Levy Noise Research Articles

Averaging principle for McKean-Vlasov SDEs with Levy noise and Holder coefficients

Averaging principle for McKean-Vlasov SDEs with Levy noise and Holder coefficients

The switch behavior in a gene transcription regulation system driven by Levy noise and Gaussian noise

The switch behavior in a gene transcription regulation system driven by Levy noise and Gaussian noise

Lévy noise impact on a stochastic epidemic model with healing, relapse and vaccination

In this work, we propose and analyze a stochastic Susceptible-Infected-Recovered-Healed (SIRH) epidemic model with healing, relapse, and jump effects to investigate the impact of vaccination on the disease dynamics. The stochastic perturbations are introduced into the model to account for the inherent randomness in the disease transmission and recovery processes. The existence and uniqueness of the positive global solution for the stochastic model are established. Further, we study the conditions for the extinction and persistence of the disease. The impact of vaccination is incorporated into the model by considering the vaccination rate as a control parameter. Numerical simulations are carried out to illustrate the dynamical behavior of the stochastic SIRH model and to analyze the effects of vaccination on the disease spread. The graphical representations of the approximate solutions for the susceptible, infected, recovered, and healed populations are provided under different vaccination scenarios.

Impact of supervise neural network on a stochastic epidemic model with Levy noise

This paper primarily focused on analyzing a stochastic $ \mathcal{SVIR} $ epidemic model that incorporates Levy noises. The population may be divided into four distinct compartments: vulnerable class ($ \mathcal{S} $), vaccinated individuals ($ \mathcal{V} $), infected individuals ($ \mathcal{I} $), and recovered individuals ($ \mathcal{R} $). To achieve this, we chose existing and unique techniques as the most feasible solution. In the nexus, the stochastic model was theoretically analyzed using a suitable Lyapunov function. This analysis broadly covered the existence and uniqueness of the non-negative solution, as well as the dynamic properties related to both the disease-free equilibrium and the endemic equilibrium. In order to eradicate diseases, a stochastic threshold value denoted as "$\textbf{R}_0$" was used to determine if they may be eradicated. If $ \textbf{R}_0 < 1, $ it means that the illnesses have the potential to become extinct. Moreover, we provided numerical performance results of the proposed model using the artificial neural networks technique combined with the Bayesian regularization method. We firmly believe that this study will establish a solid theoretical foundation for comprehending the spread of an epidemic, the implementation of effective control strategies, and addressing real-world issues across various academic disciplines.

4FSK signal processing based on adaptive tristable stochastic resonance under levy noise

Aiming at the problem of low quality of coherent demodulation transmission signal of digital signal in the background of Levy noise, this paper proposes a new method of 4FSK signal reception based on adaptive tristable stochastic resonance (SR) system. The bit error rate (BER) of output signals of adaptive tristable SR system model and the traditional modulation model are compared by simulation. The numerical simulation results show that compared with the traditional model, the output BER of the 4FSK signal after demodulation of the adaptive tristable SR system model is reduced by 4% compared with the traditional model. The time domain image of 4FSK signal is clearer, the burrs are significantly reduced and the spectral amplitude of 4FSK signal is greatly improved. The success of this experiment suggests a strong promise of SR systems for digital signal detection in impact noise.

Data-driven method to extract mean exit time and escape probability for dynamical systems driven by Lévy noises

Complex dynamical systems have been investigated through many data-driven methods with easily accessible and massive data from observations, experiments or simulations in recent decades. However, few works dealt with the stochastical non-Gaussian perturbation case. In this paper, for a class of systems perturbed by non-Gaussian [Formula: see text]-stable Lévy noises, we devise a data-driven approach to extract the mean exit time and escape probability of rare transition dynamics. The theories are based on the non-local Kramers–Moyal formulas and non-local partial differential equations generated by Kolmogorov backward operator, accompanied with the corresponding numerical algorithms. Specifically, we first identify governing laws by a machine learning framework according to non-local Kramers–Moyal formulas. With learned systems, the mean exit time and escape probability are obtained by solving corresponding partial differential equations. The feasibility and accuracy of the method are checked by one- and two-dimensional examples. This method will serve as an example to study stochastic systems with non-Gaussian perturbations from data and illuminate some insights into the extraction of other dynamical indicators like the maximum likelihood transition path.

Observer-based SMC design for stochastic systems with Levy noise

Abstract This work addresses the problem of sliding mode control (SMC) design for a continuous-time non-linear stochastic system with Levy-type noise. A state observer model is constructed to estimate the unavailable state information. Furthermore, Levy-type noise is considered to analyse small perturbations and to characterize the appearance of large samples that will occur in the system. Lyapunov stability and SMC theory are used to provide some sufficient conditions that ensure the stochastic stability of the error system and reachability of the predefined sliding surface. Finally, an example is given to demonstrate the feasibility of the proposed approach.

Stochastic delay differential equations: Analysis and simulation studies

Stochastic delay differential equations play an important role in modelling scientific and engineering systems. In this paper the partial differential equation satisfying the time evolution of the probability density function of the state variable governed by the stochastic delay differential equation (SDDE) with additive white noise and coloured noise perturbation is obtained under small time delay and small correlation time approximation using path integral formalism. This partial differential equation reduces to a Fokker–Planck equation for particular linear and non-linear SDDE. Fokker–Planck equation is then solved with reflecting boundary conditions to get the analytical solutions for Stationary Probability Density Function (SPDF). The analytical solutions for SPDF is compared with the SPDF obtained using simulation. Further the Mean First Passage Time (MFPT) of the bistable system is calculated for various values of time delay and noise strength and compared with that of the simulation. The MFPT for the time delayed system for the case of cubic potential driven by Gaussian and Levy noise as well as asymmetric bistable potential driven by a noise and periodic driving force has been investigated.

Robust Passivity Analysis of Stochastic Genetic Regulatory Networks with Levy Noise

Robust Passivity Analysis of Stochastic Genetic Regulatory Networks with Levy Noise

Impact of Levy noise on a stochastic Norovirus epidemic model with information intervention

In this article, we study the dynamics of Norovirus infection by developing a stochastic epidemic model having Levy noise. The study shows that Levy noise and informative interventions have more influence on the said dynamics. Firstly, we show that the model has a unique global positive solution. After this, we formulated a stochastic threshold value as a necessary condition for the extinction and persistence in the mean of the proposed epidemic model. Finally, numerical simulation are drawn to verify our obtained results of the dynamics. The perturbed stochastic approach may affect the dynamical properties of the model and large noises will be greatly significant to minimize its transmission.

Analysis of Applied Mathematics

Mathematics applied to applications involves using mathematics for issues that arise in various fields, e.g., science, engineering, engineering, or other areas, and developing new or better techniques to address the demands of the unique challenges. We consider it applied math to apply maths to problems in the real world with the double purpose of describing observed phenomena and forecasting new yet unknown phenomena. Thus, the focus is on math, e.g., creating new techniques to tackle the issues of the unique challenges and the actual world. The issues arise from a variety of applications, including biological and physical sciences as well as engineering and social sciences. They require knowledge of different branches of mathematics including the analysis of differential equations and stochastics. They are based on mathematical and numerical techniques. Most of our faculty and students work directly with the experimentalists to watch their research findings come to life. This research team investigates mathematical issues arising out of geophysical, chemical, physical, and biophysical sciences. The majority of these problems are explained by time-dependent partial integral or ordinary differential equations. They are also accompanied by complex boundary conditions, interface conditions, and external forces. Nonlinear dynamical systems provide an underlying geometrical and topological model for understanding, identifying, and quantifying the complex phenomena in these equations. The theory of partial differential equations lets us correctly formulate well-posed problems and study the behavior of solutions, which sets the stage for effective numerical simulations. Nonlocal equations result from the macroscopically modeling stochastic dynamical systems characterized by Levy noise and the modeling of long-range interactions. They also provide a better understanding of anomalous diffusions.

Characteristics of piecewise linear symmetric tri-stable stochastic resonance system and its application under different noises

Weak signal detection has become an important means of mechanical fault detections. In order to solve the problem of poor signal detection performance in classical tristable stochastic resonance system (CTSR), a novel unsaturated piecewise linear symmetric tristable stochastic resonance system (PLSTSR) is proposed. Firstly, by making the analysis and comparison of the output and input relationship between CTSR and PLSTSR, it is verified that the PLSTSR has good unsaturation characteristics. Then, on the basis of adiabatic approximation theory, the Kramers escape rate, the mean first-passage time (MFPT), and output signal-to-noise ratio (SNR) of PLSTSR are deduced, and the influences of different system parameters on them are studied. Combined with the adaptive genetic algorithm to synergistically optimize the system parameters, the PLSTSR and CTSR are used for numerically simulating the verification and detection of low-frequency, high-frequency, and multi-frequency signals. And the results show that the SNR and output amplitude of the PLSTSR are greatly improved compared with those of the CTSR, and the detection effect is better. Finally, the PLSTSR and CTSR are applied to the bearing fault detection under Gaussian white noise and Levy noise. The experimental results also show that the PLSTSR can obtain larger output amplitude and SNR, and can detect fault signals more easily, which proves that the system has better performance than other systems in bearing fault detection, and has good theoretical significance and practical value.

Influence of levy noise on subthreshold synchronization of spintronic stochastic neurons

Superparamagnetic tunnel junctions (SMTJs) are spintronic nanodevices that can oscillate spontaneously under the influence of thermal noise. Studies have proposed that SMTJs can be modeled as magnetic neurons by subjecting them to electrical Gaussian white noise and can be subsequently synchronized under subthreshold driving. Considering the non-Gaussian background noise in industrial applications, this study investigated the influence of levy noise on SMTJ subthreshold synchronization. The results revealed that levy noise with weak pulse characteristics can be used to promote the subthreshold synchronization of the SMTJs, whereas levy noise with enhanced pulse characteristics cannot be used. Under the same conditions, enhanced levy noise asymmetry is advantageous for achieving SMTJ synchronization under subthreshold driving.

Effective Reduction for a Nonlocal Zakai Stochastic Partial Differential Equation in Data Assimilation

We study the effective reduction for a nonlocal stochastic partial differential equation with oscillating coefficients. The nonlocal operator in this stochastic partial differential equation is the generator of non-Gaussian Levy processes, with either integrable or non-integrable jump kernels. We examine the limiting behavior of this equation as a scaling parameter tends to zero, and derive a reduced (local or nonlocal) effective equation. In particular, this work leads to an effective reduction for a data assimilation system with Levy noise, by examining the corresponding nonlocal Zakai stochastic partial differential equation. We show that the probability density for the reduced data assimilation system approximates that for the original system.

Exponential Synchronization of Stochastic Neural Networks with Time-Varying Delays and Lévy Noises via Event-Triggered Control

This study is related to the exponential synchronization problem of stochastic neural networks. A dynamic model of master-slave neural networks is established, which contains time-varying delays and Levy noises. The main purpose is to enable the slave system to follow the master system under the condition of limited communication capacity. Both the master system and the slave system are affected by random noises. Some sufficient conditions are given by means of linear matrix inequality methods which are established by applying Lyapunov functional together with the generalized Dynkin’s formula. Furthermore, a discrete event-triggered control is adopted in master-slave systems, which not only reduces the transmission resources but also avoids the Zeno phenomenon. At last, a numerical example is provided to verify the usefulness of judgment conditions in this study.

Rotation-translation coupling of a double-headed Brownian motor in a traveling-wave potential

Considering a double-headed Brownian motor moving with both translational and rotational degrees of freedom, we investigate the directed transport properties of the system in a traveling-wave potential. It is found that the traveling wave provides the essential condition of the directed transport for the system, and at an appropriate angular frequency, the positive current can be optimized. A general current reversal appears by modulating the angular frequency of the traveling wave, noise intensity, external driving force and the rod length. By transforming the dynamical equation in traveling-wave potential into that in a tilted potential, the mechanism of current reversal is analyzed. For both cases of Gaussian and Levy noises, the currents show similar dependence on the parameters. Moreover, the current in the tilted potential shows a typical stochastic resonance effect. The external driving force has also a resonance-like effect on the current in the tilted potential. But the current in the traveling-wave potential exhibits the reverse behaviors of that in the tilted potential. Besides, the currents obviously depend on the stability index of the Levy noise under certain conditions.

Lévy noise-induced transition and stochastic resonance in Brusselator system

The noise-induced transition and stochastic resonance (SR) phenomenon of Brusselator system under the excitation of Levy noise is investigated in this paper. The Janicki–Weron algorithm is used to simulate Levy noise, and the stationary probability density is obtained by solving the system equation through the fourth-order stochastic Runge–Kutta algorithm. The influences of the noise intensity, stability index and skewness parameter on the stationary probability density are analyzed. The numerical results imply that the stability index, skewness parameter and noise intensity can induce the transition. Moreover, the effects of the stability index and skewness parameter of Levy noise, as well as the amplitude of external signal on the signal-to-noise ratio are discussed. The research results show that the larger stability index, skewness parameter and signal amplitude are beneficial to the occurrence of SR.

Active disturbance rejection control approach to output-feedback stabilization of nonlinear system with Lévy noises

Abstract In this paper, the active disturbance rejection control (ADRC) approach, an emerging control approach, is applied to output-feedback stabilization for uncertain nonlinear systems disturbed by external stochastic disturbance. Here the disturbance has the form of an uncertain system driven by both Brownian motion and Levy noises. Firstly, for the sake of estimation for all the states and stochastic disturbance, an efficient extended state observer (ESO) is designed. Then an ESO-based output-feedback control approach and the stability in the mean squared sense are finally developed, respectively. A numerical simulation is also presented to demonstrate the effectiveness of our approach.

Efficient estimation methods for non-Gaussian regression models in continuous time

In this paper, we develop an efficient nonparametric estimation theory for continuous time regression models with non-Gaussian Levy noises in the case when the unknown functions belong to Sobolev ellipses. Using the Pinsker’s approach, we provide a sharp lower bound for the normalized asymptotic mean square accuracy. However, the main result obtained by Pinsker for the Gaussian white noise model is not correct without additional conditions for the ellipse coefficients. We find such constructive sufficient conditions under which we develop efficient estimation methods. We show that the obtained conditions hold for the ellipse coefficients of an exponential form. For exponential coefficients, the sharp lower bound is calculated in explicit form. Finally, we apply this result to signals number detection problems in multi-pass connection channels and we obtain an almost parametric convergence rate that is natural for this case, which significantly improves the rate with respect to power-form coefficients.

Transport of coupled particles in rough ratchet driven by Lévy noise.

This paper studies the transport of coupled particles in a tilted rough ratchet potential. The relationship between particles transport and roughness, noise intensity, external force, coupling strength, and free length is explored numerically by calculating the average velocity of coupled particles. Related investigations have found that rough potential can accelerate the process of crossing the barrier by increasing the particles velocity compared with smooth potential. It is based on the fact that the roughness on the potential surface is like a "ladder," which helps particles climb up and blocks them from sliding down. Moreover, superimposing an appropriate external force on the coupled particles or strengthening the Lévy noise leads to the particles velocity to increase. It is worth emphasizing that when the external force is selected properly, an optimal roughness can be found to maximize the particles velocity. For a given roughness, an optimal coupling coefficient is discovered to match the maximum velocity. And once the coupling coefficient is greater than the optimal value, the particles velocity drops sharply to zero. Furthermore, our results also indicate that choosing an appropriate free length between particles can also speed up transport.