scholarly journals Stability of Stochastic Differential Delay Systems with Delayed Impulses

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yanlei Wu
Keyword(s):  
Delay System ◽  
Delay Systems ◽  
Stochastic Delay ◽  
Differential Delay ◽  
Almost Sure Exponential Stability ◽  
Continuous Dynamic System ◽  
Continuous Dynamic ◽  
The Stability ◽  
Delay Differential ◽  
Delayed Impulses

We investigate the stability of stochastic delay differential systems with delayed impulses by Razumikhin methods. Some criteria on thepth moment and almost sure exponential stability are obtained. It is shown that an unstable stochastic delay system can be successfully stabilized by delayed impulses. Moreover, it is also shown that if a continuous dynamic system is stable, then, under some conditions, the delayed impulses do not destroy the stability of the systems. The effectiveness of the proposed results is illustrated by two examples.

scholarly journals Exponential Stability of Impulsive Stochastic Delay Differential Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Xiaotai Wu ◽  
Litan Yan ◽  
Wenbing Zhang ◽  
Liang Chen
Keyword(s):  
Exponential Stability ◽  
The Other ◽  
Stability Criteria ◽  
Differential Systems ◽  
Stochastic Delay ◽  
Almost Sure Exponential Stability ◽  
Delay Differential Systems ◽  
The Stability ◽  
Delay Differential ◽  
Average Impulsive Interval

This paper investigates the stability of stochastic delay differential systems with two kinds of impulses, that is, destabilizing impulses and stabilizing impulses. Both thepth moment and almost sure exponential stability criteria are established by using the average impulsive interval. When the impulses are regarded as disturbances, a lower bound of average impulsive interval is obtained; it means that the impulses should not happen too frequently. On the other hand, when the impulses are used to stabilize the system, an upper bound of average impulsive interval is derived; namely, enough impulses are needed to stabilize the system. The effectiveness of the proposed results is illustrated by two examples.

scholarly journals Stability and Stabilization of Impulsive Stochastic Delay Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Kaining Wu ◽  
Xiaohua Ding
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Almost Sure Exponential Stability ◽  
Stability And Stabilization ◽  
The Stability ◽  
Delay Differential

We consider the stability and stabilization of impulsive stochastic delay differential equations (ISDDEs). Using the Lyapunov-Razumikhin method, we obtain the sufficient conditions to guarantee thepth moment exponential stability of ISDDEs. Then the almost sure exponential stability is considered and the sufficient conditions of the almost sure exponential stability are obtained. Moreover, the stabilization problem of ISDDEs is studied and the criterion on impulsive stabilization of ISDDEs is established. At last, examples are presented to illustrate the correctness of our results.

scholarly journals Exponential Stability of Impulsive Stochastic Functional Differential Systems with Delayed Impulses

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Fengqi Yao ◽  
Feiqi Deng ◽  
Pei Cheng
Keyword(s):  
Exponential Stability ◽  
Delay Systems ◽  
Differential Systems ◽  
Stochastic Delay ◽  
Functional Differential Systems ◽  
Two Measures ◽  
Functional Differential ◽  
The Stability ◽  
Delayed Impulses ◽  
Stochastic Functional

A class of generalized impulsive stochastic functional differential systems with delayed impulses is considered. By employing piecewise continuous Lyapunov functions and the Razumikhin techniques, several criteria on the exponential stability and uniform stability in terms of two measures for the mentioned systems are obtained, which show that unstable stochastic functional differential systems may be stabilized by appropriate delayed impulses. Based on the stability results, delayed impulsive controllers which mean square exponentially stabilize linear stochastic delay systems are proposed. Finally, numerical examples are given to verify the effectiveness and advantages of our results.

scholarly journals On the Number of Periodic Orbits to Odd Order Differential Delay Systems

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1731
Author(s):  
Weigao Ge ◽  
Lin Li
Keyword(s):  
Variational Method ◽  
Periodic Orbits ◽  
Index Theory ◽  
Accurate Method ◽  
Delay System ◽  
Delay Systems ◽  
Differential Delay ◽  
Odd Order

In this paper, we study the periodic orbits of a type of odd order differential delay system with 2k−1 lags via the S1 index theory and the variational method. This type of system has not been studied by others. Our results provide a new and more accurate method for counting the number of periodic orbits.

scholarly journals Dynamic Behaviors Analysis of Asymmetric Stochastic Delay Differential Equations with Noise and Application to Weak Signal Detection

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1428
Author(s):  
Qiubao Wang ◽  
Xing Zhang ◽  
Yuejuan Yang
Keyword(s):  
Duffing Oscillator ◽  
Delay System ◽  
Weak Signal ◽  
Stochastic Delay Differential Equations ◽  
Weak Signals ◽  
Dynamic Behaviors ◽  
Asymmetric System ◽  
Stochastic Delay ◽  
Periodic State ◽  
Delay Differential

This paper presents the dynamic behaviors of a second-order asymmetric stochastic delay system with a Duffing oscillator as well as through the detection of weak signals, which are analyzed theoretically and numerically. The dynamic behaviors of the asymmetric system are analyzed based on the stochastic center manifold, together with Hopf bifurcation. Numerical analysis revealed that the time delay could enhance the noise immunity of the asymmetric system so as to enhance the asymmetric system’s ability to detect weak signals. The frequency of the weak signal under noise excitation was detected through the ‘act-and-wait’ method. The small amplitude was detected through the transition from the chaotic to the periodic state. Theoretical analysis and numerical simulation indicate that the application of the asymmetric Duffing oscillator with delay to detect weak signal is feasible.

scholarly journals The Semimartingale Approach to Almost Sure Stability Analysis of a Two-Stage Numerical Method for Stochastic Delay Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qian Guo ◽  
Xueyin Tao
Keyword(s):  
Differential Equation ◽  
Numerical Experiments ◽  
Time Varying ◽  
Almost Sure Stability ◽  
Stochastic Delay ◽  
Almost Sure Exponential Stability ◽  
Time Varying Delay ◽  
Backward Euler ◽  
Delay Differential ◽  
Varying Delay

Almost sure exponential stability of the split-step backward Euler (SSBE) method applied to an Itô-type stochastic differential equation with time-varying delay is discussed by the techniques based on Doob-Mayer decomposition and semimartingale convergence theorem. Numerical experiments confirm the theoretical analysis.

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