Stationary Probability Density Function Research Articles

Impact of Non-Gaussian Noise and Time Delay on Stability and Stochastic Resonance for a FitzHugh-Nagumo Neural System Subjected to a Multiplicative Periodic Signal

In this paper, we focus on the investigations on the stochastic stability and the stochastic resonance (SR) phenomena for a FitzHugh-Nagumo system with time delay induced by a multiplicative non-Gaussian colored noise and an additive Gaussian colored noise. By use of the fast descent method, the unified colored noise approximation and the two-state theory for the SR, the stationary probability density function (SPDF) and the signal-to-noise ratio (SNR) caused by different noise terms and time delay are explored. The investigation results indicate that the two noise intensities, time delay and the departure parameter from the Gaussian noise can all reduce the probability density around the two stable states and destroy the stability of the neural system; while the two noise correlation times [Formula: see text] and [Formula: see text] can both improve the probability density around both stable states and reinforce the biological stability of the neural system. As regards the SNR, it is found that the two noise intensities and the departure coefficient can all weaken the SR effect, while time delay [Formula: see text] and the correlation time [Formula: see text] of the multiplicative noise will always magnify the SR phenomenon. It is worth to mention that the correlation time [Formula: see text] of the additive noise can stimulate the SR effect, but not alter the maximum of the SNR.

Response and Bifurcation of Fractional Duffing Oscillator under Combined Recycling Noise and Time-Delayed Feedback Control

Response and bifurcation of fractional Duffing oscillator under recycling noise and time-delayed feedback control are investigated. Firstly, based on the principle of the minimum mean square error and small time-delayed approximation and linearize the cubic stiffness term, the fractional derivative is equivalent to a linear combination of damping and restoring forces, and the original system is simplified into an equivalent integer order system. Secondly, the Itô differential equations and one-dimensional Markov process are obtained according to stochastic averaging method, and the stochastic stability and stochastic bifurcation of the system are analyzed. Lastly, through joint probability density function diagram and the stationary probability density function diagram, the stochastic bifurcation behavior of system under the different time-delay, fractional order and noise intensity are discussed respectively, the validity of the theory and the occurrence of bifurcation phenomenon are verified.

An Improved Stochastic Averaging on the Piezoelectric Vibrational Harvester Model with Stiffness and Inertia Nonlinearities

In this paper, an improved stochastic averaging method is introduced to solve a monostable piezoelectric vibration energy harvester (VEH) model subject to basal Gaussian white and colored noises. First, the electric voltage is expressed as perturbations to the linear stiffness term and the damping term of the vibrational equation. Second, the original model is simplified into an Itô differential equation by the improved stochastic averaging method. Third, stationary solution of the Fokker–Plank–Kolmogorov function driven from the Itô equation is obtained, so that the stationary probability density functions (PDFs) of the displacement, velocity, and electrical voltage are obtained. With the help of the Monte Carlo, the effectiveness of these theoretical predictions is verified. Finally, the impacts of the nonlinear stiffness coefficient and nonlinear inertia coefficient on the responses are compared. It is found that the stiffness nonlinearity has stronger effect on the outputn electric voltage than the inertia nonlinearity.

Design and analysis of a galloping energy harvester with V-shape spring structure under Gaussian white noise

This paper presents the design of an innovative galloping energy harvester (GEH) that incorporates an elastic structure to reconstructed the traditional GEH. The mathematical model of the GEH is derived based on the Euler–Bernoulli beam theory and Kirchhoff’s law. Due to the nonlinear properties of the model, the partial linearization technique and Fokker–Planck–Kolmogorov equation are used to further explore the joint probability density function (PDF) and the stationary PDF of displacement and velocity. Subsequently, Monte Carlo simulation is used to illustrate the behavior of the theoretical results. The simulation results show that, under different conditions, the stationary PDF exhibits double-peak or single-peak shape, which is caused by the compression or elongation state. Meanwhile, higher wind speeds correspond to larger displacement amplitudes. Then, the stationary PDF of displacement amplitude is obtained using the stochastic average method, and the expressions for mean square voltage and mean output power are derived. Numerical results indicate that when the wind speed is below the critical wind speed for galloping phenomenon, the vibration of the GEH is mainly driven by noise. When the wind speed exceeds the critical wind speed, the GEH is mainly affected by the wind speed. Higher wind speeds correspond to higher mean output power. An optimal electromechanical coupling coefficient and the ratio of the mechanical and electrical can be derived based on the GEH parameters when galloping occurs.

A Markov chain approximation of switched Fokker–Planck equations for a model of on–off intermittency in the postural control during quiet standing

The intermittent on–off switching of feedback control is considered as a major mechanism of postural stabilization during human quiet standing, which can be modeled by switched-type hybrid stochastic delay differential equations with unstable subsystems. Dynamics of the model can also be described by the corresponding switched-type Fokker–Planck (FP) equations. Here, we develop a comprehensive numerical recipe to simulate switched-type FP equations in the case that the probability current is conserved at the switching boundary, as is the case for noise-free models exhibiting C0-continuity for solutions at the boundary, but in a way extendable to cases with discontinuous jumps. Specifically, the FP equations are approximated by a finite state Markov chain model using the finite element method. Then, dynamics of the Markov chain model, including time evolution of probability density function (PDF), stationary PDF, and power spectrum of postural sway are analyzed. We further investigate how the stationary PDF alters as values of important parameters of the model change. Dynamics of the Markov chain model are compared with Monte Carlo-based dynamics of the model, by which the developed numerical recipe is validated. The obtained Markov chain model forms a basis of our future investigations of the intermittent postural control as a Markov decision process.

Fractional stochastic vibration system under recycling noise

The fractional stochastic vibration system is quite different from the traditional one, and its application potential is enormous if the noise can be deployed correctly and the connection between the fractional order and the noise property is unlocked. This article uses a fractional modification of the well-known van der Pol oscillator with multiplicative and additive recycling noises as an example to study its stationary response and its stochastic bifurcation. First, based on the principle of the minimum mean square error, the fractional derivative is equivalent to a linear combination of damping and restoring forces, and the original system is simplified into an equivalent integer order system. Second, the Itô differential equations and One-dimensional Markov process are obtained according to the stochastic averaging method, using Oseledec multiplicative ergodic theorem and maximal Lyapunov exponent to judge local stability, and judging global stability is done by using the singularity theory. Lastly, the stochastic D-bifurcation behavior of the model is analyzed by using the Lyapunov exponent of the dynamical system invariant measure, and the stationary probability density function of the system is solved according to the FPK equation. The results show that the fractional order and noise property can greatly affect the system’s dynamical properties. This paper offers a profound, original, and challenging window for investigating fractional stochastic vibration systems.

Time-dependent probability density function for partial resetting dynamics

Stochastic resetting is a rapidly developing topic in the field of stochastic processes and their applications. It denotes the occasional reset of a diffusing particle to its starting point and effects, inter alia, optimal first-passage times to a target. Recently the concept of partial resetting, in which the particle is reset to a given fraction of the current value of the process, has been established and the associated search behaviour analysed. Here we go one step further and we develop a general technique to determine the time-dependent probability density function (PDF) for Markov processes with partial resetting. We obtain an exact representation of the PDF in the case of general symmetric Lévy flights with stable index . For Cauchy and Brownian motions (i.e. ), this PDF can be expressed in terms of elementary functions in position space. We also determine the stationary PDF. Our numerical analysis of the PDF demonstrates intricate crossover behaviours as function of time.

A Dynamic Behavior Analysis of a Rolling Mill’s Main Drive System with Fractional Derivative and Stochastic Disturbance

Taking the random factors into account, a fractional main drive system of a rolling mill with Gaussian white noise is developed. First, the potential deterministic bifurcation is investigated by a linearized stability analysis. The results indicate that the fractional order changes the system from a stable point to a limit cycle with symmetric phase trajectories. Then, the stochastic response is obtained with the aid of the equivalent transformation of the fractional derivative and stochastic averaging methods. It is found that the joint stationary probability density function appears to have symmetric distribution. Finally, the influence of the fractional order and noise intensity on system dynamics behavior is discussed. The study is beneficial to understand the intrinsic mechanisms of vibration abatement.

Impact of Double Time Delays on Regime Shift and Stochastic Resonance for a Species Population System Driven by Colored Correlated Multiplicative and Additive Noises

In this paper, the characteristics for the state transition between the boom state and the extinction one, varius stochastic resonance (SR) phenomena for a species population system induced by double time delays and colored cross-correlated Gaussian noises are investigated. The control of the species population system has an important effect on ecological balance and the development of human living environment. By means of the numerical calculation of the mean extinction time and the stationary probability density function (SPDF), we can figure out some fact: both external and internal noises, the noise correlation strength and its correlation time, one time delay can all reduce the ecological stability and shorten the survival time of the system; while the other time delay can enhance significantly the biological stability and postpone the extinction of the population. For the mean extinction time (MET) of the species population, it is found that noise correlation strength, time delay, the noise correlation time can all reduce the peak value of the MET, which leads to the reduction of the life cycle. Inversely, the other time delay will not change the maximum of the MET, and induce the peak value to shift to the left. With respect to the SNR, it is discovered that in the noise correlation strength and its correlation time, one-time delay mainly plays a prominent role in inhibiting the SR effect. On the contrary, the multiplicative noise can result in the enhancement of the resonant peak. In particular, the other time delay and the additive noise can stimulate the SR effect for the SNR during the initial period. However, the further increase of them will bring about the reduction of the resonant phenomenon.

Tumor state transitions driven by Gaussian and non-Gaussian noises

Tumor state transitions between the excited (high-concentration) and nonexcited (low-concentration) basins under the Gaussian white noise and non-Gaussian colored noise are investigated via the most probable steady states (MPSS) and the first escape probability (FEP)-based stochastic basin of attraction (SBA), respectively. Reducing the non-Gaussian colored noise and then utilizing the unified colored noise approximation (UCNA), the Markov system is derived. The extremal controlling equation of stationary probability density function (SPDF) is derived to analyze the impacts of noise on transitions in terms of MPSS. The existence of the ‘color’ of the non-Gaussian colored noise induces the reappearance of the uncorrelated additive white noise parameter that had vanished from the extremal controlling equation, completely reversing the inability of the uncorrelated additive Gaussian white noise to operate on transitions. The FEP-dependent SBA characterizing the excited basin stability is performed to further analyze the role of noise on the likelihood of escaping to the nonexcited state. Results show that the cross-correlated noises play a dual role in regulating SBA. The increased SBA indicating more difficulty to escape to the nonexcited state reflects a worse therapeutic effect. Therefore, enhancing the negatively correlated noise intensities and augmenting the non-Gaussian noise correlation time is essential for destabilizing the excited basin and achieving optimal therapeutic efficacy.

On the Impact of Recharging Behavior on Mobility

We consider the behavior of mobile users that, upon depletion of their device's energy, can detour from their regular path, in order to reach a location where the device can be recharged. We develop abstractions of this recharging behavior and analytically derive a charging-aware mobility model. The device is viewed as a mobile node. Based on the Palm inversion formula, we derive the (integral-form) stationary probability density function of the node's location, subject to attraction exerted by one charger in a bounded convex area. The analytical and numerical results demonstrate the distortion effect the charger has on the spatial node distribution. We observe, and explain, a counter-intuitive effect whereby the node density exhibits peaks that are asymmetric, coupled with a relative decrease of the density around the charger area. In addition, we provide a method to approximate the probability density function when multiple chargers are deployed. We examine the accuracy of this approximation and comment on its limitations.

Higher-order averaging procedure for performance analysis of a mono-stable Duffing piezoelectric energy harvesting system under white noise excitation

This work is concerned with the energy harvesting of mono-stable Duffing oscillators with piezoelectric coupling under white noise excitation. This is the first time that the higher-order averaging procedure is extended to find the first-, second- and third-order approximate stationary joint probability density functions of amplitude and full phase by solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation. Then the mean-square electric voltage and mean output power are derived through the approximate relation between the electric voltage and the mechanical states. The analytical results are verified by Monte Carlo simulation. The effects of system parameters, such as excitation intensity, mechanical damping term, nonlinear stiffness coefficients, and the resistance ratio of the resistive–capacitive circuit on the mean-square displacement and mean output power are discussed in detail. It has been shown that the averaging procedure of the averaging method of coefficients in FPK equation can be considered as a particular case of the proposed higher-order averaging procedure and the higher-order approximate solutions are much more accurate than the one obtained by the averaging method of coefficients in FPK equation.© 2022 The Authors. Published by Elsevier Ltd.

Uncertainty Propagation and Filtering via the Koopman Operator in Astrodynamics

The Koopman Operator (KO) theory is applied to generate an analytical solution of dynamical systems. The approach proposed in this work exploits a novel derivation of the KO with orthogonal polynomials to represent and propagate uncertainties, where the polynomials are modified to work with stochastic variables. Thus, a new uncertainty quantification technique is proposed in which the KO solution is expanded to include the prediction of central moments, up to an arbitrary order. The propagation of uncertainties is then expanded to develop a new filtering algorithm, for which the measurements are considered as additional observables in the KO mathematics. The uncertainty propagation technique is tested by predicting the state probability density function of a spacecraft in a halo orbit. The performance of the technique is assessed with a Monte Carlo analysis and proven to obtain accurate estimates for the state covariance, skewness, and kurtosis. The novel filtering methodology is then applied to an orbit determination application regarding a Lyapunov orbit, where an analysis on the filter accuracy and consistency shows that the new KO filter outperforms other common estimators.

A Note on the Stationary Probability Density Function and Covariance Matrix of a Stochastic Chemostat Model with Distributed Delay

A Note on the Stationary Probability Density Function and Covariance Matrix of a Stochastic Chemostat Model with Distributed Delay

Density of instantaneous frequencies in the Kuramoto–Sakaguchi model

We obtain a formula for the statistical distribution of instantaneous frequencies in the Kuramoto–Sakaguchi model. This work is based on the Kuramoto–Sakaguchi’s theory of globally coupled phase oscillators, which we review in full detail by discussing its assumptions and showing all steps behind the derivation of its main results. Our formula is a stationary probability density function with a complex mathematical structure, is consistent with numerical simulations and gives a description of the stationary collective states of the Kuramoto–Sakaguchi model.

A variational Bayesian approach for ensemble filtering of stochastically parametrized systems

AbstractModern climate models use both deterministic and stochastic parametrization schemes to represent uncertainties in their physics and inputs. This work considers the problem of estimating the involved parameters of such systems simultaneously with their state through data assimilation. Standard state‐parameter filtering schemes cannot be applied to such systems, owing to the posterior dependence between the stochastic parameters and the “dynamical” augmented state, defined as the state augmented by the deterministic parameters. We resort to the variational Bayesian (VB) approach to break this dependence, by approximating the joint posterior probability density function (pdf) of the augmented state and the stochastic parameters with a separable product of two marginal pdfs that minimizes the Kullback–Leibler divergence. The resulting marginal pdf of the augmented state is then sampled using a one‐step‐ahead smoothing‐based ensemble Kalman filter (EnKF‐OSAS), whereas a closed form is derived for the marginal pdf of the stochastic parameters. The proposed approach combines the effectiveness of the OSAS filtering approach to mitigate inconsistency issues that often arise with the joint ensemble Kalman filter (EnKF), with the advantage of obtaining a full posterior pdf for the stochastic parameters, which is not possible with the traditional maximum‐likelihood method. We demonstrate the relevance of the proposed approach through extensive numerical experiments with a one‐scale Lorenz‐96 model, which includes a stochastic parametrization representing subgrid‐scale effects.

The improvement of stochastic averaging method to solve the ship rolling response excited by narrow-band waves

The stochastic averaging method of energy envelope can accurately solve the random motion response under the excitation of white noise. The noise had to fulfill the condition of strong mixing which roughly means that the process has to be broad-band. However, the sea conditions in practical engineering conditions are narrow-band spectra. Research on applicability of the stochastic averaging method of energy envelope shows that for narrow-band excitation, the damping of the system and excitation intensity should be small, and the excitation bandwidth should be large. Based on the above background, the improvement of stochastic averaging method of energy envelope under narrow-band waves is studied. The phase angle θ is introduced to characterize the relationship between kinetic energy and potential energy of the system. The drift coefficients and diffusion coefficients are averaged with respect to the period of θ. When the excitation frequency is in the lock-in field, large parametric rolling appears. The stationary probability density function (PDF) of the roll angle obtained by solving the Fokker-Planck-Kolmogorov (FPK) equation compares well with Monte Carlo simulation in the lock-in field. The improved method can relax the limitation of broad-band excitation required by stochastic averaging method. It can also consider both the nonlinear damping and nonlinear restoring force of roll motion. Therefore, the improved method is an efficient way to calculate the nonlinear roll response of ships in real sea conditions.

Control Mechanisms for Mobile Devices

In this paper we consider control mechanisms for mobile devices in a stochastic environment. In particular, we consider a device in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -dimensional space subject to Brownian perturbations where a control mechanism moves the device towards its target location at a speed which is a function of its displacement. For this scenario, we construct stochastic differential equations for the mobility process and solve for the steady state probability density function of displacement. From this we are able to give general solutions to key metrics such as displacement outage (the long term probability of exceeding a given distance from the target), connectivity probability (derived from the SNR distribution in a Rayleigh channel with pathloss), the mean time at which the device first exceeds a given distance from the target, and the mean kinetic energy required by the control mechanism. We evaluate these metrics for important special cases of the control mechanism and also study the optimization problem of minimizing kinetic energy over the parameters of the control function.

Using SPI and SPEI for baseline probabilities and seasonal drought prediction in two agricultural regions of the Western Cape, South Africa

Drought is one of the most hazardous natural disasters in terms of the number of people directly affected. An important characteristic of drought is the prolonged absence of rainfall relative to the long-term average. The intrinsic persistence of drought conditions continuing from one month to the next can be utilized for drought monitoring and early warning systems. This study sought to better understand drought probabilities and baselines for two agriculturally important rainfall regions in the Western Cape, South Africa – one with a distinct rainfall season and one which receives year-round rainfall. The drought indices, Standardised Precipitation and Evapotranspiration Index (SPEI) and Standardised Precipitation Index (SPI), were assessed to obtain predictive information and establish a set of baseline probabilities for drought. Two sets of synthetic time-series data were used (one where seasonality was retained and one where seasonality was removed), along with observed data of monthly rainfall and minimum and maximum temperature. Based on the inherent persistence characteristics, autocorrelation was used to obtain a probability density function of the future state of the various SPI start and lead times. Optimal persistence was also established. The validity of the methodology was then examined by application to the recent Cape Town drought (2015–2018). Results showed potential for this methodology to be applied in drought early warning systems and decision support tools for the province.

Stationary distribution and probability density function of a stochastic waterborne pathogen model with saturated direct and indirect transmissions

We investigate the dynamical behavior of deterministic and stochastic waterborne pathogen models with saturated incidence rates. In particular, the indirect transmission via person–water–person contact is half‐saturated and the direct transmission by person–person contact takes the saturated form. Possible equilibrium points of the model are investigated, and their stability criterion is discussed. Basic reproduction number of the model is obtained through the next‐generation matrix method. It has been shown that the disease‐free equilibrium is locally stable when and unstable for . Furthermore, a stochastic Lyapunov method is adopted to establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the proposed stochastic model, which reveals that the infection will persist if . We also obtain an exact expression of the probability density function near the quasi‐endemic equilibrium of stochastic system. Finally, numerical simulations are carried out to support our analytical findings. Results suggest that saturation constants in both direct and indirect transmissions play a positive role in controlling disease. Our results may provide some new insights for the elimination of waterborne disease.