Multiplicative Gaussian White Noise Research Articles

An efficient method for solving high-dimension stationary FPK equation of strongly nonlinear systems under additive and/or multiplicative white noise

Engineering structures may suffer from drastic nonlinear random vibrations in harsh environments. Random vibration has been extensively studied since 1960s, but is still an open problem for large-scale strongly nonlinear systems. In this paper, a random vibration analysis method based on the Neural Networks for large-scale strongly nonlinear systems under additive and/or multiplicative Gaussian white noise (GWN) excitations is proposed. In the proposed method, the high-dimensional steady-state Fokker–Planck-Kolmogorov (FPK) equation governing the state’s probability density function (PDF) is firstly reduced to low-dimensional FPK equation involving only the interested state variables, generally one or two dimensions. The equivalent drift coefficients (EDCs) and diffusion coefficients (EDFs) in the low-dimensional FPK equation are proven to be the conditional mean of the coefficients given the interested variables. Furthermore, it is shown that the conditional mean can be optimally estimated by regression. Subsequently, the EDCs and EDFs, as functions of the retained variables, are approximated by the semi-analytical Radial Basis Functions Neural Networks trained with samples generated by a few deterministic analyses. Finally, the Physics Informed Neural Network is employed to solve the reduced steady-state FPK equation, and the PDF of the system responses is obtained. Four typical examples under additive and/or multiplicative GWN excitations are used to examine the accuracy and efficiency of the proposed method by comparing its results with the exact solution (if available) or Monte Carlo simulations. The proposed method also exhibits greater accuracy than the globally-evolving-based generalized density evolution equation scheme, a similar method of its kind, especially for strongly nonlinear systems. Notably, even though steady-state systems are applied in this paper, there is no obstacle to extending the proposed framework to transient systems.

Correlated non-Gaussian and Gaussian noises induced transition and mean first-passage time in a fractional-order bistable system

The transition and mean first-passage time(MFPT) in a fractional-order bistable system, excited by multiplicative non-Gaussian noise and additive Gaussian white noise, are investigated. Utilizing the fractional-order minimum mean square error criterion and the path integral approach, we derive approximate expressions for both the steady-state probability distribution function and the MFPT. The results show that the noise correlation strength, non-Gaussian parameter, Gaussian noise strength and fractional order can induce phase transitions. The MFPT exhibits a peak as a function of non-Gaussian noise intensity, demonstrating the noise-enhanced stability phenomenon, while the MFPT displays a consistent, monotonic relationship as a function of Gaussian white noise intensity. Numerical simulations are conducted to validate theoretical results, which show a reasonably high degree of consistency.

Impact of white noise in artificial neural networks trained for classification: Performance and noise mitigation strategies.

In recent years, the hardware implementation of neural networks, leveraging physical coupling and analog neurons has substantially increased in relevance. Such nonlinear and complex physical networks provide significant advantages in speed and energy efficiency, but are potentially susceptible to internal noise when compared to digital emulations of such networks. In this work, we consider how additive and multiplicative Gaussian white noise on the neuronal level can affect the accuracy of the network when applied for specific tasks and including a softmax function in the readout layer. We adapt several noise reduction techniques to the essential setting of classification tasks, which represent a large fraction of neural network computing. We find that these adjusted concepts are highly effective in mitigating the detrimental impact of noise.

On a Class of Stochastic Damped Wave Equation

The present work considers a wave equation with multiplicative Gaussian white noise and weak dissipative term on a bounded domain. We first give a theorem including the local existence of mild solutions. An energy bound and a differential inequality are used to give sufficient conditions that provide the blow-up of mild local solutions of the stochastic wave equation. The paper's main contribution comes from handling a multiplicative noise and a general source term contrary to the articles that exist in the literature.

On Dirac structure of infinite-dimensional stochastic port-Hamiltonian systems

Stochastic infinite-dimensional port-Hamiltonian systems (SPHSs) with multiplicative Gaussian white noise are considered. In this article we extend the notion of Dirac structure for deterministic distributed parameter port-Hamiltonian systems to a stochastic ones by adding some additional stochastic ports. Using the Stratonovich formalism of the stochastic integral, the proposed extended interconnection of ports for SPHSs is proved to still form a Dirac structure. This constitutes our main contribution. We then deduce that the interconnection between (stochastic) Dirac structures is again a (stochastic) Dirac structure under some assumptions. These interconnection results are applied on a system composed of a stochastic vibrating string actuated at the boundary by a mass–spring system with external input and output. This work is motivated by the problem of boundary control of SPHSs and will serve as a foundation to the development of stabilizing methods.

Cross-correlated sine-Wiener noises-induced transitions in a tumor growth system

Cross-correlated sine-Wiener (CCSW) noises-induced transitions in a stochastic tumor growth dynamical system are studied. Utilizing Fox’s approach and Hänggi ansatz, the role of CCSW noises on dynamical systems is equivalently transformed into that of multiplicative Gaussian white noise, effectively solving the dynamics puzzles caused by CCSW noises and facilitating the intractable theoretical investigation related to CCSW noises. The impacts of CCSW noises on transitions between the high-population (undesirable) basin B1 and low-population (desirable) basin B2, available to describe the process of tumor elimination and recurrence, are examined theoretically and numerically from the perspective of basin stability through the first escape probability (FEP) and mean first exit time (MFET). Lower FEP or longer MFET characterizing stronger stability reflects a lower likelihood of transitions. Four stability indexes, δE, δSBA, δR, and δSBA∗, are considered to fully explore the role of CCSW noises on transitions between B1 and B2 to reveal its contribution during different tumor treatment stages. We discover that (i) the CCSW noise intensities exert a dual effect on transitions: for λ≤0 (cross-correlation strength), enhanced noise intensities lead to smaller δE (/δSBA) or larger δR (or smaller δSBA∗), acting to weaken the undesirable or desirable basin stability and favor transitions; for λ>0, critical noise intensities that maximize δE (/δSBA) or minimize δR (or maximize δSBA∗) exist, indicating that noise-enhanced stability occurs, which can be reinforced by enhancing λ. (ii) Increasing the correlation time τ leads to weaker stability and higher transition probability, while λ acts oppositely in inducing transitions. Therefore, to achieve better tumor eradication efficacy, destabilizing the undesirable basin B1 by enhancing the negatively correlated SW noise intensities and increasing τ is urgent. During the stage of suppressing tumor recurrence, the unsafe parameter domain that causes a larger δR, implying a high risk of re-entry into B1, should be eschewed. Instead, the positively correlated critical noise intensities with a larger λ and smaller τ for stabilizing the desirable basin B2 are preferable.

The transient characteristics of an underdamped periodic potential system excited by two different kinds of noise

In this paper, the transient characteristics of an underdamped periodic potential system excited by multiplicative Gaussian white noise and additive Lévy noise are studied in terms of the mean first-passage time(MFPT) and the probability density function(PDF) of the first-passage time. The second-order underdamped periodic potential system is equivalent to two first-order stochastic differential equations. The MFPT was obtained by averaging the response value of the first-passage time, and the PDF image of the first-passage time was drawn under different parameter values. It was found that the increase in damping coefficient and stability index will inhibit the particle crossing, while the increase of multiplicative noise intensity, additive noise intensity and skewness parameter will promote the particle crossing to a certain extent.

Probabilistic response analysis of nonlinear vibro-impact systems with two correlated Gaussian white noises

The response distributions of a dynamic system under the effect of correlation noise not only demonstrated a Gaussian symmetric distribution, but also exhibited a non-Gaussian skewed distribution. Thus, correlation noise can induce more complex dynamic phenomena in the response of a nonlinear system. The purpose of this study is to explore the stochastic response and bifurcation of nonlinear vibro-impact systems driven by correlated additive and multiplicative Gaussian white noises using the path integral (PI) method. First, we derived an equivalent correlated stochastic system without collision conditions in the new ancillary phase space using the non-smooth Ivanov transform method. We then used Novikov’s theorem to derive an effective Fokker–Planck–Kolmogorov equation for the equivalent correlated stochastic system. Second, we used PI technology, which combines the three-point Gauss–Legendre quadrature rule to solve the effective Fokker–Planck–Kolmogorov equation and obtained the probability density functions (PDFs) of the equivalent correlated stochastic system. Subsequently, we obtained the PDFs of the original vibro-impact stochastic system using the relationship between the initial state space and the new auxiliary phase space. Finally, the proposed method was used to investigate the transient and stationary responses of a Duffing-Van der Pol Vibro-impact system perturbed by additive and multiplicative random noises. In addition, this technique can also assess the effects of restitution coefficients and barrier position on stochastic P-bifurcation. The accuracy of this method was verified by comparing it with a Monte Carlo simulation.

An integrate-and-fire approach to Ca2+ signaling. Part I: Renewal model

In computational neuroscience integrate-and-fire models capture the spike generation by a subthreshold dynamics supplemented by a simple fire-and-reset rule; they allow for a numerically efficient and analytically tractable description of stochastic single cell as well as network dynamics. Stochastic spiking is also a prominent feature of Ca2+ signaling which suggests to adopt the integrate-and-fire approach for this fundamental biophysical process. The model introduced here consists of two components describing 1) activity of clusters of inositol-trisphosphate receptor channels and 2) dynamics of the global Ca2+ concentrations in the cytosol. The cluster dynamics is given in terms of a cyclic Markov chain, capturing the puff, i.e., the punctuated release of Ca2+ from intracellular stores. The cytosolic Ca2+ concentration is described by an integrate-and-fire dynamics driven by the puff current. For the cyclic Markov chain we derive expressions for the statistics of the interpuff interval, the single-puff strength and the puff current assuming constant cytosolic Ca2+. The latter condition is often well approximated because cytosolic Ca2+ varies much slower than the cluster activity does. Furthermore, because the detailed two-component model is numerically expensive to simulate and difficult to treat analytically, we develop an analytical framework to approximate the driving puff current of the stochastic cytosolic Ca2+ dynamics by a temporally uncorrelated Gaussian noise. This approximation reduces our two-component system to an integrate-and-fire model with a nonlinear drift function and a multiplicative Gaussian white noise, a model that is known to generate a renewal spike train, i.e., a point process with statistically independent interspike intervals. The model allows for fast numerical simulations, permits to derive analytical expressions for the rate of Ca2+ spiking and the coefficient of variation of the interspike interval, and to approximate the interspike interval density and the spike train power spectrum. Comparison of these statistics to experimental data is discussed.

STOCHASTIC STABILITY AND PARAMETRIC CONTROL IN A GENERALIZED AND TRI-STABLE VAN DER POL SYSTEM WITH FRACTIONAL ELEMENT DRIVEN BY MULTIPLICATIVE NOISE

The stochastic transition behavior of tri-stable states in a fractional-order generalized Van der Pol (VDP) system under multiplicative Gaussian white noise (GWN) excitation is investigated. First, according to the minimal mean square error (MMSE) concept, the fractional derivative can be equivalent to a linear combination of damping and restoring forces, and the original system can be simplified into an equivalent integer-order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and based on singularity theory, the critical parameters for stochastic [Formula: see text]-bifurcation of the system are found. Finally, the properties of stationary PDF curves of the system amplitude are qualitatively analyzed by choosing the corresponding parameters in each sub-region divided by the transition set curves. The consistency between numerical results obtained by Monte-Carlo simulation and analytical solutions verified the accuracy of the theoretical analysis process and the method used in this paper has a direct guidance in the design of fractional-order controller to adjust the system behavior.

Unsaturated piecewise bistable stochastic resonance with three kinds of asymmetries driven by multiplicative and additive noise

The performance of an unsaturated piecewise bistable stochastic resonance (UPBSR) with three kinds of asymmetries driven by multiplicative and additive Gaussian white noise is investigated. Firstly, by introducing an asymmetric factor into UPBSR potential, three kinds of asymmetries are constructed, i.e., well-depth asymmetry (WDA), well-width asymmetry (WWA), and both well-depth and well-width asymmetry (BDWA). Then, by deriving the output signal-to-noise ratio (SNR), the behavior of asymmetric UPBSR under two independent noises and correlated noises is analyzed. Enhancement ability and noise immunity are adopted as SR evaluation index. Moreover, the mean first passage time (MFPT), is adopted to discuss the transient properties of UPBSR system. Finally, for SNR analysis, the results show that asymmetric SR outperforms symmetric SR when the asymmetric factor is in the appropriate range, and the BDWA dominates the three cases. When subject to two independent noises, multiplicative noises always shows a suppression on SR. However, when the two noises are correlated, the effect of each parameter on SNR becomes complicated, the same parameter may have the exact opposite effect on SNR in different situations. In addition, the correlation intensity λ play an important role in SR. It can excite SR to appear as a double peak, which is different from the conventional SR. Despite the effects of individual parameters on SNR are highly coupled, the introduction of three kinds of asymmetries and correlated noise can improve the performance of SR. As for MFPT analysis, when driven by independent multiplicative and additive noise, MFPT exhibits similar monotonically decreasing behavior in various cases. However, the noise-enhanced stability (NES) phenomenon is excited when driven by the correlated multiplicative and additive noise. In the case of WDA and BDWA, both appropriate λ and noise intensity ratio R favor the NES phenomenon, while in the case of WWA, the NES phenomenon is suppressed.

Transition behaviors of system energy in a bi-stable van Ver Pol oscillator with fractional derivative element driven by multiplicative Gaussian white noise

The stochastic P-bifurcation behavior of system energy in a bi-stable Van der Pol oscillator with fractional damping under multiplicative Gaussian white noise excitation is investigated. Firstly, using the principle of minimal mean square error, the non-linear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order Van der Pol system. Secondly, the system amplitude?s stationary probability density function is obtained by stochastic averaging. Then, according to the singularity theory, the critical parametric conditions for the system amplitude?s stochastic P-bifurcation are found. Finally, the types of the system?s stationary probability density function curves of amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical results and the numerical results obtained from Monte-Carlo simulation verifies the theoretical analysis in this paper, and the method used in this paper can directly guide the design of the fractional-order controller to adjust the response of the system.

Hypoellipticity and the Mori–Zwanzig formulation of stochastic differential equations

We develop a thorough mathematical analysis of the effective Mori–Zwanzig (EMZ) equation governing the dynamics of noise-averaged observables in stochastic differential equations driven by multiplicative Gaussian white noise. Building upon recent work on hypoelliptic operators, we prove that the EMZ memory kernel and fluctuation terms converge exponentially fast in time to a unique equilibrium state that admits an explicit representation. We apply the new theoretical results to the Langevin dynamics of a high-dimensional particle system with smooth interaction potential.

Transient response of the time-delay system excited by Gaussian noise based on complex fractional moments.

In this paper, the transient response of the time-delay system under additive and multiplicative Gaussian white noise is investigated. Based on the approximate transformation method, we convert the time-delay system into an equivalent system without time delay. The one-dimensional Ito stochastic differential equation with respect to the amplitude response is derived by the stochastic averaging method, and Mellin transformation is utilized to transform the related Fokker-Planck-Kolmogorov equation in the real numbers field into a first-order ordinary differential equation (ODE) of complex fractional moments (CFM) in the complex number field. By solving the ODE of CFM, the transient probability density function can be constructed. Numerical methods are used to ascertain the effectiveness of the CFM method, the effects of system parameters on system response and the level of error vary with time as well as noise intensity are investigated. In addition, the CFM method is first implemented to analyze transient bifurcation, and the relation between CFM and bifurcation is discussed for the first time. Furthermore, the imperfect symmetry property appear on the projection map of joint probability density function.

A Stochastic Theory of the Hierarchical Clustering. II. Halo Progenitor Mass Function and Large-scale Bias

Abstract We generalize the stochastic theory of hierarchical clustering presented in Paper I by Lapi & Danese to derive the (conditional) halo progenitor mass function and the related large-scale bias. Specifically, we present a stochastic differential equation that describes fluctuations in the mass growth of progenitor halos of given descendant mass and redshift, as driven by a multiplicative Gaussian white noise involving the power spectrum and the spherical collapse threshold of density perturbations. We demonstrate that, as cosmic time passes, the noise yields an average drift of the progenitors toward larger masses, which quantitatively renders the expectation from the standard extended Press and Schechter (EPS) theory. We solve the Fokker–Planck equation associated with the stochastic dynamics, and obtain as an exact, stationary solution, the EPS progenitor mass function. Then we introduce a modification of the stochastic equation in terms of a mass-dependent collapse threshold modulating the noise, and solve analytically the associated Fokker–Planck equation for the progenitor mass function. The latter is found to be in excellent agreement with the outcomes of N-body simulations; even more remarkably, this is achieved with the same shape of the collapse threshold used in Paper I to reproduce the halo mass function. Finally, we exploit the above results to compute the large-scale halo bias, and find it in pleasing agreement with the N-body outcomes. All in all, the present paper illustrates that the stochastic theory of hierarchical clustering introduced in Paper I can describe effectively not only halos’ abundance, but also their progenitor distribution and their correlation with the large-scale environment across cosmic times.

The analysis of particles behavior under a delayed tristable system driven by multiplicative and additive noises

In this paper, the motion of Brownian particles driven by a delayed tristable system with multiplicative and additive Gaussian white noise is mainly studied. First, the effective potential function and stable state probability density function (PDF) are derived by using the theory of small-time delay approximation and the approximate Fokker–Planck equation (FPE), and the expression of mean first-passage times (MFPTs) is obtained by using the definition of the MFPTs and the steepest descent method. Then, the effects of the parameters which include noise intensities of multiplicative and additive noise, and correlation strength between two noises, and time delay, and the strength of time-delayed feedback on PDF and MFPTs are analyzed. Results demonstrate that the additive noise intensity has a more profound influence on PDF than the multiplicative noise intensity. The non-equilibrium phase transition of the system can be produced by the correlation strength of noises. In addition, in the behavior of the MFPTs, we can observe the noise-enhanced stability (NES) phenomenon induced by multiplicative noise intensity. Besides, delayed time plays an important role in MFPTs. Moreover, MFPT [Formula: see text] (stands for the Brownian particle moving from the left well to the middle well) is greater than [Formula: see text] (stands for the Brownian particle moving from the middle well to the left one).

The stochastic P-bifurcation analysis of the impact system via the most probable response

Abstract The most probable response, which acts as a deterministic geometric tool for the response analysis of stochastic systems, offers an attractive alternative to traditional methods for analyzing the P-bifurcation of the stochastic impact system. Specifically, the stochastic impact system perturbed by multiplicative Gaussian white noises is considered to research the P-bifurcations under the most probable response angle. Firstly, the non-smooth coordinate transformation of state variables is applied to convert the impact system into an equivalent system without the velocity jump. Then, the stochastic averaging method of energy envelope is exploited to the transformed system and the most probable response is obtained by the combination of the Fokker-Planck equation and the extreme value theory. Finally, based on the most probable response, the bifurcation behavior of the stochastic impact system is investigated qualitatively from a new perspective. It is found that the stochastic P-bifurcation can be induced or suppressed by modulating the noise intensity D 2 or the restitution coefficient r in the stochastic impact system. However, there is no influence of the noise intensity D 1 on the most probable response of the stochastic impact system. Therefore, the noise intensity D 1 will not trigger the P-bifurcation of the stochastic impact system. Meanwhile, the validity of the proposed procedure is verified by numerical simulation.

Stochastic P-bifurcations of a noisy nonlinear system with fractional derivative element

This paper investigates the stochastic P-bifurcation (SPB) of a fractionally damped oscillator subjected to additive and multiplicative Gaussian white noise. Variable transformation and the stochastic averaging technique are applied to derive the expression of probability density function (PDF) of the system response. Critical conditions of the stochastic bifurcation induced by system parameters are presented based on the change in the number of extreme points of the probability density function. Numerical results are given to show the effectiveness of the proposed approach. Stochastic P-bifurcations for additive and multiplicative noise are studied in detail according to the critical conditions. Stochastic P-bifurcation (SPB) for additive noise can be investigated by the critical conditions. SPB induced by the fractional order is described in the bifurcation plane, see Fig. a. Figure b shows that the probability density function (PDF) for (0.1, 0.005) ∈ R2 has one peak, while the PDF for (0.8, 0.005) ∈ R2 has two peaks. It implies that the change of the fractional order leads to the variation of the structure of the PDF, which indicates the occurrence of the SPB.

Stochastic transition behaviors in a generalized Van der Pol oscillator with fractional damping under Gaussian white noise

The stochastic P-bifurcation behavior of bi-stability in a generalized Van der Pol oscillator with a fractional damping under multiplicative Gaussian white noise excitation is investigated. Firstly, using the principle of minimal mean square error, the non-linear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order Van der Pol system. Secondly, the system amplitude?s stationary probability density function is obtained by stochastic averaging. Then according to the singularity theory, the critical parametric conditions for the sys-tem amplitude?s stochastic P-bifurcation are found. Finally, the types of the system?s stationary probability density function curves of amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical results and the numerical results obtained from Monte-Carlo simulation verifies the theoretical analysis in this paper and the method used in this paper can directly guide the design of the fractional order controller to adjust the response of the system.

Covariant formulation of nonlinear Langevin theory with multiplicative Gaussian white noises

The multi-dimensional non-linear Langevin equation with multiplicative Gaussian white noises in Ito's sense is made covariant with respect to non-linear transform of variables. The formalism involves no metric or affine connection, works for systems with or without detailed balance, and is substantially simpler than previous theories. Its relation with deterministic theory is clarified. The unitary limit and Hermitian limit of the theory are examined. Some implications on the choices of stochastic calculus are also discussed.