics induced by a sinusoidal perturbation and by small-dispersion Gaussian noise. Besides considerable differences between the effect of the two types of perturbation we also find important similarities. For small amplitude, both the sinusoidal and the white noise perturbations can induce the system to change from chaotic to regular dynamics. In the case of sinusoidal perturbation, the transition time from the chaotic to regular state has a definite duration that depends on the values of the perturbation parameters. The suppression of the perturbation has no influence on the state - the system remains in the same regular state. Subsequent reinstatement of the same type of perturbation with the same amplitude does not change the periodic state of the system but, for considerably higher amplitude, the system is switched back to its chaotic state. For moderate-amplitude sinusoidal perturbation, intermittent transitions between the chaotic and regular states is observed. Most of these predictions of the model have been observed experimentally in a system of two coupled electrical discharges. Our results suggest practical methods that can be used for controlling the discharge plasma dynamics.

Stability and robust stabilization to linear stochastic systems described by differential equations with markovian jumping and multiplicative white noise

In this paper we consider linear controlled stochastic systems subjected both to white noise disturbance and Markovian jumping. Our aim is to provide a mathematical background in order to give unified approach for a large class of problems associated to linear controlled systems subjected both to multiplicative white noise perturbations and Markovian jumping. First we prove an Itô type formula. Our result extends the result of Ref. [24], to the case when the stochastic process x(t) has not all moments bounded. Necessary and sufficient conditions assuring the exponential stability in mean square for the zero solution of a linear stochastic system with multiplicative white noise and Markovian jumping are provided. Some estimates for solutions of affine stochastic systems are derived, and necessary and sufficient conditions assuring the stochastic stabilizability and stochastic detectability are given. A stochastic version of Bounded Real Lemma is proved and several aspects of the problem of robust stabilization by state feedback for a class of linear systems with multiplicative white noise and Markovian jumping are investigated.

Asymptotic Properties of Input-Output Operators Norm Associated with Singularly Perturbed Systems with Multiplicative White Noise

In this paper, we study the problem of the asymptotic property of the norm of input-output operators related to a class of singularly perturbed stochastic linear systems. The system is under perturbation of multiplicative white noise. By using reduction order and boundary layer techniques, it is shown that the norm of the operator of the perturbed system is less than a given number $\gamma$ when the small perturbation $\varepsilon$ tends to zero if both the related norms of the reduced subsystem and the boundary layer subsystem are less than $\gamma$. Furthermore, a stabilizing robust controller is designed, which is independent of perturbation $\varepsilon$.

Stochastic resonance with explicit internal signal

A new type of internal signal stochastic resonance (ISSR) is found and investigated numerically by simulating a chemical model, which is proposed to account for chaos in the Belousov–Zhabotinsky reaction. When the system is in the dynamical region of period-1 oscillation, Gaussian white noise perturbation of the control parameter causes maxima to appear on the curve of signal-to-noise ratio versus noise intensity, showing the characteristic of stochastic resonance. The differences between this new type ISSR and the previous one studied by Hu et al. [Phys. Rev. Lett. 71, 807 (1993)] and Xin et al. [J. Chem. Phys. 111, 721 (1999)] are discussed.

Solitons in random media with long-range correlation

A statistical approach of the propagation of solitons in media with a spatially random potential is developed. Applying the inverse scattering transform several regimes are demonstrated which are determined by the mass and the velocity of the incoming soliton as well as by the correlation length of the random potential. Namely, the mass of the soliton is conserved if its initial amplitude is large enough. If the initial mass is small, then the mass decays with the length of the system. The decay rate is exponential in the case of a white noise perturbation, but it obeys a power law if the carrier wavenumber of the soliton lies in the tail of the spectrum of the potential. Furthermore, the scattered radiation propagates in a backward direction in the case of a white noise perturbation, while it propagates in a forward direction (with the same carrier wavenumber as the soliton) in the case of a coloured noise with long-range correlation.

Optimal one-step-ahead stochastic adaptive control

An optimal indirect stochastic adaptive control is obtained explicitly for linear time-varying discrete-time systems with general delay and white noise perturbation, while minimizing the variance of the output around a desired value one step ahead. The resulting controller incorporates parameter uncertainties. The solution unifies and generalizes previously known results, which become special cases. The results are compared with a certainty equivalence controller in an example.

TOPICAL REVIEW: Rigorous solutions for electromagnetic scattering from rough surfaces

A statistical approach of the propagation of solitons in media with a spatially random potential is developed. Applying the inverse scattering transform several regimes are demonstrated which are determined by the mass and the velocity of the incoming soliton as well as by the correlation length of the random potential. Namely, the mass of the soliton is conserved if its initial amplitude is large enough. If the initial mass is small, then the mass decays with the length of the system. The decay rate is exponential in the case of a white noise perturbation, but it obeys a power law if the carrier wavenumber of the soliton lies in the tail of the spectrum of the potential. Furthermore, the scattered radiation propagates in a backward direction in the case of a white noise perturbation, while it propagates in a forward direction (with the same carrier wavenumber as the soliton) in the case of a coloured noise with long-range correlation.

Delayed stochastic differential model for quiet standing.

A physiological quiet standing model, described by a delayed differential equation, subject to a white noise perturbation, is proposed to study the postural control system of human beings. It has been found that the white noise destabilizes the equilibrium state, and inertia accelerates the destabilizing process, and that the position of a person is detected and processed by the person's nervous system with a delay. This paper focuses on the analysis of Hopf bifurcation and its stability in this context. Based on the analytical predictions confirmed by numerical simulations, it has been shown that the posture of a person is controlled in such a way that possible amplitude oscillations are minimized.

LES of spatially developing 3D compressible mixing layer

The spatial development of a turbulent compressible mixing layer is investigated by means of large eddy simulation (LES). The subgrid viscosity is represented by the so-called mixed-scale model, adapted to compressible flows. Two different shock capturing schemes and three sets of inlet white-noise perturbations are investigated. The comparison between numerical and experimental results gives an overall good agreement.

CHAOS COMMUNICATION OVER NOISY CHANNELS

The problem of transmitting digital information using chaotic signals over a channel with Gaussian white noise perturbation is introduced rigorously. It is shown that discrete time base-band chaotic communication systems with discrete time Gaussian white noise in the channel are sufficiently general in this context. The optimal receiver is given explicitly in terms of conditional probabilities. For the example of chaos shift keying using iterations of the tent map, the optimal classifier is constructed explicitly. Finally, it is shown how previously published methods, in particular those based on chaos synchronization, fit into this framework.

Solitons in media with random dispersive perturbations

A statistical approach of the propagation of solitons in media with spatially random dispersive perturbations is developed. Applying the inverse scattering transform several regimes are put into evidence which are determined by the mass and the velocity of the incoming soliton and also by the correlation length of the perturbation. Namely, the mass of the soliton is almost conserved if it is initially large. If the initial mass is too small, then the mass decays with the length of the system. The decay rate is exponential in case of a white noise perturbation, but the mass will decrease as the inverse of the square root of the length if the central wave number of the soliton lies in the tail of the spectrum of the perturbation.

An iterative method for maximum entropy regularization reconstruction in MRI

Many problems in physics involve imaging objects with high spatial frequency content in a limited amount of time. The limitation of available experimental data leads to the infamous problem of diffraction limited data, which manifests itself by causing ringing in the image. This ringing is due to interference phenomena in optics and is known as the Gibbs phenomenon in engineering. In this paper, an iterative maximum entropy regularization (IMER) algorithm for magnetic resonance imaging (MRI) is developed, which produces a super-resolution and optimal signal-to-noise solution to the problem of reconstructing a source from partial Fourier transform data. This method is capable, in principle, of unlimited resolution and is robust with respect to Gaussian white noise perturbation. Comparisons of the IMER method with the conventional Fourier transform method are carried out with the real magnetic resonance data to illustrate the performance of the proposed method. © 2000 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 10, 427–431, 1999

Propagation and interaction of optical solitons in random media

The propagation of an optical soliton is studied in a fiber with randomly varying core diameter or random amplification along the fiber. The adiabatic dynamics of the soliton and radiative processes are investigated by a perturbation method based on the inverse scattering transform. The mean emitted power and the damping rate of the soliton are calculated. The interaction of solitons in random media is investigated by the Karpman–Solov’ev perturbation theory. Numerical simulations of the full nonlinear Schrodinger equation with multiplicative white- and colored-noise perturbations are performed for initial conditions corresponding to a single soliton and to two interacting solitons. The results obtained are in good agreement with the analytical estimates for white noise in the initial stage of the propagation. The numerical simulations reveal a new phenomenon in which single solitons disintegrate or split under white-noise perturbation but stabilize under colored-noise action. Finally, the existence of a bound state of two interacting solitons is observed in numerical simulations in media with colored-noise perturbation.

Accumulation of cAMP augments dynamic vagal control of heart rate.

Recent investigations in our laboratory using a Gaussian white noise perturbation technique have shown that simultaneous sympathetic stimulation augmented the gain of the transfer function from vagal stimulation frequency to heart rate response. However, the mechanism of that augmentation remains to be elucidated. In this study, we examined in anesthetized rabbits how three pharmacological interventions known to cause intracellular accumulation of cAMP affected the transfer function. Isoproterenol (0.3 microg . kg-1 . min-1 iv) increased the dynamic gain of transfer function from 7.12 +/- 0.67 to 12.4 +/- 1.21 beats . min-1 . Hz-1 (P < 0.05) without changing the corner frequency or the lag time. Similar augmentations were observed when forskolin (5 microg . kg-1 . min-1 iv) or theophylline (20 mg/kg iv) was administered under conditions of beta-adrenergic blockade. These results suggest that the accumulation of cAMP at postjunctional effector sites contributes, at least in part, to the sympathetic augmentation of the dynamic vagal control of heart rate.

Random perturbations of nonlinear oscillators

Degenerate white noise perturbations of Hamiltonian systems in $R^2$ are studied. In particular, perturbations of a nonlinear oscillator with 1 degree of freedom are considered. If the oscillator has more than one stable equilibrium, the long time behavior of the system is defined by a diffusion process on a graph. Inside the edges the process is defined by a standard averaging procedure, but to define the process for all $t > 0$ one should add gluing conditions at the vertices. Calculation of the gluing conditions is based on delicate Hormander-type estimates.

BUOYANCY-INDUCED SECONDARY VORTICES IN THE ENTRANCE SECTION OF A HORIZONTAL STRAIGHT TUBE

Developing mixed connective flow in the entrance section of a horizontal straight lube is analyzed with special emphasis on the buoyancy-induced secondary vortices. The flow is subject to a constant wall temperature. Elliptic calculations on a half cross-section grid an performed for various Grashof numbers and two Prandtl numbers. It appears that for air (Pr = 0.7) the flow exhibits a two-vortex solution for all Grashof numbers and axial positions investigated, whereas for water (Pr = 5.0), secondary flow evolves from a two-vortex solution, via a four-vortex solution toward a six-vortex solution for increasing Grashof numbers and axial distances. Calculations on a full cross-section grid reveal that these solutions are insensitive to white noise perturbations in the “initial” flow field.

Robust H∞ infinity control in the presence of stochastic uncertainty

This paper considers a robust H infinity state feedback control problem for linear uncertain systems with stochastic uncertainty. The uncertainty class considered in the paper involves uncertain multiplicative white noise perturbations which satisfy a certain variance constraint. A state feedback controller is presented which guarantees a prescribed level of disturbance attenuation for all admissible stochastic uncertainties.