scholarly journals Random response of a simple system with stochastic uncertainty and noise in parameters

2021 ◽  
Vol 15 (2) ◽  
Author(s):  
Jiří Náprstek ◽  
Cyril Fischer
Keyword(s):  
White Noise ◽  
Probability Density ◽  
Random Perturbation ◽  
Random Response ◽  
Additive White Noise ◽  
Normal Probability ◽  
Probability Densities ◽  
Parametric Noise ◽  
Truncated Normal ◽  
General Explanation

The paper is concerned with the analysis of the simultaneous effect of a random perturbation and white noise in the coefficient of the system on its response. The excitation of the system of the 1st order is described by the sum of a deterministic signal and additive white noise, which is partly correlated with a parametric noise. The random perturbation in the parameter is considered statistics in a set of realizations. It reveals that the probability density of these perturbations must be limited in the phase space, otherwise the system would lose the stochastic stability in probability, either immediately or after a certain time. The width of the permissible zone depends on the intensity of the parametric noise, the extent of correlation with the additive excitation noise, and the type of probability density. The general explanation is demonstrated on cases of normal, uniform, and truncated normal probability densities.

Exact Stationary Response Solution for Second Order Nonlinear Systems Under Parametric and External White Noise Excitations: Part II

1988 ◽  
Vol 55 (3) ◽  
pp. 702-705 ◽  
Author(s):  
Y. K. Lin ◽  
Guoqiang Cai
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Probability Distribution ◽  
White Noise ◽  
Probability Density ◽  
Detailed Balance ◽  
Stationary Probability ◽  
Probability Flow ◽  
Parametric And External Excitations ◽  
White Noises

A systematic procedure is developed to obtain the stationary probability density for the response of a nonlinear system under parametric and external excitations of Gaussian white noises. The procedure is devised by separating the circulatory portion of the probability flow from the noncirculatory flow, thus obtaining two sets of equations that must be satisfied by the probability potential. It is shown that these equations are identical to two of the conditions established previously under the assumption of detailed balance; therefore, one remaining condition for detailed balance is superfluous. Three examples are given for illustration, one of which is capable of exhibiting limit cycle and bifurcation behaviors, while another is selected to show that two different systems under two differents sets of excitations may result in the same probability distribution for their responses.

Virtual Instrument for Weak Signal Detecting on Monostable Stochastic Resonance

2011 ◽  
Vol 279 ◽  
pp. 361-366
Author(s):  
Quan Yuan ◽  
Yan Shen ◽  
Liang Chen
Keyword(s):  
Power Spectrum ◽  
White Noise ◽  
Stochastic Resonance ◽  
Virtual Instrument ◽  
Stable System ◽  
Periodic Signal ◽  
Nonlinear Phenomenon ◽  
Weak Signal ◽  
Additive White Noise ◽  
Periodic Signals

Stochastic resonance (SR) is a nonlinear phenomenon which can be used to detect weak signal. The theory of SR in a biased mono-stable system driven by multiplicative and additive white noise as well as a weak periodic signal is investigated. The virtual instrument (VI) for weak signal detecting based on this theory is designed with LabVIEW. This instrument can be used to detect weak periodic signals which meets the conditions given and can greatly improved the power spectrum of the weak signal. The results that related to different sets of parameters are given and the features of these results are in accordance with the theory of mono-stable SR. Thus, the application of this theory in the detecting of weak signal is proven to be valid.

The structure of internal intermittency in turbulent flows at large Reynolds number: experiments on scale similarity

1973 ◽  
Vol 59 (3) ◽  
pp. 537-559 ◽  
Author(s):  
C. W. Van Atta ◽  
T. T. Yeh
Keyword(s):  
Reynolds Number ◽  
Probability Density ◽  
Streamwise Velocity ◽  
Statistical Characteristics ◽  
Large Reynolds Number ◽  
Higher Moments ◽  
Scale Similarity ◽  
Probability Densities ◽  
Scale Ratio ◽  
Velocity Derivative

Some of the statistical characteristics of the breakdown coefficient, defined as the ratio of averages over different spatial regions of positive variables characterizing the fine structure and internal intermittency in high Reynolds number turbulence, have been investigated using experimental data for the streamwise velocity derivative ∂u/∂tmeasured in an atmospheric boundary layer. The assumptions and predictions of the hypothesis of scale similarity developed by Novikov and by Gurvich & Yaglom do not adequately describe or predict the statistical characteristics of the breakdown coefficientqr,lof the square of the streamwise velocity derivative. Systematic variations in the measured probability densities and consistent variations in the measured moments show that the assumption that the probability density of the breakdown coefficient is a function only of the scale ratio is not satisfied. The small positive correlation between adjoint values ofqr,land measurements of higher moments indicate that the assumption that the probability densities for adjoint values ofqr,lare statistically independent is also not satisfied. The moments ofqr,ldo not have the simple power-law character that is a consequence of scale similarity.As the scale ratiol/rchanges, the probability density ofqr,levolves from a sharply peaked, highly negatively skewed density for large values of the scale ratio to a very symmetrical distribution when the scale ratio is equal to two, and then to a highly positively skewed density as the scale ratio approaches one. There is a considerable effect of heterogeneity on the values of the higher moments, and a small but measurable effect on the mean value. The moments are roughly symmetrical functions of the displacement of the shorter segment from the centre of the larger one, with a minimum value when the shorter segment is centrally located within the larger one.

scholarly journals Computation of Power Loss in Likelihood Ratio Tests for Probability Densities Extended by Lehmann Alternatives

2018 ◽  
Author(s):  
Lucas Gallindo
Keyword(s):  
Probability Density ◽  
Likelihood Ratio ◽  
Power Loss ◽  
Likelihood Ratio Tests ◽  
Probability Densities ◽  
Lehmann Alternative

We compute the loss of power in likelihood ratio tests when we test the original parameter of a probability density extended by the first Lehmann alternative.

scholarly journals A search for a stochastic archetype of quantum probability

2021 ◽  
Author(s):  
Tim C Jenkins
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Probability Density ◽  
Random Variables ◽  
Quantum Probability ◽  
Interference Term ◽  
Mathematical Foundation ◽  
Quantum Process ◽  
Hidden Variable ◽  
Probability Densities

Abstract Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory. In recent years, Man’ko and coauthors have successfully reconciled quantum and classic probability using a symplectic tomographic model. Nevertheless, there remains an unexplained coincidence in quantum mechanics, namely, that mathematically, the interference term in the squared amplitude of superposed wavefunctions gives the squared amplitude the form of a variance of a sum of correlated random variables, and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classic probability directly. The properties that would need to be satisfied for this to be the case are identified, and a generic hidden variable that satisfies them is found that would be present everywhere, transforming into a process-specific variable wherever a quantum process is active. Uncovering this variable confirms the possibility that it could be the stochastic archetype of quantum probability.

scholarly journals A search for a stochastic archetype of quantum probability

2021 ◽  
Author(s):  
Tim C Jenkins
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Probability Density ◽  
Random Variables ◽  
Quantum Probability ◽  
Interference Term ◽  
Mathematical Foundation ◽  
Classical Probability ◽  
Quantum Process ◽  
Probability Densities

Abstract Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory. In recent years Man’ko and co-authors have successfully reconciled quantum and classical probability using a symplectic tomographic model. Nevertheless, there remains an unexplained coincidence in quantum mechanics, namely that mathematically the interference term in the squared amplitude of superposed wavefunctions has the form of a variance of a sum of correlated random variables and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classical probability directly. The properties that would need to be satisfied for this to be the case are identified, and a generic variable that satisfies them is found that would be present everywhere, transforming into a process-specific variable wherever a quantum process is active. This hidden generic variable appears to be such an archetype.

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