Stochastic Differential Delay Systems Research Articles

Boundedness and stability of nonlinear hybrid neutral stochastic delay differential equation with Lévy jumps under different structures

This paper investigates the boundedness and stability of a class of nonlinear hybrid neutral stochastic differential delay systems with Lévy jumps and different structures. The coefficients in this system satisfy the local Lipschitz condition and a suitable Khasminskii-type condition, and the state space of the system is separated into two subsets, the existence uniqueness, asymptotic boundedness, and exponential stability of the system are obtained by designing a new Lyapunov function and applying the M-matrix technique as well as dealing with the non-differentiable delay function. Different with the existing work, we not only consider the neutral term, but also the case of the delay function being bounded and non-differentiable. At last, numerical examples are performed to manifest the obtained results.

Hyers–Ulam Stability of Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

In this paper, we explore the stability of a new class of Caputo-type fractional stochastic delay differential systems with Poisson jumps. We prove the Hyers–Ulam stability of the solution by utilizing a version of fixed point theorem, fractional calculus, Cauchy–Schwartz inequality, Jensen inequality, and some stochastic analysis techniques. Finally, an example is provided to illustrate the effectiveness of the results.

The Existence and Averaging Principle for Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

In this paper, we obtain the existence and uniqueness theorem for solutions of Caputo-type fractional stochastic delay differential systems(FSDDSs) with Poisson jumps by utilizing the delayed perturbation of the Mittag–Leffler function. Moreover, by using the Burkholder–Davis–Gundy inequality, Doob’s martingale inequality, and Hölder inequality, we prove that the solution of the averaged FSDDSs converges to that of the standard FSDDSs in the sense of Lp. Some known results in the literature are extended.

A Novel Approach to Discriminate Mean-Square Exponential Stability of Hybrid Stochastic Delay Differential Systems

A Novel Approach to Discriminate Mean-Square Exponential Stability of Hybrid Stochastic Delay Differential Systems

Results on approximate controllability for fractional stochastic delay differential systems of order r ∈ (1,2)

In this paper, we deal with the approximate controllability of fractional stochastic delay differential inclusions of order [Formula: see text]. By using fractional calculus, stochastic analysis, the theory of cosine family and Dhage fixed point techniques, a new set of necessary and sufficient conditions are formulated which guarantees the approximate controllability of the nonlinear fractional stochastic system. In particular, the results are established with the assumption that the associated linear part of the system is approximately controllable. Further, the result is extended to obtain the conditions for the solvability of controllability results for fractional inclusions with nonlocal conditions. Finally, an example is presented to illustrate the theory of the obtained result.

Stochastic dynamics of stochastic delay Duffing oscillator and sensitivity analysis to different types of signals

In this paper, a stochastic delay differential system driven by the square wave signal is presented. The sensitivity of the system to different types of signals is evaluated by the dynamic behaviors of the system. The stochastic dynamic behaviors of the system is analyzed theoretically, and the theoretical chaotic threshold under different external excitation signals is deduced. The numerical simulation results show that the stochastic delay differential system is more sensitive to the square wave signals with the same frequency than the harmonic signal and the square wave signal with the different frequency.

Impact of Delay on Stochastic Predator–Prey Models

Ordinary differential equations (ODE) have long been an important tool for modelling and understanding the dynamics of many real systems. However, mathematical modelling in several areas of the life sciences requires the use of time-delayed differential models (DDEs). The time delays in these models refer to the time required for the manifestation of certain hidden processes, such as the time between the onset of cell infection and the production of new viruses (incubation periods), the infection period, or the immune period. Since real biological systems are always subject to perturbations that are not fully understood or cannot be explicitly modeled, stochastic delay differential systems (SDDEs) provide a more realistic approximation to these models. In this work, we study the predator–prey system considering three time-delay models: one deterministic and two types of stochastic models. Our numerical results allow us to distinguish between different asymptotic behaviours depending on whether the system is deterministic or stochastic, and in particular, when considering stochasticity, we see that both the nature of the stochastic systems and the magnitude of the delay play a crucial role in determining the dynamics of the system.

Mean Square Exponential Stability of Stochastic Delay Differential Systems with Logic Impulses

This paper focuses on the mean square exponential stability of stochastic delay differential systems with logic impulses. Firstly, a class of nonlinear stochastic delay differential systems with logic impulses is constructed. Then, the logic impulses are transformed into an equivalent algebraic expression by using the semi-tensor product method. Thirdly, the mean square exponential stability criteria of nonlinear stochastic delay differential systems with logic impulses are given. Finally, two kinds of stochastic delay differential systems with logic impulses and uncertain parameters are discussed, and the coefficient conditions guaranteeing the mean square exponential stability of these systems are obtained.

The existence and averaging principle for Caputo fractional stochastic delay differential systems

In this paper, we first establish the existence and uniqueness theorem for solutions of Caputo type fractional stochastic delay differential systems by using delayed perturbation of Mittag-Leffler function. Secondly, we obtain an averaging principle for the solution of the considered system under some suitable assumptions. Finally, two simulation examples are given to verify the theoretical results.

Approximate controllability of stochastic delay differential systems driven by Poisson jumps with instantaneous and noninstantaneous impulses

AbstractControl systems in which instantaneous and noninstantaneous impulses occur simultaneously are difficult to handle. In this article, we investigate the solvability and approximate controllability for a new category of stochastic differential equations steered by Poisson jumps with instantaneous and noninstantaneous impulses. Utilizing the theory of fundamental solution, stochastic analysis, the measure of noncompactness, and the fixed‐point approach, we establish the presence of a mild solution for the proposed system. We have also constructed a new set of sufficient constraints that assures approximate controllability of the considered system. Next, we discuss the existence of a solution and approximate controllability for an impulsive deterministic control system in which the nonlinear term contains spatial derivatives. Lastly, two examples are presented to encapsulate the abstract results.

(2010-6217) Solvability of fuzzy fractional stochastic Pantograph differential system

In this paper, a new type of equation namely fuzzy fractional stochastic Pantograph delay differential system (FSPDDS) is proposed. In our previous work, a first extension of fuzzy stochastic differential system into fuzzy fractional stochasticdifferential system by using Granular differentiability has been established. Here we study the existence and uniqueness results for the fuzzy FSPDDS which are obtained by using generalized Granular differentiability and contraction principlewith weaker conditions. This kind of equation is used in many real world problems. Finally, we provide two numerical examples for the effectiveness of the theoretical results.

Mean-square stability of the zero equilibrium of the nonlinear delay differential equation: Nicholson’s blowflies application

We are concerned about the stochastic nonlinear delay differential equation. The stochasticity arises from the white Gaussian noise, which is the time derivative of the standard Brownian motion. The main objective of this paper is to introduce a new technique using the Lyapunov functional for the study of stability of the zero solution of the stochastic delay differential system. Constructing a new appropriate deterministic system in the neighborhood of the origin is an effective way to investigate the necessary and sufficient conditions of stability in the sense of the mean square. Nicholson’s blowflies equation is one of the major problems in ecology; necessary conditions for the possible extinction of the Nicholson’s blowflies population are investigated. We support our theoretical results by providing areas of stability and some numerical simulations of the solution of the system using the Euler–Maruyama scheme, which is mean square stable Maruyama (Rendiconti del Circolo Matematico di Palermo 4(1):48, 1955), Cao et al. (Appl Math Comput 159(1):127–135, 2004).

Retraction Note: Existence of weak solutions of stochastic delay differential systems with Schr\xf6dinger\u2013Brownian motions

Retraction Note: Existence of weak solutions of stochastic delay differential systems with Schr\xf6dinger\u2013Brownian motions

Exponential stability of non-linear neutral stochastic delay differential system with generalized delay-dependent impulsive points

This work aims to analyze the exponential stability of a non-linear impulsive neutral stochastic delay differential system. In this study, impulse perturbation is considered a delay-dependent state variable. The solution of the delay-dependent impulsive neutral stochastic delay differential system is associated with the solution of the system without impulses. First, we developed a relation connecting the solution of the neutral stochastic delay differential system without impulses and the solution of the corresponding system with impulses. Then, the conditions of the exponential stability of the proposed impulsive system are derived by determining the stability analysis of the respective system without impulse. The numerical approach for the neutral stochastic delay system without impulses is generated using the Euler-Maruyama method and adopted for the corresponding impulsive system. Finally, the achieved theoretical results are illustrated for applying the Malthusian single species neutral stochastic delay population model with immigration impulses.

Stochastic delay differential neoclassical growth system

In the stochastic delay differential neoclassical growth system, the white noises stochastically perturb some parameters. we show that the stochastic system has the positive solutions, which will not explode to infinity in a finite time and, in fact, will be ultimately bounded. A numerical test shows the effectiveness of the theoretical results.

Stability of Stochastic Delay Differential Systems With Variable Impulses Due to Logic Choice

This paper is concerned with stability problems of stochastic delay differential systems with variable impulses due to logic choice. Firstly, a class of variable impulses due to logic choice is introduced in this paper which is more general than the logic impulses established by (Suo and Sun, 2015) and (Zhang et al., 2018). Then, by establishing a connection between the stochastic delay differential system with logic impulses and a corresponding stochastic delay differential system without logic impulses, some sufficient conditions for stability of the systems are obtained. Finally, the application in a class of linear stochastic delay differential systems with logic impulses is discussed, and several stability criteria together with two numerical examples are given.

Almost sure exponential stability of semi-Euler numerical scheme for nonlinear stochastic functional differential equation

It is generally known that explicit solution can rarely be obtained for stochastic differential equations (SDEs), not to mention the nonlinear stochastic functional differential equation (SFDEs). Therefore, this paper proposes a numerical scheme called semi-Euler numerical scheme for general SFDEs, which adapts better to the connotation of the Itô differential equations. Under a generalized polynomial growth condition, the scheme admits high nonlinearity of the system, the almost sure exponential stability of the continuous model and the numerical scheme is investigated by contrast. It is confirmed that the numerical scheme preserves the stability property of the continuous model with no restriction to the step size. Compared to the literature, the polynomial growth condition in [L. Liu, F. Deng, and T. Hou, Almost sure exponential stability of implicit numerical solution for stochastic functional differential equation with extended polynomial growth condition, Applied Mathematics and Computation. 330 (2018), pp. 201–212.], [W. Fei, L. Hu, X. Mao, and M. Shen, Delay dependent stability of highly nonlinear hybrid stochastic systems, Automatica, 82 (2017), pp. 165–170.], [M. Shen, W. Fei, X. Mao, and Y. Liang, Stability of highly nonlinear neutral stochastic differential delay equations, Systems & Control Letters, 115 (2018), pp. 1–8.], etc. is generalized to that described with Lyapunov function. Besides, to provide an approach for the solvability of the implicit scheme, the generalized monotonicity of the vectorial functions is introduced. An application is given for the scheme and the stability conclusion, by virtue of the generalized monotonicity condition, the scheme is solvable also without restriction to the step size. At the end of the paper, a high order example is proposed to illustrate the theory of this paper, and a further research direction from this work is pointed out.

The TVICMs method for weak signal detection based on a nonlinear stochastic delay differential system

A nonlinear stochastic delay differential system with symmetry SD oscillator is investigated to detect weak signal. Firstly, the stochastic Melnikov function and the chaos threshold are obtained by using the stochastic mean square criterion. Secondly, the stochastic center manifold and the stochastic averaging method are used to obtain the one-dimensional Itoˆ equation and the corresponding FPK equation. The stationary probability density of the FPK equation is studied and the corresponding nonlinear stochastic dynamical behaviors are analyzed. Finally, the highlight of this paper is to detect the weak signal with ‘transient vacancy in chaotic motions’ (TVICMs) method successfully and extract the features information such as frequency and amplitude on the premise of unknown features information. Astonishingly, numerical simulation shows that the time delay can enhance the detection capability of the system and the effectiveness of the new method.

Exponential stability of non-linear stochastic delay differential system with generalized delay-dependent impulsive points

Abstract This paper is concerned with a non-linear stochastic delay differential system with delay-dependent impulsive perturbations. In this work, the size of the jump is defined as a general non-linear delay-dependent state variable and the solution of the impulsive stochastic delay differential system corresponding to the system without impulsive perturbations is given. This work is based on the relation between the solution of the equivalent model of stochastic delay differential system without impulses corresponding to the solution of the system with impulses. Then the conditions of the exponential stability of the proposed impulsive system are obtained by deriving stability criteria of the corresponding system without impulses. The numerical approximation for the stochastic delay system without impulses is developed using the Runge-Kutta-Maruyama method and it is suitably applied for the corresponding impulsive system. Finally, the obtained theoretical results are illustrated graphically for a stochastic delay system with impulses.

Some Improved Razumikhin Stability Criteria for Impulsive Stochastic Delay Differential Systems

This paper is devoted to study the Razumikhin stability theorem for a class of impulsive stochastic delay differential systems. By developing a new lemma, stochastic analysis technique, and Razumikhin approach, several novel criteria of the $p$ th moment exponential stability are derived for the related systems. The key feature of the criteria is that time-derivatives of the Razumikhin functions are allowed to be indefinite, which loosens the constraints of the existing results greatly. Finally, two examples are given to illustrate the usefulness and significance of the theoretical results.