scholarly journals A Class of Stochastic Nonlinear Delay System with Jumps

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Ling Bai ◽  
Kai Zhang ◽  
Wenju Zhao
Keyword(s):  
Sufficient Conditions ◽  
Delay System ◽  
Lévy Noise ◽  
Local Martingale ◽  
Martingale Inequality ◽  
Strong Law ◽  
Delay Differential System ◽  
Delay Differential ◽  
Levy Noise ◽  
Nonlinear Delay

We consider stochastic suppression and stabilization for nonlinear delay differential system. The system is assumed to satisfy local Lipschitz condition and one-side polynomial growth condition. Since the system may explode in a finite time, we stochastically perturb this system by introducing independent Brownian noises and Lévy noise feedbacks. The contributions of this paper are as follows. (a) We show that Brownian noises or Lévy noise may suppress potential explosion of the solution for some appropriate parameters. (b) Using the exponential martingale inequality with jumps, we discuss the fact that the sample Lyapunov exponent is nonpositive. (c) Considering linear Lévy processes, by the strong law of large number for local martingale, sufficient conditions for a.s. exponentially stability are investigated in Theorem 13.

scholarly journals Positivity of Fundamental Matrix and Exponential Stability of Delay Differential System

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Alexander Domoshnitsky ◽  
Roman Shklyar ◽  
Mikhail Gitman ◽  
Valery Stolbov
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Sufficient Conditions ◽  
Delay System ◽  
Necessary Condition ◽  
Cauchy Matrix ◽  
Linear Ordinary Differential Equations ◽  
Delay Differential System ◽  
Delay Differential ◽  
Necessary And Sufficient

The classical Wazewski theorem established that nonpositivity of all nondiagonal elementspij  (i≠j,  i,j=1,…,n)is necessary and sufficient for nonnegativity of the fundamental (Cauchy) matrix and consequently for applicability of the Chaplygin approach of approximate integration for system of linear ordinary differential equationsxi′t+∑j=1n‍pijtxjt=fit,   i=1,…,n.Results on nonnegativity of the Cauchy matrix for system of delay differential equationsxi′t+∑j=1n‍pijtxjhijt=fit,   i=1,…,n,which were based on nonpositivity of all diagonal elements, were presented in the previous works. Then examples, which demonstrated that nonpositivity of nondiagonal coefficientspijis not necessary for systems of delay equations, were found. In this paper first sufficient results about nonnegativity of the Cauchy matrix of the delay system without this assumption are proven. A necessary condition of nonnegativity of the Cauchy matrix is proposed. On the basis of these results on nonnegativity of the Cauchy matrix, necessary and sufficient conditions of the exponential stability of the delay system are obtained.

Stability and boundedness of solutions of a certain n-dimensional nonlinear delay differential system of third-order

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Ayman M. Mahmoud ◽  
Cemil Tunç
Keyword(s):  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Zero Solution ◽  
Third Order ◽  
All Solutions ◽  
Uniform Ultimate Boundedness ◽  
Delay Differential System ◽  
Nonlinear Vector ◽  
Delay Differential ◽  
Nonlinear Delay

AbstractIn this paper, by defining Lyapunov functionals, we investigate proper sufficient conditions for the uniform stability of the zero solution, and also for the uniform boundedness and uniform ultimate boundedness of all solutions of a certain third-order nonlinear vector delay differential equation of the type

Adaptive state estimation of Markov switched neural networks driven by Lévy noise

2019 ◽  
Vol 42 (2) ◽  
pp. 330-336
Author(s):  
Dongbing Tong ◽  
Qiaoyu Chen ◽  
Wuneng Zhou ◽  
Yuhua Xu
Keyword(s):  
Neural Networks ◽  
State Estimation ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Lévy Noise ◽  
Linear Matrix ◽  
Delayed Neural Networks ◽  
Switched Neural Networks ◽  
Inequality Technique ◽  
Levy Noise

This paper proposes the [Formula: see text]-matrix method to achieve state estimation in Markov switched neural networks with Lévy noise, and the method is very distinct from the linear matrix inequality technique. Meanwhile, in light of the Lyapunov stability theory, some sufficient conditions of the exponential stability are derived for delayed neural networks, and the adaptive update law is obtained. An example verifies the condition of state estimation and confirms the effectiveness of results.

A stochastic vaccinated epidemic model incorporating Lévy processes with a general awareness-induced incidence

2021 ◽  
pp. 2150044
Author(s):  
Khadija Akdim ◽  
Adil Ez-Zetouni ◽  
Mehdi Zahid
Keyword(s):  
Numerical Simulations ◽  
Positive Solution ◽  
Dynamic Behavior ◽  
Epidemic Model ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Lévy Noise ◽  
Global Positive Solution ◽  
Levy Noise ◽  
Disease Free

In this paper, we investigate a stochastic vaccinated epidemic model with a general awareness-induced incidence perturbed by Lévy noise. First, we show that this model has a unique global positive solution. Therefore, we establish the dynamic behavior of the solution around both disease-free and endemic equilibrium points. Furthermore, when [Formula: see text], we give sufficient conditions for the existence of an ergodic stationary distribution to the model when the jump part in the Lévy noise is null. Finally, we present some examples to illustrate the analytical results by numerical simulations.

scholarly journals Boundedness of Stochastic Delay Differential Systems with Impulsive Control and Impulsive Disturbance

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Liming Wang ◽  
Baoqing Yang ◽  
Xiaohua Ding ◽  
Kai-Ning Wu
Keyword(s):  
Stochastic Analysis ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
Stochastic Delay ◽  
Analysis Techniques ◽  
Moment Boundedness ◽  
Delay Differential System ◽  
Delay Differential Systems ◽  
Delay Differential

This paper considers thep-moment boundedness of nonlinear impulsive stochastic delay differential systems (ISDDSs). Using the Lyapunov-Razumikhin method and stochastic analysis techniques, we obtain sufficient conditions which guarantee thep-moment boundedness of ISDDSs. Two cases are considered, one is that the stochastic delay differential system (SDDS) may not be bounded, and how an impulsive strategy should be taken to make the SDDS be bounded. The other is that the SDDS is bounded, and an impulsive disturbance appears in this SDDS, then what restrictions on the impulsive disturbance should be adopted to maintain the boundedness of the SDDS. Our results provide sufficient criteria for these two cases. At last, two examples are given to illustrate the correctness of our results.

Asymptotic behavior, attracting and quasi-invariant sets for impulsive neutral SPFDE driven by Lévy noise

2017 ◽  
Vol 18 (01) ◽  
pp. 1850010 ◽  
Author(s):  
Diem Dang Huan ◽  
Ravi P. Agarwal
Keyword(s):  
Exponential Stability ◽  
Mild Solution ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Invariant Sets ◽  
Lévy Noise ◽  
Almost Surely Exponential Stability ◽  
Functional Differential ◽  
Levy Noise ◽  
Moment Exponential Stability

By establishing two new impulsive-integral inequalities, the attracting and quasi-invariant sets of the mild solution for impulsive neutral stochastic partial functional differential equations driven by Lévy noise are obtained, respectively. Moreover, we shall derive some sufficient conditions to ensure stability of this mild solution in the sense of both moment exponential stability and almost surely exponential stability.

Analysis of a stochastic predator–prey model with prey subject to disease and Lévy noise

2019 ◽  
Vol 19 (05) ◽  
pp. 1950038
Author(s):  
Meihong Qiao ◽  
Shenglan Yuan
Keyword(s):  
Positive Solution ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Lévy Noise ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Stochastic Boundedness ◽  
Prey Model ◽  
Global Positive Solution ◽  
Levy Noise

We consider a non-autonomous predator–prey model, with prey subject to the disease and Lévy noise. We show the existence of global positive solution and stochastic boundedness. Then, we examine the asymptotic properties of the solution. Finally, we offer sufficient conditions for persistence and extinction.

scholarly journals Asymptotic Stability of Stochastic Differential Equations Driven by Lévy Noise

2009 ◽  
Vol 46 (4) ◽  
pp. 1116-1129 ◽  
Author(s):  
David Applebaum ◽  
Michailina Siakalli
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lévy Noise ◽  
Stochastic Integrals ◽  
Martingale Inequality ◽  
Moment Estimates ◽  
Exponential Martingale ◽  
Exponentially Stable ◽  
Levy Noise

Using key tools such as Itô's formula for general semimartingales, Kunita's moment estimates for Lévy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the stochastic differential equations (SDEs) driven by Lévy noise are stable in probability, almost surely and moment exponentially stable.

scholarly journals Global Attractivity of Positive Periodic Solutions of Delay Differential Equations with Feedback Control

2007 ◽  
Vol 2007 ◽  
pp. 1-11 ◽  
Author(s):  
Zhi-Long Jin
Keyword(s):  
Feedback Control ◽  
Periodic Solutions ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Upper And Lower Bounds ◽  
Positive Periodic Solutions ◽  
Liapunov Functionals ◽  
Delay Differential System ◽  
Several Delays ◽  
Delay Differential

By constructing suitable Liapunov functionals and estimating uniform upper and lower bounds of solutions, sufficient conditions are obtained for the global attractivity of positive periodic solutions of the delay differential system with feedback controldy/dt=y(t)F(t,y(t−τ1(t)),…,y(t−τn(t)),u(t−δ(t))),du/dt=−η(t)u(t)+a(t)y(t−σ(t)). When these results are applied to the periodic logistic equation with several delays and feedback control, some new results are obtained.

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