An Averaging Principle for Stochastic Fractional Differential Equations Driven by fBm Involving Impulses

Liu, Jiankang; Wei, Wei; Xu, Wei

doi:10.3390/fractalfract6050256

Cited by 23 publications

(9 citation statements)

References 35 publications

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“…In recent years, averaging principles for fractional stochastic differential equations (FSDEs) have attracted great attention since they can provide approximate analytical methods to reduce such systems [2]. Xu et al [3,4] developed averaging results for various types of FSDEs early.…”

Section: Introductionmentioning

confidence: 99%

A Note on Averaging Principles for Fractional Stochastic Differential Equations

Liu,

Zhang,

Wang

et al. 2024

Fractal Fract

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Over the past few years, many scholars began to study averaging principles for fractional stochastic differential equations since they can provide an approximate analytical method to reduce such systems. However, in the most previous studies, there is a misunderstanding of the standard form of fractional stochastic differential equations, which consequently causes the wrong estimation of the convergence rate. In this note, we take fractional stochastic differential equations with Lévy noise as an example to clarify these two issues. The corrections herein have no effect on the main proofs except the two points mentioned above. The innovation of this paper lies in three aspects: (i) the standard form of the fractional stochastic differential equations is derived under natural time scale; (ii) it is first proved that the convergence interval and rate are related to the fractional order; and (iii) the presented results contain and improve some well known research achievements.

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Section: Introductionmentioning

confidence: 99%

A Note on Averaging Principles for Fractional Stochastic Differential Equations

Liu,

Zhang,

Wang

et al. 2024

Fractal Fract

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“…We refer the interested reader to [13], which provides a concise overview and comparison of these frameworks. More recent studies of discontinuous dynamics include results for delay [14][15][16], fractional order [17][18][19], stochastic equations [20][21][22], fuzzy differential systems [23,24], and applications of impulses for stabilization and control purposes [25][26][27].…”

Section: Introductionmentioning

confidence: 99%

ω-Limit Sets of Impulsive Semigroups for Hyperbolic Equations

Feketa,

Fedorenko,

Bezushchak

et al. 2023

Axioms

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In this paper, we investigate the qualitative behavior of an evolutionary problem consisting of a hyperbolic dissipative equation whose trajectories undergo instantaneous impulsive discontinuities at the moments when the energy functional reaches a certain threshold value. The novelty of the current study is that we consider the case in which the entire infinite-dimensional phase vector undergoes an impulsive disturbance. This substantially broadens the existing results, which admit discontinuities for only a finite subset of phase coordinates. Under fairly general conditions on the system parameters, we prove that such a problem generates an impulsive dynamical system in the natural phase space, and its trajectories have nonempty compact ω-limit sets.

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“…In addition, random factors are ubiquitous and non-negligible [24][25][26][27][28][29]. Actually, there are lots of random factors in rolling mill systems and rolling process [30,31].…”

Section: Introductionmentioning

confidence: 99%

A Dynamic Behavior Analysis of a Rolling Mill’s Main Drive System with Fractional Derivative and Stochastic Disturbance

Wang,

2023

Symmetry

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Taking the random factors into account, a fractional main drive system of a rolling mill with Gaussian white noise is developed. First, the potential deterministic bifurcation is investigated by a linearized stability analysis. The results indicate that the fractional order changes the system from a stable point to a limit cycle with symmetric phase trajectories. Then, the stochastic response is obtained with the aid of the equivalent transformation of the fractional derivative and stochastic averaging methods. It is found that the joint stationary probability density function appears to have symmetric distribution. Finally, the influence of the fractional order and noise intensity on system dynamics behavior is discussed. The study is beneficial to understand the intrinsic mechanisms of vibration abatement.

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An Averaging Principle for Stochastic Fractional Differential Equations Driven by fBm Involving Impulses

Cited by 23 publications

References 35 publications

A Note on Averaging Principles for Fractional Stochastic Differential Equations

A Note on Averaging Principles for Fractional Stochastic Differential Equations

ω-Limit Sets of Impulsive Semigroups for Hyperbolic Equations

A Dynamic Behavior Analysis of a Rolling Mill’s Main Drive System with Fractional Derivative and Stochastic Disturbance

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