Nonlinear Drift Term Research Articles

Mean random dynamical systems and random attractors of stochastic Hopfield neural models with locally Lipschitz noise and time delay

In this paper, we investigate the long term dynamical behaviour of stochastic Hopfield neural network models driven by white noise with time delay, where the nonlinear drift and diffusion terms are both locally Lipschitz continuous. We establish the global-in-time solvability of these systems with locally Lipschitz continuous nonlinear drift and diffusion terms by using mean square uniform estimates of solutions. Furthermore, we demonstrate that the mean random dynamical system generated by the solution operators possesses a unique weak pullback mean random attractor in the Hilbert space L 2 ( Ω ~ , F ι ; l 2 ) × L 2 ( Ω ~ , F ι ; L 2 ( ( − ϱ , 0 ) , l 2 ) ) .

Exponential contraction rates for a class of degenerate SDEs with Lévy noises

Given a separable and real Hilbert space H, we consider the following stochastic differential equation (SDE) on H:dXt=−Xtdt+b(Xt)dt+dZt, where Z:=(Zt)t≥0 is a cylindrical pure jump Lévy process on H which may be degenerate in the sense that the support of Z is contained in a finite dimensional space. When the nonlinear drift term b(x) is contractive with respect to some proper modified norm of H for large distances, we obtain explicit exponential contraction rates of the SDE above in terms of Wasserstein distance under mild assumptions on the Lévy process Z. The approach is based on the refined basic coupling of Lévy noises, and it also works well when the so-called Lyapunov condition is satisfied.

Large deviation principle for stochastic FitzHugh–Nagumo lattice systems

This paper is devoted to the large deviation principle of stochastic FitzHugh–Nagumo lattice system, where the nonlinear drift and diffusion terms are locally Lipschitz continuous. Based on the well-posedness and the convergence of the solutions of the controlled system associated with the considered system, by the weak convergence method, we establish the large deviation principle in C([0,T],l2×l2) for such infinite dimensional system.

Invariant measures of stochastic delay complex Ginzburg-Landau equations

This paper is concerned with the existence of invariant measures for stochastic delay complex Ginzburg-Landau equations defined on the entire integer set. When the nonlinear drift and diffusion terms are globally Lipschitz continous, the existence of invariant measures of the equation is proved by estiblishing the tightness of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem.

Electrical Response Estimation of Vibratory Energy Harvesters via Hilbert Transform Based Stochastic Averaging

Abstract Converting mechanical vibrations into electrical power with vibratory energy harvesters can ensure the portability, efficiency, and sustainability of electronic devices and batteries. Vibratory energy harvesters are typically modeled as nonlinear oscillators subject to random excitation, and their design requires a complete characterization of their probabilistic responses. However, simulation techniques such as Monte Carlo are computationally prohibitive when the accurate estimation of the response probability distribution is needed. Alternatively, approximate methods such as stochastic averaging can estimate the probabilistic response of such systems at a reduced computational cost. In this paper, the Hilbert transform based stochastic averaging is used to model the output voltage amplitude as a Markovian stochastic process with dynamics governed by a stochastic differential equation with nonlinear drift and diffusion terms. Moreover, the voltage amplitude dependent damping and stiffness terms are determined via an appropriate equivalent linearization, and the stationary probability distribution of the output voltage amplitude is obtained analytically by solving the corresponding Fokker–Plank equation. Two examples are used to demonstrate the accuracy of the obtained analytical probability distributions via comparisons with Monte Carlo simulation data.

Periodic measures for the stochastic delay modified Swift–Hohenberg lattice systems

In this paper, the existence and the limiting behavior of periodic measures for the periodic stochastic modified Swift–Hohenberg lattice systems with variable delays are analyzed. We first prove the existence and uniqueness of global solution when the nonlinear T-periodic drift and diffusion terms are locally Lipchitz continuous and linearly growing. Then we show the existence of periodic measures of the system under some assumptions. Finally, by strengthening the assumptions, we prove that the set of all periodic measures is weakly compact, and we also show that every limit point of a sequence of periodic measures of the original system must be a periodic measure of the limiting system when the noise intensity tends to zero.

Weak mean attractors of stochastic p-Laplacian delay lattice systems driven by nonlinear noise

This paper deals with the existence and uniqueness of weak pullback mean random attractors for a class non-autonomous stochastic p-Laplacian delay lattice systems with local Lipschitz nonlinear terms driven by nonlinear noise defined on the entire integer set Z. We first establish the global well-posedness to non-autonomous stochastic p-Laplacian delay lattice system in C([τ,∞),L2(Ω,ℓ2)) when the nonlinear diffusion terms and drift terms are locally Lipschitz continuous functions, based on the uniform estimates of approximate solutions in a finite time-interval as well as an appropriate stopping time. We then define a mean random dynamical system via the solution operators and prove the existence and uniqueness of weak pullback mean random attractors in product Hilbert space L2(Ω,F;ℓ2)×L2(Ω,F;L2((−ρ,0),ℓ2)) under the condition that the constant λ in the system is relatively large.

A positivity-preserving numerical algorithm for stochastic age-dependent population system with Lévy noise in a polluted environment

This paper develops a model for describing a stochastic age-dependent population system (SADPS) with Lévy noise in a polluted environment. As the model includes a nonlinear drift term and Lévy noise, it is difficult, if not impossible, to derive an analytical solution of the model. This study thus aims at developing a numerical algorithm that is based on a modified truncated Euler-Maruyama (EM) method and positivity preserving. We first study global positivity for the solution of the SADPS model in a polluted environment. Subsequently, we present a semi-discrete Galerkin finite element method in an age variable, and construct a full-discrete numerical approximation using modified truncated EM scheme on time to preserve positivity. Under certain suitable regularity conditions, estimates of convergence error of the full-discrete and a semi-discrete scheme are presented. Finally, a numerical example is given to illustrate the theoretical results.

Well-Posedness of an Integro-Differential Model for Active Brownian Particles

We propose a general strategy for solving nonlinear integro-differential evolution problems with periodic boundary conditions, where no direct maximum/minimum principle is available. This is motivated by the study of recent macroscopic models for active Brownian particles with repulsive interactions, consisting of advection-diffusion processes in the space of particle position and orientation. We focus on one of such models, namely a semilinear parabolic equation with a nonlinear active drift term, whereby the velocity depends on the particle orientation and angle-independent overall particle density (leading to a nonlocal term by integrating out the angular variable). The main idea of the existence analysis is to exploit a-priori estimates from (approximate) entropy dissipation. The global existence and uniqueness of weak solutions is shown using a two-step Galerkin approximation with appropriate cutoff in order to obtain nonnegativity, an upper bound on the overall density and preserve a-priori estimates. Our anyalysis naturally includes the case of finite systems, corresponding to the case of a finite number of directions. The Duhamel principle is then used to obtain additional regularity of the solution, namely continuity in time-space. Motivated by the class of initial data relevant for the application, which includes perfectly aligned particles (same orientation), we extend the well-posedness result to very weak solutions allowing distributional initial data with low regularity.

Optimal filtering equations in state space model of the two factors mean reverting Ornstein-Uhlenbech process

This paper investigates a nonlinear state space model for energy markets ruled by a two factor mean reverting Ornstein Uhlenbeck process with a stochastic nonlinear autoregressive drift term. A recursive approach using Taylor series based approximations for filtering, prediction, and smoothing problem of the hidden mean reverting factor from the noisy observations is proposed. The closed-form solutions are obtained for estimating the hidden mean reverting factor. Also, the existence and uniqueness of the solution for this nonlinear system are proved. Finally, the performance of the developed methods is checked in a simulation study.

Three-factor mean reverting Ornstein-Uhlenbeck process with stochastic drift term innovations: Nonlinear autoregressive approach with dependent error

This paper introduces a novel approach, withen the context of energy market, by employing a three-factor mean reverting Ornstein-Uhlenbeck process with a stochastic nonlinear autoregressive drift term having a dependent error. Initially the unique solvability for the given nonlinear system is investigated. Then, to estimate the nonlinear regression function, a semiparametric method, based on the conditional least square estimator for the parametric approach, and the nonparametric kernel method for autoregressive modification estimation have been presented . A maximum likelihood estimator has been used for parameter estimation of the Ornstein-Uhlenbeck process. Finally, some numerical simulations and real data studies have been provided to support the main conclusions of the study.

Limiting behavior of invariant measures of highly nonlinear stochastic retarded lattice systems

<p style='text-indent:20px;'>This paper deals with the limiting behavior of invariant measures of the highly nonlinear stochastic retarded lattice systems. Although invariant measures of stochastic retarded lattice system has been studied recently, there is so far no result of invariant measure of stochastic retarded lattice systems with highly nonlinear drift or diffusion terms. We first show the existence of invariant measures of the systems. We then prove that any limit point of a tight sequence of invariant measures of the stochastic retarded lattice systems must be an invariant measure of the corresponding limiting system as the intensity of noise converges or the time-delay approaches zero.</p>

An iterative model of the generalized Cauchy process for predicting the remaining useful life of lithium-ion batteries

The degradation process of lithium-ion batteries has memory, i.e. it has long-range dependence (LRD). In this paper, an iterative model of the generalized Cauchy (GC) process with LRD characteristics is proposed for the remaining useful life (RUL) prediction of lithium-ion batteries. The GC process uses two independent parameters, fractal dimension and Hurst exponent, to measure the LRD of the degradation process. The diffusion term of the GC iterative model is replaced by the increment of the GC time sequences, constructed via the autocorrelation function (ACF) to describe uncertainty and the LRD characteristics of the lithium-ion batteries capacity degradation. Linear and nonlinear drift terms are used to explain the degradation trend of the lithium-ion batteries capacity. A comparison is made with fractional Brownian motion (FBM) and long-short-term memory (LSTM) network models to show how the GC iterative model has the best performance in RUL prediction of lithium-ion batteries.

Attractors for multi-valued lattice dynamical systems with nonlinear diffusion terms

This paper is devoted to the long-term behavior of nonautonomous random lattice dynamical systems with nonlinear diffusion terms. The nonlinear drift and diffusion terms are not expected to be Lipschitz continuous but satisfy the continuity and growth conditions. We first prove the existence of solutions, and establish the existence of a multi-valued nonautonomous cocycle. We then show the existence and uniqueness of pullback attractors parameterized by sample parameters. Finally, we establish the measurability of this pullback attractor by the method based on the weak upper semicontinuity of the solutions.

Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise

We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L2(G), where G is an open bounded domain in mathbb {R}^{d} with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function that is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.

Neuro-adaptive fast integral terminal sliding mode control design with variable gain robust exact differentiator for under-actuated quadcopter UAV

In this paper, a robust global fast terminal attractor based full flight trajectory tracking control law has been developed for the available regular form which is operated under matched uncertainties. Based on the hierarchical control principle, the aforesaid model is first subdivided into two subsystems, i.e., a fully-actuated subsystem and an under-actuated subsystem. In other words, the under-actuated subsystem is further transformed into a regular form whereby the under-actuated characteristics are decoupled in terms of control inputs. In the proposed design, the nonlinear drift terms, which certainly varies in full flight, are estimated via functional link neural networks to improve the performance of the controller in full flight. Besides, a variable gain robust exact differentiator (VG-RED) is designed to provide us with estimated flight velocities. It has consequently reduced the noise in system’s velocities and has mapped this controller as a practical one. The finite-time sliding mode enforcement and the states’ convergence are shown, for all flight loops, i.e., forward flight and backward flight, via the Lyapunov approach. All these claims are verified via numerical simulations and experimental implementation of the quadcopter system in a Matlab environment. For a more impressive presentation, the developed simulation results are compared with standard literature.

Higher order pathwise approximation for the stochastic Burgers' equation with additive noise

This article aims to investigate the pathwise convergence of the higher order scheme, introduced by Jentzen (2011) [9], for the stochastic Burgers' equation (SBE) driven by space-time white noise. In particular, first and second order derivatives of the non-linear drift term of the SBE are assumed to be defined and bounded in Sobolev spaces using the definition of distribution derivative i.e. Lemma 4.7 in Blömker and Jentzen (2013) [2] is extended. Based on this extension, temporal convergence analysis of the higher order scheme is carried out for the SBE with additive noise. As a result, minimum temporal convergence order is improved from θ (Theorem 4.1 in Blömker et al. (2013) [3]) to 2θ, where every θ∈(0,12)). Numerical experiments are performed to validate the theoretical findings.

Random dynamics of non-autonomous fractional stochastic p-Laplacian equations on $${\mathbb {R}}^N$$

This article is concerned with the random dynamics of a wide class of non-autonomous, non-local, fractional, stochastic p-Laplacian equations driven by multiplicative white noise on the entire space $${\mathbb {R}}^N$$ . We first establish the well-posedness of the equations when the time-dependent non-linear drift terms have polynomial growth of arbitrary orders $$p,q\ge 2$$ . We then prove that the equation has a unique bi-spatial pullback random attractor that is measurable, compact in $$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$$ and attracts all random subsets of $$L^2({\mathbb {R}}^N)$$ under the topology of $$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$$ . In addition, we establish the upper semi-continuity of these attractors in $$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$$ when the density of noise shrinks to zero. The idea of uniform tail estimates and the method of asymptotic a priori estimates are applied to prove the pullback asymptotic compactness of the solutions in $$L^2({\mathbb {R}}^N)\cap L^p({\mathbb {R}}^N)\cap L^q({\mathbb {R}}^N)$$ to overcome the non-compactness of Sobolev embeddings on $${\mathbb {R}}^N$$ as well as the almost sure nondifferentiability of the sample paths of the Wiener process.

Random dynamics of [formula omitted]-Laplacian lattice systems driven by infinite-dimensional nonlinear noise

This article is concerned with the global existence and random dynamics of the non-autonomous p-Laplacian lattice system defined on the entire integer set driven by infinite-dimensional nonlinear noise. The existence and uniqueness of mean square solutions to the equations are proved when the nonlinear drift and diffusion terms are locally Lipschitz continuous. It is shown that the mean random dynamical system generated by the solution operators has a unique tempered weak pullback random attractor in a Bochner space. The existence of invariant measures for the stochastic equations in the space of square summable sequences is also established. The idea of uniform tail-estimates of solutions is employed to show the tightness of a family of distribution laws of the solutions.

Invariant measures of stochastic delay lattice systems

<p style='text-indent:20px;'>This paper is concerned with the existence and uniqueness of invariant measures for infinite-dimensional stochastic delay lattice systems defined on the entire integer set. For Lipschitz drift and diffusion terms, we prove the existence of invariant measures of the systems by showing the tightness of a family of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem. We also show the uniqueness of invariant measures when the Lipschitz coefficients of the nonlinear drift and diffusion terms are sufficiently small.