Stochastic Perturbation Research Articles

Induced polar perturbations with stochastic effects in dense matter relativistic stars: A theoretical probe at intermediate sub-hydro mesoscopic scales

A linear response relation between metric and fluid perturbations driven by a background internal noise source is used as a framework for addressing stochastic effects in order to establish a mesoscopic theory for dense matter relativistic stars. In this paper, nonradial polar perturbations are worked out, which are important from the point of view of detection in future. We present qualitative first results in this paper, numerical estimates have to await further progress in theoretical modeling. These perturbations carry a new generalized stochastic nature and are obtained as solutions of the classical Einstein–Langevin equation which has been recently proposed. The significance of these stochastic nonradial polar perturbations lies at probing the intermediate sub-hydro scales inside the dense fluid. This formalism extends towards a mesoscopic scale nonequilibrium/near-equilibrium statistical mechanics study for relativistic star interiors. The generalized stochastic noise originates as the remnant of collapse mechanism and dynamical effects at intermediate scales in isolated massive stars which drives these polar perturbations. More specifically, for cold dense matter which is our focus in this paper, it is either the interplay between the degeneracy pressure and the gravitational pressure, or the multiscale phenomena like turbulences giving rise to the seeds of stochastic effects in the gravitating body. Characterizing such stochastic effects can lead to an improved understanding of the nature of dense matter and help to probe multiple scales which are yet untouched.

Discontinuous Adaptive Impulsive Control of Uncertain System With Extension in Stochastic Perturbation and Actuator Saturation.

This article investigates the discontinuous adaptive impulsive control of uncertain linear and nonlinear systems with stochastic perturbations and actuator saturation. Existing literature on adaptive impulsive control schemes adopt continuous state information in designing the continuous adaptive law, which loses the advantages of impulsive control completely. In this article, the discontinuous adaptive law is proposed which only requires the state information be transmitted at impulsive instants, therefore, the communication cost could be reduced and the control system is more practical in implementation. Furthermore, a discontinuous adaptive impulsive control law is derived to realize stabilization of uncertain nonlinear systems with stochastic perturbations and actuator saturation, and the robustness of the closed-loop system with the discontinuous adaptive impulsive control scheme is proved to be effective. Finally, two simulation examples for adaptive impulsive control are presented to verify the accuracy of our results.

Synchronization robustness analysis of memristive-based neural networks with deviating arguments and stochastic perturbations

<abstract><p>In this article, we investigate the robustness of memristive-based neural networks (MNNs) with deviating arguments (DAs) and stochastic perturbations (SPs). Based on the set-valued mapping method, differential inclusion theory and Gronwall inequalities, we derive the upper bounds for the width of DAs and the intensity of SPs. When the DAs and SPs are smaller than these upper bounds, the MNNs maintains exponential synchronization. Finally, several specific simulation examples demonstrate the effectiveness of the results.</p></abstract>

Stability of asymptotically Hamiltonian systems with damped oscillatory and stochastic perturbations

Stability of asymptotically Hamiltonian systems with damped oscillatory and stochastic perturbations

Synchronization of Multiplex Networks With Stochastic Perturbations via Intermittent Control

Synchronization of Multiplex Networks With Stochastic Perturbations via Intermittent Control

Stochastic stability of an autoresonance model with a center–saddle bifurcation

The purpose of this work is to investigate the effect of stochastic perturbations of the white noise type on the stability of capture into autoresonance in oscillating systems with a variable pumping amplitude and frequency such that a center–saddle bifurcation occurs in the corresponding limiting autonomous system. The another purpose is determine the dependence of the intervals of stochastic stability of the autoresonance on the noise intensity. Methods. The existence of autoresonant regimes with increasing amplitude is proved by constructing and justificating asymptotic solutions in the form of power series with constant coefficients. The stability of solutions in terms of probability with respect to noise is substantiated using stochastic Lyapunov functions. Results. The conditions are described under which the autoresonant regime is preserved and disappears when the parameters pass through bifurcation values. The dependence of the intervals of stochastic stability of autoresonance on the degree of damping of the noise intensity is found. It is shown that more stringent restrictions are required to preserve the stability of solutions for the bifurcation values of the parameters. Conclusion. At the level of differential equations describing capture into autoresonance, the effect of damped stochastic perturbations on the center–saddle bifurcation is studied. The results obtained indicate the possibility of using damped oscillating perturbations for stable control of nonlinear systems.

Splitting integrators for linear Vlasov equations with stochastic perturbations

Splitting integrators for linear Vlasov equations with stochastic perturbations

Dynamics and numerical simulations of a generalized mosquito-borne epidemic model using the Ornstein-Uhlenbeck process: Stability, stationary distribution, and probability density function

<abstract><p>In this paper, we proposed a generalized mosquito-borne epidemic model with a general nonlinear incidence rate, which was studied from both deterministic and stochastic insights. In the deterministic model, we proved that the endemic equilibrium was globally asymptotically stable when the basic reproduction number $ R_0 $ was greater than unity and the disease free equilibrium was globally asymptotically stable when $ R_0 $ was lower than unity. In addition, considering the effect of environmental noise on the spread of infectious diseases, we developed a stochastic model in which the infection rates were assumed to satisfy the mean-reverting log-normal Ornstein-Uhlenbeck process. For this stochastic model, two critical values, known as $ R_0^s $ and $ R_0^E $, were introduced to determine whether the disease will persist or die out. Additionally, the exact probability density function of the stationary distribution near the quasi-equilibrium point was obtained. Numerical simulations were conducted to validate the results obtained and to examine the impact of stochastic perturbations on the model.</p></abstract>

Longtime evolution and stationary response of a stochastic tumor-immune system with resting T cells.

In this paper, we take the resting T cells into account and interpret the progression and regression of tumors by a predator-prey like tumor-immune system. First, we construct an appropriate Lyapunov function to prove the existence and uniqueness of the global positive solution to the system. Then, by utilizing the stochastic comparison theorem, we prove the moment boundedness of tumor cells and two types of T cells. Furthermore, we analyze the impact of stochastic perturbations on the extinction and persistence of tumor cells and obtain the stationary probability density of the tumor cells in the persistent state. The results indicate that when the noise intensity of tumor perturbation is low, tumor cells remain in a persistent state. As this intensity gradually increases, the population of tumors moves towards a lower level, and the stochastic bifurcation phenomena occurs. When it reaches a certain threshold, instead the number of tumor cells eventually enter into an extinct state, and further increasing of the noise intensity will accelerate this process.

The Algebra and Calculus of Stochastically Perturbed Spacetime with Classical and Quantum Applications

We consider an alternative to dark matter as a potential solution to various remaining problems in physics: the addition of stochastic perturbations to spacetime to effectively enforce a minimum length and establish a fundamental uncertainty at minimum length (ML) scale. To explore the symmetry of spacetime to such perturbations both in classical and quantum theories, we develop some new tools of stochastic calculus. We derive the generators of rotations and boosts, along with the connection, for stochastically perturbed, minimum length spacetime (“ML spacetime”). We find the metric, the directional derivative, and the canonical commutator preserved. ML spacetime follows the Lie algebra of the Poincare group, now expressed in terms of the two-point functions of the stochastic fields (per Ito’s lemma). With the fundamental uncertainty at ML scale a symmetry of spacetime, we require the translational invariance of any classical theory in classical spacetime to also include the stochastic spacetime perturbations. As an application of these ideas, we consider galaxy rotation curves for massive bodies to find that—under the Robertson–Walker minimum length theory—rotational velocity becomes constant as the distance to the center of the galaxy becomes very large. The new tools of stochastic calculus also set the stage to explore new frontiers at the quantum level. We consider a massless scalar field to derive the Ward-like identity for ML currents.

Toward a Stochastic Complete Active Space Second-Order Perturbation Theory.

In this work, an internally contracted stochastic complete active space second-order perturbation theory, stochastic-CASPT2, is reported. The method relies on stochastically sampled reduced density matrices (RDMs) up to rank four and contractions thereof with the generalized Fock matrix. A new protocol for calculating higher-order RDMs in full configuration interaction quantum Monte Carlo (FCIQMC) has been designed based on (1) restricting sampling of the corresponding excitations to a deterministic subspace, (2) averaging the RDMs from independent dynamics and (3) projecting them onto the closest positive semi-definite matrix. Our protocol avoids previously encountered numerical conditioning problems in the orthogonalization of the perturber overlap matrix stemming from numerical noise. The chromium dimer CASSCF(12,12)/CASPT2 binding curve is computed as a proof of concept.

About an Unsolved Problem of Stabilization by Noise for Difference Equations

The paper is devoted to the effect of “stabilization by noise”. The essence of this effect is that an unstable deterministic system is stabilized by stochastic perturbations of sufficiently high intensity. The problem is that the effect of “stabilization by noise”, well-known already for more than 50 years for stochastic differential equations, still has no analogue for stochastic difference equations. Here, a corresponding hypothesis is formulated and discussed, the truth of which is illustrated and confirmed by numerical simulation of solutions of stochastic linear and nonlinear difference equations. However, a problem of a formal proof of this hypothesis remains open.

Stationary distribution and near‐optimal control of a stochastic reaction–diffusion HIV model

Considering stochastic perturbations and spatial diffusion in both virus‐to‐cell and cell‐to‐cell transmissions, a stochastic reaction–diffusion HIV model is developed. Firstly, the existence and uniqueness of a global positive solution and stationary distribution are demonstrated. Secondly, considering the effect of drug therapy on the disease, the control strategy is introduced into the stochastic HIV model. By employing Pontryagin's stochastic maximum principle, the sufficient and necessary conditions are obtained for near‐optimal control. Finally, numerical simulations are reported to support and supplement our theoretical results.

Exact Response Theory for Time-Dependent and Stochastic Perturbations.

The exact, non perturbative, response theory developed within the field of non-equilibrium molecular dynamics, also known as TTCF (transient time correlation function), applies to quite general dynamical systems. Its key element is called the dissipation function because it represents the power dissipated by external fields acting on the particle system of interest, whose coupling with the environment is given by deterministic thermostats. This theory has been initially developed for time-independent external perturbations, and then it has been extended to time-dependent perturbations. It has also been applied to dynamical systems of different nature, and to oscillator models undergoing phase transitions, which cannot be treated with, e.g., linear response theory. The present work includes time-dependent stochastic perturbations in the theory using the Karhunen-Loève theorem. This leads to three different investigations of a given process. In the first, a single realization of the stochastic coefficients is fixed, and averages are taken only over the initial conditions, as in a deterministic process. In the second, the initial condition is fixed, and averages are taken with respect to the distribution of stochastic coefficients. In the last investigation, one averages over both initial conditions and stochastic coefficients. We conclude by illustrating the applicability of the resulting exact response theory with simple examples.

On–off intermittency in population outbreaks: Reactive equilibria and propagation on networks

Ecological systems are subject to environmental variability and fluctuations: understanding the role of such stochastic perturbations in inducing on–off intermittency is the central motivation for this study. This research extends the exploration of parameters leading to the emergence of on–off intermittency within a discrete Beddington-Free-Lawton host-parasitoid model. We introduce random perturbation factors that impact both the grazing intensity and the growth rate of the host population. An intriguing aspect of this study is the numerical evidence of the reactivity of the free-parasitoid fixed point as a route to on–off intermittency. This finding is significant because it sheds light on how stable ecological equilibria can transition into intermittency before progressing toward chaotic behaviour. Moreover, our study explores the host-parasitoid coupling within the Beddington-Free-Lawton model when it is applied to a complex network, a significant framework for modelling ecological interactions. The paper reveals that such network-based interactions induce parasitoid bursts that are not observed in a single population scenario.

On the existence and uniqueness of solution to a stochastic Chemotaxis–Navier–Stokes model

In this article, we study a mathematical system modelling the dynamic of the collective behaviour of oxygen-driven swimming bacteria in an aquatic fluid flowing in a two dimensional bounded domain under stochastic perturbation. This model can be seen as a stochastic version of Chemotaxis–Navier–Stokes model. We prove the existence of a unique (probabilistic) strong solution. In addition, we establish some properties of the strong solution. More precisely, we prove that the unique solution is positive and satisfies the mass conservation property and an energy inequality.

From Cell States to Cell Fates: Control of Cell State Transitions.

We examine the coordinated behavior of thousands of genes in cell fate transitions through genome expression as an integrated dynamical system using theconcepts of self-organized criticality and coherent stochastic behavior. To quantify theeffects of thecollective behavior of genes, we adopted the flux balance approach and developed it in a new tool termed expression flux analysis (EFA). Here we describe this tool and demonstrate how its application to specific experimental genome-wide expression data provides new insights into the dynamics of the cell-fate transitions. Particularly, we show that in cell fate change, specific stochastic perturbations can spread over the entire system to guide distinct cell fate transitions through switching cyclic flux flow in the genome engine. Utilization of EFA enables us to elucidate a unified genomic mechanism for when and how cell-fate change occurs through critical transitions.

Effects of stochastic perturbations on the tree–grass coexistence in savannas

Effects of stochastic perturbations on the tree–grass coexistence in savannas

Eigenvibrations of Kirchhoff Rectangular Random Plates on Time-Fractional Viscoelastic Supports via the Stochastic Finite Element Method.

The present work's main objective is to investigate the natural vibrations of the thin (Kirchhoff-Love) plate resting on time-fractional viscoelastic supports in terms of the Stochastic Finite Element Method (SFEM). The behavior of the supports is described by the fractional order derivatives of the Riemann-Liouville type. The subspace iteration method, in conjunction with the continuation method, is used as a tool to solve the non-linear eigenproblem. A deterministic core for solving structural eigenvibrations is the Finite Element Method. The probabilistic analysis includes the Monte-Carlo simulation and the semi-analytical approach, as well as the iterative generalized stochastic perturbation method. Probabilistic structural response in the form of up to the second-order characteristics is investigated numerically in addition to the input uncertainty level. Finally, the probabilistic relative entropy and the safety measure are estimated. The presented investigations can be applied to the dynamics of foundation plates resting on viscoelastic soil.

Dynamical noise can enhance high-order statistical structure in complex systems.

Recent research has provided a wealth of evidence highlighting the pivotal role of high-order interdependencies in supporting the information-processing capabilities of distributed complex systems. These findings may suggest that high-order interdependencies constitute a powerful resource that is, however, challenging to harness and can be readily disrupted. In this paper, we contest this perspective by demonstrating that high-order interdependencies can not only exhibit robustness to stochastic perturbations, but can in fact be enhanced by them. Using elementary cellular automata as a general testbed, our results unveil the capacity of dynamical noise to enhance the statistical regularities between agents and, intriguingly, even alter the prevailing character of their interdependencies. Furthermore, our results show that these effects are related to the high-order structure of the local rules, which affect the system's susceptibility to noise and characteristic time scales. These results deepen our understanding of how high-order interdependencies may spontaneously emerge within distributed systems interacting with stochastic environments, thus providing an initial step toward elucidating their origin and function in complex systems like the human brain.