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The well-posedness and exponential ergodicity are proved for stochastic Hamiltonian systems containing a singular drift term which is locally integrable in the component with noise. As an application, the well-posedness and uniform exponential ergodicity are derived for a class of singular degenerated McKean-Vlasov SDEs. [ABSTRACT FROM AUTHOR]

ABSTRACT This study proposes a filtering based maximum likelihood generalized extended gradient algorithm for multivariable nonlinear systems with autoregressive moving average noises. The parameter estimates are obtained by minimizing the half squared residual measurement which can approach the true values. An auxiliary model is also established with the measurable information of the system, and the output of the auxiliary model is used to replace the unmeasurable variables of the system, so that the output of the auxiliary model approximates these unmeasurable variables, so as to obtain the consistent estimation of the system parameters. A maximum likelihood generalized extended gradient algorithm is derived for comparison and a numerical example is provided to show the effectiveness of the proposed method and the estimates converge to the actual values quickly. [ABSTRACT FROM AUTHOR]

This manuscript explores the stability theory of several stochastic/random models. It delves into analyzing the stability of equilibrium states in systems influenced by standard Brownian motion and exhibit random variable coefficients. By constructing appropriate Lyapunov functions, various types of stability are identified, each associated with distinct stability conditions. The manuscript establishes the necessary criteria for asymptotic mean-square stability, stability in probability, and stochastic global exponential stability for the equilibrium points within these models. Building upon this comprehensive stability investigation, the manuscript delves into two distinct fields. Firstly, it examines the dynamics of HIV/AIDS disease persistence, particularly emphasizing the stochastic global exponential stability of the endemic equilibrium point denoted as , where the underlying basic reproductive number is greater than one (). Secondly, the paper shifts its focus to finance, deriving sufficient conditions for both the stochastic market model and the random Ornstein–Uhlenbeck model. To enhance the validity of the theoretical findings, a series of numerical examples showcasing stability regions, alongside computer simulations that provide practical insights into the discussed concepts are provided. [ABSTRACT FROM AUTHOR]

This article discusses a finite-source stock-dependent stochastic inventory system with multiple servers and a retrial facility. The system can store a maximum of S items, and the lifetime of each item is exponentially distributed. The primary customer arrives at the waiting hall from the finite source and receives service from multi-servers. The rate at which customers arrive depends on the current stock level. If the waiting hall is full during the primary customer's arrival, he enters the finite orbit. Additionally, customers in the waiting hall may lose patience and enter the orbit. To replenish the stock, we follow the (s, Q) ordering policy. We calculate the joint probability distribution of the number of inventory items, busy servers, and number of customers in the waiting hall and orbit at a steady state. We conduct a comparative numerical analysis to determine the impact of heterogeneous and homogeneous service rates on various metrics, such as the average impatient customer rate, the fraction of successful retrials, and the average number of customers in the waiting hall and orbit. [ABSTRACT FROM AUTHOR]

ABSTRACT This article considers a distributed identification problem for linear stochastic systems whose input and output observations are scheduled by an adaptive event‐triggered scheme. An event detector with time‐varying thresholds is designed to control the transmission of measurements from the sensors to the estimators, which leads to that only a subset of input and output data is available for identification. The estimators exchange information over a network and cooperatively identify the unknown parameters. A distributed recursive identification algorithm under the event‐triggered scheme is proposed based on the distributed stochastic approximation algorithm with expanding truncations (DSAAWET). Under mild assumptions, the strong consistency of the algorithm is proved, that is, the estimates generated from each estimator achieve consensus and converge to the true parameters with probability one. Finally, two numerical examples are provided to validate the theoretical results of the algorithm. [ABSTRACT FROM AUTHOR]

This study is concerned with the influence of astronomical forcing and stochastic disturbances on non-linear dynamics of the Earth's climate. As a starting point, we take the system of climate equations derived by Saltzman and Maasch for late Cenozoic climate changes. This system contains variations of three prognostic variables: the global ice mass, carbon dioxide concentration, and deep ocean temperature. The bifurcation diagram of deterministic system shows possible existence/coexistence of stable equilibria and limit cycle leading either to monostability or bistability. Fitting the astronomical forcing by an oscillatory function and representing the deep ocean temperature deviations by means of white Gaussian noise of various intensities, we analyze the corresponding stochastic system of Saltzman and Maasch equations for the deviations of prognostic variables from their average values (equilibrium state). The main conclusions of our study are as follows: (i) astronomical forcing causes the climate system transitions from large-amplitude oscillations to small-amplitude ones and vice versa; (ii) astronomical and stochastic forcings together cause the mixed-mode climate oscillations with intermittent large and small amplitudes. In this case, the Earth's climate would be shifting from one stable equilibrium with a warmer climate to another stable equilibrium with a colder climate and back again. [ABSTRACT FROM AUTHOR]

Stochastic collocation (SC) is a well‐known non‐intrusive method of constructing surrogate models for uncertainty quantification. In dynamical systems, SC is especially suited for full‐field uncertainty propagation that characterizes the distributions of the high‐dimensional solution fields of a model with stochastic input parameters. However, due to the highly nonlinear nature of the parameter‐to‐solution map in even the simplest dynamical systems, the constructed SC surrogates are often inaccurate. This work presents an alternative approach, where we apply the SC approximation over the dynamics of the model, rather than the solution. By combining the data‐driven sparse identification of nonlinear dynamics framework with SC, we construct dynamics surrogates and integrate them through time to construct the surrogate solutions. We demonstrate that the SC‐over‐dynamics framework leads to smaller errors, both in terms of the approximated system trajectories as well as the model state distributions, when compared against full‐field SC applied to the solutions directly. We present numerical evidence of this improvement using three test problems: a chaotic ordinary differential equation, and two partial differential equations from solid mechanics. [ABSTRACT FROM AUTHOR]

Although traditional active suspension can offer superior riding comfort and maneuverability over its semiactive and passive counterparts, its reliance on an external power supply has hindered its widespread applications in vehicles. To overcome this deficiency, this paper proposes an innovative self-powered active suspension design for a high-speed train (HST), by leveraging the recently emerging H-bridge circuit-based electromagnetic damper (HB-EMD), allowing bidirectional power flow between the damper and controlled system. The capability of HB-EBD to achieve unique self-powered active skyhook control was previously proved in a simplified single degree-of-freedom (SDOF) structure under harmonic excitations; however, the feasibility of employing HB-EMD to realize active vibration control for more complex structural systems under stochastic excitations remains an unanswered question. One main challenge is designing a novel control algorithm that can simultaneously realize vibration control and self-powering objectives, which is unattainable by traditional active control algorithms. In this study, an ad hoc model predictive controller (MPC) is designed to guarantee the fulfillment of these dual objectives. To evaluate the performance of the proposed active suspension design, two separate HB-EMDs are implemented on the front and rear sides of the secondary lateral suspensions of an HST model subjected to stochastic track irregularities. At a speed of 200km/h, the proposed HB-EMDs with MPC could achieve a 55% reduction in the lateral acceleration of the car body in comparison with passive suspension, meanwhile maintaining energy harvesting performance with an average output power of 25.0W. In contrast, a traditional active linear quadratic Gaussian (LQG) controller consumes 72.7W power when performing comparable vibration reduction. This study, for the first time, validates the feasibility of designing a self-powered, actively controlled secondary lateral HST suspension system without relying on an external power source, which will potentially pave the way for a new active vibration control paradigm for other generic structures. [ABSTRACT FROM AUTHOR]

The dynamics of plant-insect interactions play a crucial role in the ecosystem, influenced by complex molecular signaling pathways. This study extends existing deterministic models of plant-insect systems by incorporating stochastic elements and molecular interactions, particularly focusing on the roles of Botrytis Induced Kinase-1 (BIK1) and Phyto Alexin Deficient-4 (PAD4) proteins. The model evaluates the effects of constant inhibition, pulsed inhibition, and adaptive feedback control on plant biomass ( y 1 ) , insect herbivore density ( y 2 ) , PAD4 levels ( y 3 ) , and BIK1 levels ( y 4 ). Additionally, we examine the impact of different noise types, including deterministic, Gaussian, and Lévy noise, on system variability and stability. Results indicate that our stochastic model is superior as it shows a significant reduction in BIK1 levels, particularly under higher noise intensities, which enhances PAD4 activity and improves plant defense mechanisms. Moreover, moderate noise intensity (σ = 0.05) provides an optimal balance, sustaining PAD4 levels while effectively controlling insect herbivore populations. We also integrate MEMS-based feedback mechanisms, which dynamically adjust plant biomass and molecular signaling, further stabilizing the system's response to environmental variability. [ABSTRACT FROM AUTHOR]

ABSTRACT This article seeks the enhancement of nonlinearity tolerance in time/event‐triggered stabilization for stochastic systems. In the related results, the controller functions are required to obey global Lipschitz condition for the suppression of sampling/execution error, and the drift and/or diffusion coefficients of the stochastic systems are restricted to polynomial or, even, linear growth. These limitations deserve overcoming for enlarged applications. To this end, a distinct framework of time/event‐triggered controls is established for stochastic systems, targeted at exponential stabilization. Specifically, inclusive Lyapunov‐type feasibility conditions are proposed by capturing the effect of sampling/execution error and distinguishing the role of system nonlinearities. Particularly, different from the related results, the evolution of sampling/execution error is subtly exploited via Lyapunov functions to reveal the dynamic interaction between the sampling/execution errors and system state. Then, time‐triggered exponential stabilization via sampled‐data controller is achieved not only in the moment sense but also in the almost sure sense. Accordingly, Lyapunov function based analysis is performed for the composite dynamics of system state and sampling error, confronted with the coupling of discontinuous and stochastic features. To further reduce execution, periodic event‐triggered control is exploited to achieve exponential stabilization for stochastic nonlinear systems, by virtue of the relation with the dynamic evolution under sampled‐data control. Through typical examples, we demonstrate the potential of our framework in handling the cases with the controller functions violating global Lipschitz condition and with the system nonlinearities beyond polynomial growth. [ABSTRACT FROM AUTHOR]

ABSTRACT In this paper, we investigate observer‐based switching‐like adaptive‐triggered resilient state feedback control for discrete singular systems under deception‐injection (DI) attacks. Considering state variables are not completely measurable, a delayed state observer is engineered to reconstruct system states and the occurrence of DI attacks is described by Bernoulli random variables. The upper bound method is used to deal with the DI attacks with jumping patterns and imprecise occurrence probabilities. Taking into account the sporadic nature of DI attacks, switching‐like adaptive‐triggered resilient control method is presented, and the collaboratively designed triggered mechanism and resilient control strategy can not only automatically adjust data transmission based on the system states, but also save communication resources and network bandwidth, and enhance the tolerance to hostile attacks so as to improve safety and reliability of networked singular systems. By utilizing singular value decomposition technique, the noncausal behavior of the singular system is avoided and by applying linear matrix inequality (LMI) technology, the desired gains of controller and observer are achieved concurrently, then neoteric conditions about H∞$$ {H}_{\infty } $$ stochastic admissibility of closed‐loop discrete singular systems are provided. Ultimately, the validity of the proposed resilient coordination control approach is verified via a direct current (DC) motor model. [ABSTRACT FROM AUTHOR]

In this study, we conduct a comprehensive investigation of the novel characteristics of the (2 + 1)-dimensional stochastic Hirota–Maccari System (SHMS), which is a prominent mathematical model with significant applications in the field of nonlinear science and applied mathematics. Specifically, SHMS plays a critical role in the study of soliton dynamics, nonlinear wave propagation, and stochastic effects in complex physical systems such as fluid dynamics, optics, and plasma physics. In order to account for the abrupt and significant fluctuation, the aforementioned system is investigated using a Wiener process with multiplicative noise in the Itô sense. The considered equation is studied by the new extended direct algebraic method (NEDAM) and the modified Sardar sub-equation (MSSE) method. By solving this equation, we systematically derived the novel soliton solutions in the form of dark, dark-bright, bright-dark, singular, periodic, exponential, and rational forms. Additionally, we also categorize and analyze the W-shape, M-shape, bell shape, exponential, and hyperbolic soliton wave solutions, which are not documented by researchers. The bifurcation, chaos and sensitivity analysis has been depicted which represent the applicability of the system in different dynamics. These findings greatly advance our knowledge of nonlinear wave events in higher-dimensional stochastic systems both theoretically and in terms of possible applications. These findings are poised to open new avenues for future research into the applicability of stochastic nonlinear models in various scientific and industrial domains. [ABSTRACT FROM AUTHOR]

ABSTRACT In this work, the stability problem of nonlinear stochastic systems is investigated under exogenous disturbances through event‐triggered impulsive control (ETIC). The ETIC strategy proposed in this paper incorporates three levels of events, taking into account four indicators: threshold, control‐free index, inspection interval, and waiting time for stochastic systems, which is more practically significant and can effectively eliminate the Zeno behavior. By utilizing Lyapunov stability theory, stochastic analysis techniques, and some fundamental inequalities, sufficient conditions for the rth$$ r\mathrm{th} $$ moment input‐to‐state stability (r$$ r $$‐ISS) and exponentially r$$ r $$‐ISS of the considered system can be achieved through the implementation of the designed ETIC. Then, the theoretical results are employed in actual nonlinear stochastic systems, leading to the establishment of LMI‐based criteria for exponential r$$ r $$‐ISS. Ultimately, the feasibility of ETIC strategy is confirmed through two instances. [ABSTRACT FROM AUTHOR]

In this paper, we consider a coupled system of fractional stochastic differential equations involving the Hilfer derivative of order 1 2 < α < 1 \frac{1}{2}<\alpha<1 . Under some assumptions, we prove the existence of mild solutions for our system based on Perov’s and Schaefer’s fixed point theorems. An example illustrating our results is also provided. [ABSTRACT FROM AUTHOR]

During flight, projectiles are subject to uncertainties such as aerodynamic forces, wind gusts, and measurement errors; all of which significantly affect their stability and accuracy. As a result, studying the response of projectile systems under stochastic excitation is essential. This paper focuses on the solution and analysis of projectile system responses under stochastic excitation. We employed the path integral method to compute the transient and stationary probability density functions for projectile systems subjected to Gaussian stochastic external and parametric excitations. Based on the probabilistic responses, we analyzed the evolution of the system's probability density function over time under Gaussian white noise excitation, as well as the changes in the stationary probability density function with air density and flight speed as bifurcation parameters. The analysis results indicate that within a specific range of parameter variations, air density can induce stochastic P‐bifurcation phenomena. Furthermore, increasing air density and flight speed can enhance the stability of the projectile. [ABSTRACT FROM AUTHOR]

This manuscript critically examines existing research on cyber battle management systems (CBMS) and underscores the importance of advancing complex structure thinking, cybernetics, wicked problem-solving, and emerging behavior analysis. It advocates for a systems-thinking approach to solving complex problems by identifying and understanding associated systems, predicting their behavior, and managing changes. The manuscript explores the integration of cybernetics meta-methodology and the viable system model with metasystems reductionism to address negative emergent behavior in complex systems. The study highlights the roles of individual systems, systems of systems, and metasystems, emphasizing the deterministic nature of single systems and the stochastic characteristics of systems of systems. By integrating cybernetics, viable system models, and meta-metasystems, the manuscript explores key parameters for building intelligent systems, revealing that meta-metasystems offer superior capabilities for coordinating and integrating multiple systems. The research results demonstrate the successful development of a meta-metasystem tailored for CBMS, providing a strategic framework for the future of cyber battle management. [ABSTRACT FROM AUTHOR]

In this tutorial review paper we explain the concept of homomorphism and identify some principles that justify homomorphism construction based on the homogeneity of structure and coupling in systems with multiple components. We discuss some simple examples to show how these underlying justifying conditions can arise. Examples include brain simulation, combat attrition, and the greatly reduced computational complexity represented by Pascal's triangle. Homomorphism is also shown to be fundamental for constructing approximate low-resolution models. Models that simplify complex time-demanding simulation models are often used as surrogate or metamodels in system optimization. However, such models are fitted typically to computationally derived response surfaces and not structurally related directly to the originals using homomorphisms as described here. Along these lines, we show how homomorphism plays an essential role in a novel approach being developed to strongly control tree expansion in state space explorations of stochastic system simulation. [ABSTRACT FROM AUTHOR]

This paper concerns the optimality problem of distributed linear quadratic control in a linear stochastic multi‐agent system (MAS). The main challenge stems from MAS network topology that limits access to information from non‐neighbouring agents, imposing structural constraints on the control input space. A distributed control‐estimation synthesis is proposed which circumvents this issue by integrating distributed estimation for each agent into distributed control law. Based on the agents' state estimate information, the distributed control law allows each agent to interact with non‐neighbouring agents, thereby relaxing the structural constraint. Then, the primal optimal distributed control problem is recast to the joint distributed control‐estimation problem whose solution can be obtained through the iterative optimization procedure. The stability of the proposed method is verified and the practical effectiveness is supported by numerical simulations and real‐world experiments with multi‐quadrotor formation flight. [ABSTRACT FROM AUTHOR]

The conditional spacings are investigated here based on a series system with n heterogeneous exponential component lifetimes. First, the survival function and the joint distribution of conditional spacings are obtained. Then, the survival function of the first spacing is shown to be Schur-convex in the hazard rates of component lifetimes. Further, some stochastic comparisons of spacings are carried out. Finally, some results for the conditional sample range are discussed. [ABSTRACT FROM AUTHOR]

Let T be the regularity structure associated with a given system of singular stochastic PDEs. The paracontrolled representation of the Π map provides a linear parametrization of the nonlinear space of admissiblemodelsM= (g,Π) onT in terms of the family of pararemainders used in the representation.We give an explicit description of the action of the most general class of renormalization schemes on the parametrization space of the space of admissible models. The action is particularly simple for renormalization schemes associated with degree preserving preparation maps. The BHZ renormalization scheme has that property. [ABSTRACT FROM AUTHOR]

This research investigates the security challenges posed by worm propagation in wireless sensor networks (WSNs). A novel stochastic susceptible – infectious – vaccination – recovered model is introduced to analyse the dynamics of worm spread. Conditions for the existence of a unique global solution are examined, and necessary conditions for worm eradication are established. By incorporating random environmental fluctuations, the proposed model provides a more precise depiction of propagation dynamics than deterministic models. Empirical findings are presented to validate the model's predictive accuracy across diverse scenarios, underscoring its robustness. Numerical simulations affirm the effectiveness of the analytical approach in understanding worm propagation within WSNs. The study offers valuable insights into worm dynamics and proposes a methodological framework to enhance network security. The findings underscore the significant role of stochastic systems in modelling and provide strategic perspectives for designing resilient defensive frameworks against worm attacks in WSNs. [ABSTRACT FROM AUTHOR]

The Lyapunov approach is one of the most effective and efficient means of studying the partial stability of stochastic systems. A number of authors have analysed the partial practical stability of stochastic differential equations using Lyapunov techniques. Nevertheless, no results are concerned with the partial stability of stochastic systems based on the knowledge of the solution of the system explicitly. This paper has been concerned with the problem of partial practical stability for linear time-varying stochastic perturbed systems. Necessary and sufficient conditions for partial practical uniform exponential stability are given based on generalised Gronwall inequalities, in particular of Gamidov's type. An example is developed to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

The analysis of the stochastic bifurcation behaviors of stochastic nonlinear systems often requires artificial judgment based on the joint probability density and cannot be automated. A new calculation method for automatic calculation of random bifurcation points was proposed. The bi-stable Van der Pol system under strong noise excitation was taken as an example, the influences of damping coefficient changes on stochastic dynamic responses were analyzed. The research results show that, the joint probability density of the system bifurcates for 3 times with the increase of the damping coefficient, exhibiting 4 different types of geometric features. The proposed method can hopefully be applied to the study of stochastic bifurcation behaviors of other stochastic nonlinear systems. [ABSTRACT FROM AUTHOR]

We establish the existence of weak solutions for a decoupled system of a forward stochastic differential equation (SDE) and a backward stochastic differential equation (BSDE). The generator H ⁢ (x , y , z) is assumed continuous in (y , z) but possibly discontinuous in x. The drift of the forward component is merely measurable drift and the diffusion coefficient can be discontinuous. Our approach is based on partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a coupled system of fractional stochastic differential equations involving the Hilfer derivative of order 1 2 < α < 1 . Under some assumptions, we prove the existence of mild solutions for our system based on Perov's and Schaefer's fixed point theorems. An example illustrating our result is provided. [ABSTRACT FROM AUTHOR]

With the increase in the number of urban vehicles, various traffic problems have gradually emerged. Studying the causes of traffic congestion and proposing effective mitigation strategies have important practical significance. This paper proposes a macroscopic traffic flow model that considers the delayed speed difference. This paper applies nonlinear bifurcation to describe and predict nonlinear traffic phenomena on highways from the perspective of global stability of the traffic system. By using the traveling wave transformation, the proposed car-following model is converted into a macroscopic traffic flow model. Next, this paper employs the linear stability analysis to find the bifurcation points of the stability transition in the traffic system, exploring the qualitative characteristics of the inhomogeneous continuous traffic flow model. Theoretical derivations demonstrate the existence of bifurcation points within the model. Additionally, this paper plots the density-time space diagrams and phase plane diagrams of the system to visually present the sudden changes in traffic flow as variable parameters pass through these bifurcation points. Finally, this paper designs a feedback controller to regulate the Hopf bifurcation, aiming to delay or eliminate the occurrence of Hopf bifurcations in the stochastic system. [ABSTRACT FROM AUTHOR]

The study of long-term behavior in stochastic systems is critical for understanding the dynamics of complex processes influenced by randomness. This paper addresses the existence and uniqueness of Stepanov-like pseudo S-asymptotically (ω , c) -periodic solutions for a class of stochastic integro-differential equations. These equations model systems where the interplay between deterministic and stochastic components dictates the overall dynamics, making periodic analysis essential. The problem addressed in this study is the lack of a comprehensive framework to describe the periodic behavior of such systems in noisy environments. To tackle this, we employ advanced techniques in stochastic analysis, fixed-point theorems and the properties of L 1 - and L 2 -convolution kernels to establish conditions for the existence and uniqueness of mild solutions under these extended periodicity settings. The methodology involves leveraging the decay properties of the operator kernels and the boundedness of stochastic integrals to ensure well-posedness. The major outputs of this study include novel results on the existence, uniqueness and stability of Stepanov-like pseudo S-asymptotically (ω , c) -periodic solutions, along with illustrative example demonstrating their applicability in real-world stochastic systems. [ABSTRACT FROM AUTHOR]

This paper studies the existence and uniqueness of solution for the fractional Hall-magnetohydrodynamics system (HMH) and with two stochastic terms (SHMH). Based on the theory of Besov–Morrey spaces and the contraction principle, we will demonstrate tow main result. The first result shows the existence, uniqueness and the analyticity of solution for (HMH) in Besov–Morrey spaces N p , λ s . The second result prove the existence and uniqueness of solution for (SHMH) in L 0 1 (Ω × (0 , T) , P ; M p λ) ∩ N p , λ s . [ABSTRACT FROM AUTHOR]

Motivation Computational modeling of cell state transitions has been a great interest of many in the field of developmental biology, cancer biology, and cell fate engineering because it enables performing perturbation experiments in silico more rapidly and cheaply than could be achieved in a lab. Recent advancements in single-cell RNA-sequencing (scRNA-seq) allow the capture of high-resolution snapshots of cell states as they transition along temporal trajectories. Using these high-throughput datasets, we can train computational models to generate in silico "synthetic" cells that faithfully mimic the temporal trajectories. Results Here we present OneSC, a platform that can simulate cell state transitions using systems of stochastic differential equations govern by a regulatory network of core transcription factors (TFs). Different from many current network inference methods, OneSC prioritizes on generating Boolean network that produces faithful cell state transitions and terminal cell states that mimic real biological systems. Applying OneSC to real data, we inferred a core TF network using a mouse myeloid progenitor scRNA-seq dataset and showed that the dynamical simulations of that network generate synthetic single-cell expression profiles that faithfully recapitulate the four myeloid differentiation trajectories going into differentiated cell states (erythrocytes, megakaryocytes, granulocytes, and monocytes). Finally, through the in silico perturbations of the mouse myeloid progenitor core network, we showed that OneSC can accurately predict cell fate decision biases of TF perturbations that closely match with previous experimental observations. Availability and implementation OneSC is implemented as a Python package on GitHub (https://github.com/CahanLab/oneSC) and on Zenodo (https://zenodo.org/records/14052421). [ABSTRACT FROM AUTHOR]

We consider multiple stochastic integrals with respect to càdlàg martingales, which approximate a cylindrical Wiener process. We define a chaos expansion, analogous to the case of multiple Wiener stochastic integrals, for these integrals and use it to show moment bounds. Key tools include an iteration of the Burkholder–Davis–Gundy inequality and a multi-scale decomposition similar to the one developed in Hairer and Quastel (Forum Math Pi 6:e3, 2018). Our method can be combined with the recently developed discretisation framework for regularity structures (Hairer and Matetski in Ann Probab 46(3):1651–1709, 2018, Erhard and Hairer in Ann Inst Henri Poincaré Probab Stat 55(4):2209–2248, 2019) to prove convergence of interacting particle systems to singular stochastic PDEs. A companion article (Grazieschiet al. in The dynamical Ising–Kac model in 3D converges to Φ 3 4 , 2023. arXiv:2303.10242) applies the results of this paper to prove convergence of a rescaled Glauber dynamics for the three-dimensional Ising–Kac model near criticality to the Φ 3 4 dynamics on a torus. [ABSTRACT FROM AUTHOR]

Reliability prediction for complex systems utilizing prognostic methods has attracted increasing attention. Furthermore, achieving accurate reliability predictions for complex systems necessitates considering the interaction between components and the multivariate functional relationship that exists among them. This paper proposes a bi‐level method to evaluate the variability of degradation processes and predictive reliability based on the profile monitoring approach for multicomponent systems. Firstly, a multivariate profile structure is introduced to model the framework of degradation analysis in scenarios where there exists stochastic dependency and a multivariate functional relationship between the degradation processes of components. At the component level, the objective is to evaluate the variability of the degradation process for each component considering the presence of stochastic dependence. For the system level analysis, the proposed approach enables the prediction of degradation variability and system reliability by considering the functional relationships among components, without the need for direct calculation of individual component reliabilities. The performance of the proposed model is evaluated through a numerical study and sensitivity analysis conducted on a multicomponent system with a k‐out‐of‐n structure. The results demonstrate the model's notable flexibility and efficiency. [ABSTRACT FROM AUTHOR]

The time-delayed feedback term can improve the output of a system, while the two-dimensional stochastic resonance (SR) system has a stronger signal amplification capability. To improve the output signal-to-noise ratio (SNR) of the system, this paper proposes a two-dimensional time-delayed tri-stable stochastic resonance system (TDTDTSR) based on the advantages of the above two systems. First, the steady-state probability density function (SPD), the mean first-pass time (MFPT), and the output SNR are derived under adiabatic approximation theory, and the effects of different system parameters on them are investigated. Next, TDTDTSR and the classical two-dimensional tri-stable stochastic resonance system (CTDTSR) system are simulated numerically. The results show that the mean signal-to-noise gain (MSNRG) of TDTDTSR system is higher than that of the CTDTSR system. Finally, the system parameters are optimized using a genetic algorithm, and the application of TDTDTSR to bearing fault detection is compared with CTDTSR and the novel piecewise symmetric two-dimensional tri-stable stochastic resonance (NPSTDTSR) systems. The experimental results demonstrate that TDTDTSR system has better performance, providing valuable theoretical support and practical engineering applications for the system in subsequent analyses. [ABSTRACT FROM AUTHOR]

In this paper, we propose a stochastic model for epidemiology data. The proposed model is obtained as a random perturbation of a suitable parameter in a deterministic SEIR system. This perturbation allows us to obtain a set of coupled stochastic differential equations (SDEs) that still have the conservation law. Afterward, by using Girsanov's Theorem, we calculate the maximum likelihood estimation (MLE) for parameters that represent the symptomatic infection rate, asymptomatic infection rate, and the proportion of symptomatic individuals. These parameters are crucial to obtain information about the dynamic of the disease. We prove the consistency of the MLE for a fixed time observation window, in which the disease is in its growth phase. The proposed stochastic SEIR model improves the uncertainty quantification of an overestimated MCMC scheme based on its deterministic model to count reported-confirmed COVID-19 cases of Mexico City. Using a particular mechanism to manage missing data, we developed MLE for some parameters of the stochastic model that improves the description of variance of the actual data. [ABSTRACT FROM AUTHOR]

The Mackey–Glass system is a paradigmatic example of a delayed model whose dynamics is particularly complex due to, among other factors, its multistability involving the coexistence of many periodic and chaotic attractors. The prediction of the long-term dynamics is especially challenging in these systems, where the dimensionality is infinite and initial conditions must be specified as a function in a finite time interval. In this paper, we extend the recently proposed basin entropy to randomly sample arbitrarily high-dimensional spaces. By complementing this stochastic approach with the basin fraction of the attractors in the initial conditions space, we can understand the structure of the basins of attraction and how they are intermixed. The results reported here allow us to quantify the predictability giving us an idea about the long-term evolution of trajectories as a function of the initial conditions. The tools employed can result very useful in the study of complex systems of infinite dimension. [ABSTRACT FROM AUTHOR]

This study investigates the most probable trajectories of a birhythmic oscillator under stochastic perturbations. The distinctive feature of the birhythmic oscillator is the coexistence of two stable limit cycles with different amplitudes and frequencies, separated by an unstable limit cycle. The path integral method was utilized to compute the instantaneous probability density. Based on the theory of most probable dynamics, by maximizing the probability density function, we present the time series of the most probable trajectories starting from different initial states. Furthermore, we conducted a detailed analysis of the noise-induced transitions between the two stable limit cycles under different parameter conditions. This approach enables us to understand and track the most probable escape time and specific most probable trajectories as the system transitions from the basin of attraction of one stable limit cycle to another. This work visualizes the most probable trajectories in stochastic systems and provides an innovative solution to the complex problem of noise-induced transitions between two stable limit cycles. Our research aims to provide a new perspective for studying complex stochastic dynamical systems. [ABSTRACT FROM AUTHOR]

This paper presents a survey and critical analysis of the mathematical literature on modeling of dynamic populations living in a fluid medium. The present review paper is divided into two main parts: The first part deals with the multiscale derivation of deterministic and stochastic cross-diffusion systems governed by the incompressible Navier–Stokes equations. The derivation is obtained from the underlying description at the microscopic scale in kinetic theory models according to the micro–macro decomposition method. In the second part of this review, we are delighted to present a new variety of mathematical models describing different applications, namely, the pursuit–evasion dynamics, cancer invasion, and virus dynamics. Finally, critical analysis and future research perspectives are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study the stochastic Hamiltonian flow in Wasserstein manifold, the probability density space equipped with L 2 -Wasserstein metric tensor, via the Wong–Zakai approximation. We begin our investigation by showing that the stochastic Euler–Lagrange equation, regardless it is deduced from either the variational principle or particle dynamics, can be interpreted as the stochastic kinetic Hamiltonian flows in Wasserstein manifold. We further propose a novel variational formulation to derive more general stochastic Wasserstein Hamiltonian flows, and demonstrate that this new formulation is applicable to various systems including the stochastic Schrödinger equation, Schrödinger equation with random dispersion, and Schrödinger bridge problem with common noise. [ABSTRACT FROM AUTHOR]

This article is to propose an adaptive reinforcement learning (RL)-based optimized distributed formation control for the unknown stochastic nonlinear single-integrator dynamic multi-agent system (MAS). For solving the issue of unknown dynamic, an adaptive identifier neural network (NN) is developed to determine the stochastic MAS under expectation sense. And then, for deriving the optimized formation control, the RL is putted into effect via constructing a pair of critic and actor NNs. With regard of the traditional RL optimal controls, their algorithm exists the inherent complexity, because their adaptive RL algorithm are derived from negative gradient of the square of Hamilton–Jacobi–Bellman (HJB) equation. As a result, these methods are difficultly extended to stochastic dynamical systems. However, since this adaptive RL laws are derived from a simple positive function rather than the square of HJB equation, it can make optimal control with simple algorithm. Therefore, this optimized formation scheme can be smoothly performed to the stochastic MAS. Finally, according to theorem proof and computer simulation, the optimized method can realize the required control objective. • This optimized method can archieve the stochastic MAS formation. • This optimized method has the simple algorithm. • This optimized method not require the PE condition. [ABSTRACT FROM AUTHOR]

This paper investigates the stochastic path following control of underactuated marine vehicles (UMVs) subject to multiple disturbances and constraints. Firstly, the complex marine environment in which UMVs navigate typically contains stochastic components, thus the multiple disturbances are categorized as slow-varying deterministic disturbances and stochastic disturbances. Secondly, a position-constrained line-of-sight (PCLOS) based fractional-order sliding mode stochastic (FSMS) control strategy is established to achieve path following control of UMVs. A PCLOS guidance law based on universal barrier Lyapunov function is proposed to ensure that the position errors remain within the constraint ranges, which is versatile for systems with symmetric constraints or without constraints. An FSMS controller based on fractional-order theory and sliding mode control is designed to improve the dynamic response speed of the system and effectively attenuate chattering phenomenon. A stochastic disturbance observer is developed to estimate the slow-varying deterministic disturbances in the stochastic system, and auxiliary dynamic compensators are used to mitigate the impact of input constraints. Lastly, theoretical analysis indicates that the closed-loop system is stable and the position constraint requirements are satisfied. Comparative simulations illustrate the effectiveness of the proposed control strategy. • The multiple disturbances are categorized as slow-varying deterministic disturbances, and stochastic disturbances. • A novel PCLOS guidance law is designed to ensure that the position errors do not exceed the given constraint ranges. • The fractional-order sliding mode stochastic control method exhibits improved robustness and flexibility compared to traditional integer-order sliding mode control method. [ABSTRACT FROM AUTHOR]

This article highlights my lifelong friendship and collaboration with George Yin on developing new methodologies in diversified fields in control theory and their branches, as well as their engineering and medical applications. It explains the roots and motivations in initiating these problems and George's critical contributions in applying his extensive expertise in stochastic systems to bring these challenging pursuits to fruitful new frameworks and methodologies. [ABSTRACT FROM AUTHOR]

This paper establishes the optimality of an adaptive identification algorithm for stochastic dynamical systems with saturated output observations, which arise from various fields in engineering and social systems. We focus on a two-step Newton-type adaptive algorithm proposed in the authors' previous paper to estimate the unknown parameter. By using a Bayesian embedding approach and several powerful mathematical techniques including stochastic Lyapunov functions and limit theories for martingale, we show that the conditional mean square error of estimates can asymptotically achieve the conditional Cramér-Rao (C-R) bound for dynamical systems as can be achieved by the optimal estimates. We remark that our theory does not need the widely used independent and identically distributed (i.i.d) condition or periodicity property on system signals in the existing related literature, and does not exclude the applications to stochastic dynamical feedback systems. [ABSTRACT FROM AUTHOR]

This paper applies the Razumikhin idea to study the stabilisation of hybrid stochastic systems by discrete-time state feedback control, which works intermittently and is designed boundedly. Theoretically, the Razumikhin method is generalised in view of time-varying functions, rather than constants, where the time-inhomogeneous property of intermittent control could be fully made use of. In practice, the control cost could be reduced significantly since the controller is bounded, not observed continuously and having rest time. Moreover, there will be a wider range of applications especially for models that do not satisfy the linear growth condition (say highly nonlinear). An example of the coupled Van der PolDuffing oscillator system is hence provided to show the practicability of the developed theory. [ABSTRACT FROM AUTHOR]

This work is devoted to studying a class of biological control problems in a stochastic environment. Specifically, it focuses on stochastic Lotka-Voltera systems. Our effort is on treating average cost per unit time controlled diffusions. It is natural to use a vanishing discount argument. However, in contrast to the existing literature, neither the 'near-monotone' nor the 'stable' condition is satisfied in the current set up. In reference to one of our recent works, we divide the domain into two parts. In one sub-domain, the 'near-monotone' condition is satisfied, whereas in the other sub-domain, the 'stable' condition is satisfied. We then carefully work out the analysis in the two domains so as to obtain the desired optimal control. [ABSTRACT FROM AUTHOR]

This research aims to investigate the stabilization of highly nonlinear hybrid stochastic differential delay equations (HSDDEs) with Lévy noise by delay feedback control. The coefficients of these systems satisfy a more general polynomial growth condition instead of classical linear growth condition. Precisely, an appropriate Lyapunov functional is constructed to analyze the stabilization of such systems in the sense of H∞-stability and asymptotic stability. The theoretical analysis indicates that the delay can affect the stability of highly nonlinear hybrid stochastic systems. [ABSTRACT FROM AUTHOR]

We present a numerical method for learning the dynamics of slow components of unknown multiscale stochastic dynamical systems. While the governing equations of the systems are unknown, bursts of observation data of the slow variables are available. By utilizing the observation data, our proposed method is capable of constructing a generative stochastic model that can accurately capture the effective dynamics of the slow variables in distribution. We present a comprehensive set of numerical examples to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the long term dynamics of the supercritical stochastic wave equation driven by multiplicative noise on unbounded domains. By introducing an appropriate variable, we convert the stochastic equation into a pathwise deterministic wave equation and prove the existence and uniqueness of solutions by the uniform Strichartz estimates. Then we prove the existence and uniqueness of pullback random attractors of the non-autonomous random dynamical system associated with the stochastic equation. The pullback asymptotic compactness of solutions is established by the idea of uniform tail-ends estimates as well as the decomposition method of the solution operator in order to overcome the non-compactness of Sobolev embeddings on unbounded domains. [ABSTRACT FROM AUTHOR]

We consider the problem of obtaining effective representations for the solutions of linear, vector-valued stochastic differential equations (SDEs) driven by non-Gaussian pure-jump Lévy processes, and we show how such representations lead to efficient simulation methods. The processes considered constitute a broad class of models that find application across the physical and biological sciences, mathematics, finance, and engineering. Motivated by important relevant problems in statistical inference, we derive new, generalised shot-noise simulation methods whenever a normal variance-mean (NVM) mixture representation exists for the driving Lévy process, including the generalised hyperbolic, normal-gamma, and normal tempered stable cases. Simple, explicit conditions are identified for the convergence of the residual of a truncated shot-noise representation to a Brownian motion in the case of the pure Lévy process, and to a Brownian-driven SDE in the case of the Lévy-driven SDE. These results provide Gaussian approximations to the small jumps of the process under the NVM representation. The resulting representations are of particular importance in state inference and parameter estimation for Lévy-driven SDE models, since the resulting conditionally Gaussian structures can be readily incorporated into latent variable inference methods such as Markov chain Monte Carlo, expectation-maximisation, and sequential Monte Carlo. [ABSTRACT FROM AUTHOR]

This paper studies the exponential stability in the mean square of the stochastic complex networks with time-varying coupling under an intermittent dynamic event-triggered control. A dynamic term and an exponential function are introduced into the event-triggered mechanism to reduce the number of control updates. Simultaneously, the minimum inter-execution time for every sample path solution of the stochastic complex networks, independent mathematical expectation, is obtained. Unlike previous research, the event-triggered mechanism under the stochastic version is more reasonable due to the absence of mathematical expectations in the event-triggered function. Furthermore, using the average control rate for intermittent strategy and the Lyapunov method, sufficient conditions for exponential stability in the mean square under intermittent dynamic event-triggered control are derived. Finally, an example with numerical simulations is provided to validate the feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]

Aiming at the issue of suboptimal demodulation performance in OFDM signal enhancement processes due to slow response speeds in first-order overdamped stochastic resonance systems, this paper designs an OFDM signal enhancement and demodulation system based on second-order underdamped unsaturated bistable stochastic resonance (SUUBSR). Firstly, a nonsaturated bistable potential function model is constructed by combining a traditional monostable potential function with a Gaussian potential function, and a damping coefficient is introduced simultaneously to build the SUUBSR system. Subsequently, the transient and steady-state output response analytical expressions of the SUUBSR system under OFDM signal excitation are derived, the energy loss of OFDM symbol waveforms caused by the transient response of the system is discussed, and the relationship between the damping coefficient and the steady-state output response of the system is explored. Finally, simulations are conducted to evaluate the enhancement and demodulation process of OFDM signals using the SUUBSR system. The simulation results show that at an input signal–noise ratio of 2 dB, compared to the first-order unsaturated bistable stochastic resonance (FUBSR) system, the proposed system reduces the system response time by 3.956% of a symbol period and decreases the demodulation bit error rate for OFDM signals by approximately 35%. [ABSTRACT FROM AUTHOR]