Poisson White Noise Research Articles

Stochastic response analysis of a birhythmic oscillator under Poisson white noise excitation via path integration method

Stochastic response analysis of a birhythmic oscillator under Poisson white noise excitation via path integration method

Self-similarity and probability density function of the transient response of fractional compound motion

In recent years, there has been an increasing use of fractional differential equations due to their ability to effectively represent various engineering phenomena, including viscoelasticity, heat transport, non-local continuum, and others. These equations take into account certain effects that cannot be accurately predicted using classical differential equations.This paper provides a comprehensive analysis of the fractional compound motion, specifically focusing on the response of a one-term fractional differential equation that is excited by a Poissonian white noise process. The present study introduces a straightforward equation for the probability density function of fractional compound motion. The validity of this equation is subsequently confirmed by the execution of various numerical simulations. Furthermore, a comprehensive analysis is conducted on the self-similarity of fractional compound motion, demonstrating that the phenomenon can be regarded as self-similar in weak sense. This characteristic can be effectively employed to mitigate the loss of Markovianity in fractional differential equations.

Erratum to: Analysis of Weak Signal Detection Based on Tri-Stable System under Poisson White Noise

Erratum to: Analysis of Weak Signal Detection Based on Tri-Stable System under Poisson White Noise

Approximate averaged derivative moments and stochastic response of high-DOF quasi-non-integrable Hamiltonian systems under Poisson white noises

Stochastic response prediction is one of the principal topics in diverse branches of engineering, such as aerospace engineering, mechanical engineering, and vehicle engineering. However, the challenge of addressing high-dimensional partial differential equations or evaluating high-dimensional domain integrals in determining averaged derivative moments has limited the research. This study proposes a method to predict the stochastic response of high-degree-of-freedom (DOF) quasi-non-integrable Hamiltonian systems under Poisson white noises. The original finite DOF system is reduced to a one-dimensional one governing the Hamiltonian by applying the stochastic averaging method, and the associated reduced generalized Fokker–Planck–Kolmogorov (GFPK) equation is derived. Then, the two-step generalized elliptical coordinate transformation is extended to evaluate averaged derivative moments that are the key to obtaining the stationary probability density function (SPDF) for probabilistic information prediction. Moreover, the perturbation method is introduced to address the high-order GFPK equation. Finally, an 8-DOF strongly nonlinear system under purely additive Poisson white noises is given as an example to highlight the accuracy of the proposed method. Results from the proposed method show good agreement with those from Monte Carlo simulations, which indicates that the proposed method can be applied to assess the response of high-DOF quasi-non-integrable Hamiltonian systems under Poisson white noises. In addition, if the potential energy of an arbitrary finite n-DOF system is the same form as that in Appendix B, pertinent high-dimensional domain integrals are converted into explicit expressions when n is even.

Probability density of the solution to nonlinear systems driven by Gaussian and Poisson white noises

A new method is proposed to compute the probability density of the multi-dimensional nonlinear dynamical system perturbed by a combined excitation of Gaussian and Poisson white noises. We first deduce a probability-density solver from the Euler–Maruyama scheme of the stochastic system and the corresponding Chapman–Kolmogorov equation. This solver actually is an explicit numerical formula of the probability density of the solution to this stochastic system. To compute the probability density, we propose an efficient algorithm for this solver, which actually is the implementation of a numerical integration. Furthermore, we prove this solver is an approximated solution of the corresponding forward Kolmogorov equation. Numerical examples are conducted to illustrate our probability-density solver.

Deep neural network method to predict the dynamical system response under random excitation of combined Gaussian and Poisson white noises

Nonlinear dynamical systems excited by combined Gaussian and Poisson white noises are widely found in the scientific field. However, the introduction of Poisson white noise results in the associated forward Kolmogorov equation becoming an integro-differential equation (IDE), which makes it difficult to solve the response of these dynamical systems. At present, traditional numerical IDE solvers are limited due to both the mesh generation and high computational costs. In this context, this paper integrates the multi-element Gauss–Legendre (GL) quadrature into the physics-informed neural networks (PINNs) algorithm to predict the response of systems excited by combined Gaussian and Poisson white noises. A residual-based adaptive distribution (RAD) sampling method, which replaces Latin hypercube sampling (LHS) method, is implemented to adjust the residual points to improve the accuracy and training efficiency of the proposed algorithm. The results demonstrate that the integration of the PINNs algorithm with RAD sampling method dramatically enhances the tolerance of noisy data and expedites the convergence of the PINNs algorithm compared with the LHS sampling method. It has the ability to achieve better performance in a shorter training time and reduces the demand for data during the training process. Additionally, the Monte Carlo (MC) simulation is carried out to demonstrate the effectiveness of the numerical results in this paper.

Research on Power Angle Characteristics of One-Machine Infinite-Bus Power Systems of Mixed Gauss and Poisson Stochastic Excitation

Background: A high percentage of renewable energy and a high percentage of power electronic devices are connected to the power system, which leads to the diversification and complexity of stochastic excitation, and the traditional single-excitation stochastic model is no longer applicable. Objective: The study aimed to solve the problem that the high proportion of renewable energy and the high proportion of power electronic equipment are connected to the power system, which leads to the diversification and complexity of stochastic excitation and makes the traditional stochastic model of single excitation no longer applicable. Methods: Firstly, stochastic differential equations for power systems have been modelled with mixed Gaussian white noise and Poisson white noise excitation. Secondly, the Milstein-Euler predictor-corrector method has been developed to solve the stochastic differential equation model of the power system. Finally, the influence of Gauss white noise and Poisson white noise on the power system stability under different excitation intensities has been analyzed. The rationality and correctness of the model have been verified by the simulation of a one-machine infinitebus (OMIB) system. Results: The stochastic differential equation model of a power system with Gauss white noise and Poisson white noise excitation has been established and its angle stability has been analyzed. Increasing the Gaussian white noise and Poisson white noise excitation intensity can lead to an increase in the fluctuation of the power angle curve, as well as an increase in the standard deviation and expected value of the power angle mean curve, which may decrease the stability of the power system. Conclusion: This study provides a reference for stochastic power systems modeling and efficient simulation, and has important application value for power system stability assessment and safety evaluation as well as related patent applications.

Optogenetic Interrogation of Electrophysiological Dendritic Properties and Their Effect on Pacemaking Neurons from Acute Rodent Brain Slices.

Understanding dendritic excitability is essential for a complete and precise characterization of neurons' input-output relationships. Theoretical and experimental work demonstrates that the electrotonic and nonlinear properties of dendrites can alter the amplitude (e.g., through amplification) and latency of synaptic inputs as viewed in the axosomatic region where spike timing is determined. The gold-standard technique to study dendritic excitability is using dual-patch recordings with a high-resistance electrode used to patch a piece of distal dendrite in addition to a somatic patch electrode. However, this approach is often impractical when distal dendrites are too fine to patch. Therefore, we developed a technique that utilizes the expression of Channelrhodopsin-2 (ChR2) to study dendritic excitability in acute brain slices through the combination of a somatic patch electrode and optogenetic activation. The protocol describes how to prepare acute slices from mice that express ChR2 in specific cell types, and how to use two modes of light stimulation: proximal (which activates the soma and proximal dendrites in a ~100 µm diameter surrounding the soma) with the use of a high-magnification objective and full-field stimulation through a low-magnification objective (which activates the entire somato-dendritic field of the neuron). We use this technique in conjunction with various stimulation protocols to estimate model-based spectral components of dendritic filtering and the impact of dendrites on phase response curves, peri-stimulus time histograms, and entrainment of pacemaking neurons. This technique provides a novel use of optogenetics to study intrinsic dendritic excitability through the use of standard patch-clamp slice physiology. Key features • A method for studying the effects of electrotonic and nonlinear dendritic properties on the sub- and suprathreshold responses of pacemaking neurons. • Combines somatic patch clamp or perforated patch recordings with optogenetic activation in acute brain slices to investigate dendritic linear transformation without patching the dendrite. • Oscillatory illumination at various frequencies estimates spectral properties of the dendrite using subthreshold voltage-clamp recordings and studies entrainment of pacemakers in current clamp recordings. • This protocol uses Poisson white noise illumination to estimate dendritic phase response curves and peri-stimulus time histograms.

Probabilistic response determination of high-dimensional nonlinear dynamical systems enforced by parametric multiple Poisson white noises

Probabilistic response determination of high-dimensional nonlinear dynamical systems enforced by parametric multiple Poisson white noises

Reliability analysis of iced transmission lines under Poisson white noise excitation via path integration method

Reliability analysis of iced transmission lines under Poisson white noise excitation via path integration method

Adaptive stochastic resonance under Poisson white noise background and its application for bolt looseness detection

Adaptive stochastic resonance under Poisson white noise background and its application for bolt looseness detection

Stochastic response of nonlinear oscillators under non-homogeneous Poisson white noise excitations

Poisson white noise is a typical non-Gaussian process. It is generally utilized to model stationary loadings by the assumed constant impulse arrival rate. Much research work has been done on that. However, the actual impulse arrival times of some discrete loadings are not uniform in the given time duration. The resulting arrival rate is a time-varying function instead of a constant one. Corresponding Poisson process is non-homogeneous. Less work in the literature has been found. In this paper, a Poisson white noise process with a time-varying impulse arrival rate is considered in the excitation model. Under such excitation the response of nonlinear system is a Markov process. The resulting generalized time-dependent Fokker–Planck equation becomes more complicated. The perturbation method, combined with the exponential-polynomial closure (EPC) method, is utilized to solve this equation. Examples of typical nonlinear systems under non-homogeneous Poisson white noise excitations are investigated. The effect of time-varying impulse arrival rate on the response is illustrated. Numerical results show that the response statistical characteristics depend strongly on the impulse arrival rate. Large errors would generate if the arrival rate varies fast over time and is still treated as an average constant.

A new path integration method for the stochastic system under Poisson white noise excitation based on a probability mapping

In this paper, a new path integration method is proposed for stochastic dynamical systems excited by Poisson white noise. An efficient one-step transition probability density function (TPDF) matrix is constructed based on a decoupling probability mapping. The new method can handle the case of multiple impulses occurring in a one-step transition time interval and considers the randomness of the impulse instant, which compensates for a drawback of previous path integration methods. The probability mapping realizes the decoupling of randomness and the one-step TPDF matrix, which can be extended to general stochastic systems satisfying the Markov property. The stochastic responses of two dynamical systems excited by Poisson white noises for different mean arrival rates are obtained by using the new path integration method, and MC simulations prove that the new method is very effective, and it maintains good accuracy even for large mean arrival rates.

The occurrence mechanisms of extreme events in a class of nonlinear Duffing-type systems under random excitations.

The occurrence mechanisms of extreme events under random disturbances are relatively complex and not yet clear. In this paper, we take a class of generalized Duffing-type systems as an example to reveal three mechanisms for the occurrence of extreme events. First, it is intuitive that a very large excitation can generate extreme events, such as the Lévy noise. In such a case, extreme excitation works, while it does not require much about the systems. Second, when a system has a bifurcation structure, if the difference of the branches at the bifurcation point is large, a randomly varying bifurcation parameter can lead to extreme events. Finally, when a system has rare attractors, a random impulse excitation, such as Poisson white noise, is able to cause the system to escape from one general attractor into rare attractors. Such a kind of special regime switching behavior can lead to extreme events. These results reveal the possible mechanisms of extreme events in a class of nonlinear Duffing-type systems and provide guidance for further prediction and avoidance of extreme events.

Non-stationary non-Gaussian random vibration analysis of Duffing systems based on explicit time-domain method

Non-stationary non-Gaussian random vibration problems of structures are challenging and drawing increasing attention. In the present study, firstly, an explicit time-domain method (ETDM) is proposed to determine the higher-order response statistics of linear systems subjected to non-stationary non-Gaussian random excitations, in which the first four orders of cumulants of dynamic responses are directly formulated through the cumulant operation rule based on the explicit expressions of responses. Secondly, an equivalent linearization – explicit time-domain method (EL-ETDM) is further developed to solve the non-stationary non-Gaussian random vibration problems of Duffing systems, in which the equivalent linear system is derived discarding the assumption of Gaussian response, and the corresponding higher-order cumulant analyses of the linearized system are accomplished by the efficient ETDM. The present approach can account for non-Gaussian random excitations with arbitrary forms, and two specific applications to the Poisson white noise and the square form of Gaussian random process are investigated. Four numerical examples are presented to demonstrate the effectiveness of the proposed methods.

Parametric Poisson Bifurcated Autoregressive Process: Application to Worldwide, Regional, and Peculiar Countries’ of Automobile Production

This article introduces Bifurcated Autoregressive (BAR) process with two apart marginal distribution error terms of w2 and w2+1 of Poisson white noises to make it Poisson Bifurcated Autoregressive (PBAR) in a parametric setting. The statistical definition of PBAR (1) process with parameters B1 and B2 that must be |B1 | and |B2 |<1 for stationary process was spelt-out. Weighted Least Squares (WLS) parameter estimation technique was adopted and the process limiting distribution was carried-out via the combination methods of martingale process and Lindeberg’s condition. Monthly automobile production in Japan, Outside Japan, America, USA, Europe, Asia, and China that approximately tantamount to worldwide, regional, and peculiar countries’ of automobile production was subjected to the PBAR process. In conclusion, Japan automobile production possessed the highest and largest error correlation (w2 , w2+1 ) of 0.6582 (65%) with first order PBAR, with B1Y(t/2) , such that B1=0.2228 of degenerated two major divisions of automobile production of Registrations and Mini-Vehicles with descendant of different brands (models).

Transient response of energy harvesting systems with multi-well potential under Poisson white noise excitations

This paper studies the transient response of vibration-based energy harvesting (VEH) systems characterized by multi-well potential functions exposed to Poisson white noise excitations. With the recently developed radial basis function neural networks (RBFNN) approach, the generalized Fokker–Planck–Kolmogorov (GFPK) equation in three-dimensional state space for the transient probability density function (PDF) is solved. The trial solution is composed of a series of radial basis Gaussian functions, each multiplied by a time-varying weight coefficient. The method of Lagrange multiplier is introduced for the constraint of weight coefficients to normalized the trial solution of the PDF solution. As examples to validate the proposed solution approach, the mono-stable and tri-stable energy harvester systems are studied. Monte Carlo simulations (MCS) have been applied to check the analytical solution. Some theoretical trends of the harvested mean square voltage (MSV) as a function of system parameters are examined. The non-Gaussian excitation and time-varying nature of the system can significantly affect the MSV output. It is noted that this work provides researchers and engineers a tool to optimize energy harvesting systems operated in random environment.

Random vibration of hysteretic systems under Poisson white noise excitations

Hysteresis widely exists in civil structures, and dissipates the mechanical energy of systems. Research on the random vibration of hysteretic systems, however, is still insufficient, particularly when the excitation is non-Gaussian. In this paper, the radial basis function (RBF) neural network (RBF-NN) method is adopted as a numerical method to investigate the random vibration of the Bouc-Wen hysteretic system under the Poisson white noise excitations. The solution to the reduced generalized Fokker-Planck-Kolmogorov (GFPK) equation is expressed in terms of the RBF-NNs with the Gaussian activation functions, whose weights are determined by minimizing the loss function of the reduced GFPK equation residual and constraint associated with the normalization condition. A steel fiber reinforced ceramsite concrete (SFRCC) column loaded by the Poisson white noise is studied as an example to illustrate the solution process. The effects of several important parameters of both the system and the excitation on the stochastic response are evaluated, and the obtained results are compared with those obtained by the Monte Carlo simulations (MCSs). The numerical results show that the RBF-NN method can accurately predict the stationary response with a considerable high computational efficiency.

Heat fluctuations in chemically active systems.

Chemically active systems such as living cells are maintained out of thermal equilibrium due to chemical events which generate heat and lead to active fluctuations. A key question is to understand on which time and length scales active fluctuations dominate thermal fluctuations. Here, we formulate a stochastic field theory with Poisson white noise to describe the heat fluctuations which are generated by stochastic chemical events and lead to active temperature fluctuations. We find that on large length- and timescales, active fluctuations always dominate thermal fluctuations. However, at intermediate length- and timescales, multiple crossovers exist which highlight the different characteristics of active and thermal fluctuations. Our work provides a framework to characterize fluctuations in active systems and reveals that local equilibrium holds at certain length- and timescales.

Feedback stabilization of quasi nonintegrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A procedure for designing a feedback control to asymptotic Lyapunov stability with probability one of quasi nonintegrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations is proposed. First, a one dimensional partially averaged Itô stochastic differential equation for controlled Hamiltonian is derived from the motion equations of the system by using the stochastic averaging method. Second, the dynamical programming equation for the ergodic control problem of the averaged system with undetermined cost function is set up based on the dynamical programming principle and the jump–diffusion chain stochastic differential rules. The optimal control law is obtained by solving the dynamical programming equation. Third, the analytical expression for the largest Lyapunov exponent of the averaged system is derived. Finally, the asymptotic Lyapunov stability with probability one of the originally controlled system is analyzed approximately by using the largest Lyapunov exponent. The cost function and optimal control forces are determined by the requirements of stabilizing the system. An example is worked out in detail to illustrate the effectiveness of the proposed method for stabilization control, and the control effect of the proposed feedback stabilization varies with the change of parameters is also studied in this paper, such as, the greater the excitation intensity of Gaussian and Poisson white noise, the better the stabilization control effect.