Існування та єдиність розв’язку стохастичного диференціально-функціонального рівняння з частинними похідними спеціального вигляду та методи його комп’ютерного моделювання

Capturing uncertainty in models of complex dynamical systems is crucial to designing safe controllers. Stochastic noise causes aleatoric uncertainty, whereas imprecise knowledge of model parameters leads to epistemic uncertainty. Several approaches use formal abstractions to synthesize policies that satisfy temporal specifications related to safety and reachability. However, the underlying models exclusively capture aleatoric but not epistemic uncertainty, and thus require that model parameters are known precisely. Our contribution to overcoming this restriction is a novel abstraction-based controller synthesis method for continuous-state models with stochastic noise and uncertain parameters. By sampling techniques and robust analysis, we capture both aleatoric and epistemic uncertainty, with a user-specified confidence level, in the transition probability intervals of a so-called interval Markov decision process (iMDP). We synthesize an optimal policy on this iMDP, which translates (with the specified confidence level) to a feedback controller for the continuous model with the same performance guarantees. Our experimental benchmarks confirm that accounting for epistemic uncertainty leads to controllers that are more robust against variations in parameter values.

This paper focuses on the hybrid effects of parameter uncertainty, stochastic perturbation, and impulses on global stability of delayed neural networks. By using the Ito formula, Lyapunov function, and Halanay inequality, we established several mean-square stability criteria from which we can estimate the feasible bounds of impulses, provided that parameter uncertainty and stochastic perturbations are well-constrained. Moreover, the present method can also be applied to general differential systems with stochastic perturbation and impulses.

We review the behaviour of the Gibbs’ and conditional entropies in deterministic and stochastic systems and continue to a formulation appropriate for a stochastically perturbed system with {} dynamics. The underlying question driving these investigations: ``Is the origin of the universally observed unidirectionality of time in our universe connected to the behaviour of entropy?’’ We focus on temporal entropic behaviour with a review of previous results in deterministic and stochastic systems. Our emphasis is on the temporal behaviour of the Gibbs’ and conditional entropies as these entropies give equilibrium results in concordance with experimental findings. In invertible deterministic systems both entropies are temporally constant as has been well known for decades. The addition of stochastic perturbations (a Wiener process) leads to an indeterminate (either increasing or decreasing) behaviour of the Gibbs’ entropy, but the conditional entropy monotonically approaches equilibrium with increasing time. The presence of delays in the dynamics, whether stochastically perturbed or not, leads to situations in which the Gibbs’ and conditional entropies evolution can be oscillatory and not monotone, and may not approach equilibrium.

We investigate the effects of different stochastic noises on the dynamics of the edge-localised modes (ELMs) in magnetically confined fusion plasmas by using a time-dependent PDF method, path-dependent information geometry (information rate, information length), and entropy-related measures (entropy production, mutual information). The oscillation quenching occurs due to either stochastic particle or magnetic perturbations, although particle perturbation is more effective in this amplitude diminishment compared with magnetic perturbations. On the other hand, magnetic perturbations are more effective at altering the oscillation period; the stochastic noise acts to increase the frequency of explosive oscillations (large ELMs) while decreasing the frequency of more regular oscillations (small ELMs). These stochastic noises significantly reduce power and energy losses caused by ELMs and play a key role in reproducing the observed experimental scaling relation of the ELM power loss with the input power. Furthermore, the maximum power loss is closely linked to the maximum entropy production rate, involving irreversible energy dissipation in non-equilibrium. Notably, over one ELM cycle, the information rate appears to keep almost a constant value, indicative of a geodesic. The information rate is also shown to be useful for characterising the statistical properties of ELMs, such as distinguishing between explosive and regular oscillations and the regulation between the pressure gradient and magnetic fluctuations.

The dynamical evolution of dislocations in plastically deformed metals is controlled by both deterministic factors arising out of applied loads and stochastic effects appearing due to fluctuations of internal stress. Such types of stochastic dislocation processes and the associated spatially inhomogeneous modes lead to randomness in the observed deformation structure. Previous studies have analyzed the role of randomness in such textural evolution, but none of these models have considered the impact of a finite decay time (all previous models assumed instantaneous relaxation which is "unphysical") of the stochastic perturbations in the overall dynamics of the system. The present article bridges this knowledge gap by introducing a colored noise in the form of an Ornstein-Uhlenbeck noise in the analysis of a class of linear and nonlinear Wiener and Ornstein-Uhlenbeck processes that these structural dislocation dynamics could be mapped on to. Based on an analysis of the relevant Fokker-Planck model, our results show that linear Wiener processes remain unaffected by the second time scale in the problem, but all nonlinear processes, both the Wiener type and Ornstein-Uhlenbeck type, scale as a function of the noise decay time τ. The results are expected to ramify existing experimental observations and inspire new numerical and laboratory tests to gain further insight into the competition between deterministic and random effects in modeling plastically deformed samples.

An Interactive Ensemble (IE) platform was established based on a Standard Coupled (SC) climate model with seven atmosphere–land model realizations coupled to a single ocean model and a single sea ice model. The IE strategy reduces stochastic noise generated by atmospheric dynamics and therefore can be used to estimate the impact of atmospheric perturbations on the ocean. The atmospheric noise reduction shallows the mixed layer, modifies the oceanic upwelling/downwelling, and cools the surface. It is assumed that the deterministic part of the surface cooling is due to the reduced heat capacity of the mixed layer in high latitudes and the changed surface wind curl that causes upwelling/downwelling. The positive albedo feedback is responsible for the polar amplified changes. The weakening of meridional overturning circulation (MOC) further suppresses ocean heat transport to the Northern Hemisphere. The model results suggest that the irregular El Niño Southern Oscillation (ENSO) cycle partly arises from the stochastic perturbation in the atmosphere. Comparison between the IE and the SC simulations also indicates the overestimated atmospheric perturbations in the SC model.

Cluster synchronization of Markovian switching complex networks with hybrid couplings and stochastic perturbations

Fractional stochastic differential equations with memory effects are fundamental in modeling phenomena across physics, biology, and finance, where long‐range dependencies and random fluctuations coexist, yet their stability analysis under non‐Lipschitz conditions remains a significant challenge, particularly for systems involving Riemann–Liouville fractional operators with stochastic perturbations. To address this challenge, this article establishes sufficient conditions for the existence, uniqueness, and Ulam–Hyers–Rassias (UHR) stability of mild solutions to nonlinear Riemann–Liouville fractional stochastic differential equations (RL‐FSDEs) driven by additive white noise. Through the application of a generalized Banach contraction principle within a carefully constructed weighted metric space, our methodology relaxes the restrictive Lipschitz constraints commonly found in existing literature, thereby significantly broadening the class of admissible nonlinearities. The UHR stability is rigorously established in the mean‐square sense by deriving explicit bounds that systematically relate approximate solutions to exact trajectories. Our theoretical findings are substantiated through four illustrative examples demonstrating that solutions remain consistently close to approximate trajectories under random fluctuations, thereby significantly advancing the stability theory of stochastic fractional systems and providing a robust analytical framework for applications in control theory, financial modeling, and engineering systems governed by memory effects and stochastic noise.

Regulatory T cells (Treg) and T helper 17 cells (Th17), both derived from naïve T cells, play pivotal roles in modulating immune responses, and their dynamic balance is critical for maintaining immune homeostasis. Existing studies predominantly focus on the regulatory mechanisms of individual cell types and lack a systematic analysis of how multiparametric interactions and stochastic perturbations jointly influence cell-fate equilibrium. In this study, we investigate the gene regulatory network of Treg and Th17 cells in two major aspects: (i) elucidating the dynamical features of the network and (ii) examining the regulatory effects of Gaussian white noise on the balance between the two lineages. By integrating systems dynamics, non-equilibrium mechanics, and stochastic process theory, we propose a unified modeling framework that incorporates Gaussian white noise to simulate stochastic perturbations in gene expression, thereby establishing a mapping between parameter sets and cellular phenotypes and quantifying the regulatory weights of key factors. Our results demonstrate that parameters such as extracellular TGF-β input, foxp3 mRNA synthesis rate, and Stat3 protein degradation rate significantly modulate the differentiation balance between Treg and Th17 cells. Furthermore, within a certain range, stronger Gaussian white noise promotes the differentiation of naïve T cells toward the Th17 lineage, thereby enhancing immune responsiveness. This finding aligns with prior experimental evidence demonstrating that stochastic noise can amplify immune response efficacy. This framework uniquely couples static and dynamic perturbations, revealing stochasticity's role in cell-fate decisions and offering both a quantitative tool for studying Th17-Treg balance and a generalizable approach for other differentiation systems.

This article is mainly concerned with the fixed-time and predefined-time synchronization problem for a type of complex-valued BAM neural networks with stochastic perturbations and impulse effect. First, some previous fixed-time stability results on nonlinear impulsive systems in which stabilizing and destabilizing impulses were separately analyzed are extended to a general case in which the stabilizing and destabilizing impulses can be handled simultaneously. Additionally, using the same logic, a new predefined-time stability lemma for stochastic nonlinear systems with a general impulsive effect is obtained by using the inequality technique. Then, based on these novel results, two novel controllers are implemented to derive some simple fixed/predefined-time synchronization criteria for the considered complex-valued impulsive BAM neural networks with stochastic perturbations using the non-separation method. Finally, two numerical examples are given to demonstrate the feasibility of the obtained results.

The time evolution of prices of different financial quantities is often represented as a partial differential equation (PDE) with independent variables being time and prices of some other, often underlying, assets. Let $$V(S_t, t)$$ be the price of an option at time t when the share price of the underlying stock is $$S_t$$ . See Appendix A.1 for background on mathematical finance that is used in what follows. Under the Black–Scholes set-up, we have a risk-less asset bond $$B_t$$ and a risky asset stock $$S_t$$ . They evolve as where r is the interest rate, $$\mu $$ is the drift, $$\sigma $$ is the volatility and W is a standard Brownian motion(BM). We apply Ito’s formula to the option price to get Consider the discounted option price $$B_t^{-1}V(S_t,t)$$ . By the Fundamental Theorem of Arbitrage Pricing (see Appendix), the discounted option price must be a martingale under the risk-neutral measure . Also, under the risk-neutral measure $$\mu =r$$ . We have, from the above, For a martingale, the coefficient of the dt term has to be zero, otherwise there is a systematic drift.

Synchronization of delayed complex dynamical networks with impulsive and stochastic effects

BackgroundForecasting influenza outbreaks remains a significant challenge due to the complexity of disease transmission and the influence of environmental and behavioral factors. Traditional models based solely on the basic reproduction number often fall short in capturing the full scope of outbreak dynamics.MethodsIn this study, we employ a seasonally adjusted SEIRT model incorporating stochastic differential equations (SDEs), including Brownian motion and Lévy jump processes, to simulate random and abrupt fluctuations in transmission. A branching process approximation is used to evaluate the probability of an epidemic under the influence of seasonal variability and stochastic perturbations. The model is calibrated using weekly influenza case data from Mexico, with noise components estimated from publicly available CDC [1] and WHO [2] surveillance data.ResultsSimulation results show that the inclusion of stochastic effects and periodic transmission rates significantly enhances the model's accuracy in reflecting real-world epidemic dynamics. Numerical comparisons between deterministic, Brownian-based, and Lévy-based scenarios reveal that both the initial state of the exposed or infectious subpopulation and the seasonal transmission patterns are critical to determining outbreak probabilities. Results indicate that seasonal transmission rates and stochastic effects significantly alter epidemic probabilities, with Lévy processes capturing abrupt outbreak dynamics more accurately than deterministic models.ConclusionsThe findings underscore that deterministic models may underestimate epidemic risk when they overlook random and sudden changes in contact rates or disease introduction. The proposed stochastic modeling framework yields a deeper understanding of influenza transmission dynamics by incorporating uncertainty and seasonal variability, thereby supporting more informed and effective public health decision-making.

This work concerns the structural vibration of a bladeless wind turbine, modelled by a two-deck Euler–Bernoulli beam, due to a surrounding potential flow. The deflection is governed by the Euler–Bernoulli equation which is studied first by a linear theory and then computed numerically by a finite difference method in space with a collocation method over the arc length, and an implicit Euler method in time. The fluid motion in the presence of gravity is governed by the full Euler equations and solved by the time-dependent conformal mapping technique together with a pseudo-spectral method. Numerical experiments of excitation by a moving disturbance on the fluid surface with/without a stochastic noise are carried out. The random process involved in generating the noise on the water surface is driven by a Wiener Process. A Monte Carlo method is used for stochastic computations. The generated surface waves impinge on the beam causing structural vibration which is presented and discussed in detail. By elementary statistical analysis, the structural response subject to the stochastic hydrodynamic disturbance caused by white noise is found to be Gaussian.

Quantifying effects of stochastic focusing and defocusing on noise propagation in biochemical reaction networks