Exponential stability of singularly perturbed impulsive delay differential equations

In this paper, the exponential stability of singularly perturbed impulsive delay differential equations (SPIDDEs) is concerned. We first establish a delay differential inequality, which is useful to deal with the stability of SPIDDEs, and then by the obtained inequality, a sufficient condition is provided to ensure that any solution of SPIDDEs is exponentially stable for sufficiently small ε>0. A numerical example and the simulation result show the effectiveness of our theoretical result.     

Exponential stability for impulsive delay differential equations by Razumikhin method

In this paper, we study exponential stability for impulsive delay differential equation of the form

     

On stability and bifurcation of periodic solutions of delay differential equations

The purpose of this paper is to study a class of delay differential equations with two delays. first, we consider the existence of periodic solutions for some delay differential equations. Second, we investigate the local stability of the zero solution of the equation by analyzing the correlocal stability of the zero solution of the equation by analyzing the corresponding characteristic equation of the linearized equation. The exponential stability of a perturbed delay differential system with a bounded lag is studied. Finally, by choosing one of the delays as a bifurcation parameter, we show that the equation exhibits Hopf and saddle-node bifurcations.     

G-Brown运动驱动的中立型随机时滞微分方程的指数稳定性

研究了一类G-Brown运动驱动的中立型随机时滞微分方程的指数稳定性.在G-框架意义下,运用合适的Lyapunov-Krasovskii泛函,中立型时滞微分方程理论以及随机分析技巧,证明了所研究方程平凡解的p-阶矩指数稳定性,得到了所研究方程平凡解是p-阶矩指数稳定的充分条件.最后通过例子说明所得的结果.     

Exponential stability of Euler-Maruyama solutions for impulsive stochastic differential equations with delay

This paper establishes a method to study the exponential stability of Euler-Maruyama (EM) method for impulsive stochastic differential equations with delay. By using the properties of M-matrix and stochastic analysis technique, some conditions under which the EM solution is exponentially mean-square stable are obtained. Some examples are provided to illustrate the results.     

Exponential stability of equilibria of differential equations with time-dependent delay and non-Lipschitz nonlinearity

This paper studies stability of equilibria of differential equations with time-dependent delay and non-Lipschitz nonlinearity. For this class of problems, we develop a novel method of analysis, the relative nonlinear measure method. Using this method, we obtain a sufficient condition for exponential stability. Moreover, this condition is used to study the stability of the equilibrium of a neural network model. Finally, some examples illustrate that our results are improvement and extension of some existing ones.     

Bifurcations of periodic solutions of delay differential equations

In this paper we develop Kaplan-Yorke's method and consider the existence of periodic solutions for some delay differential equations. We especially study Hopf and saddle-node bifurcations of periodic solutions with certain periods for these equations with parameters, and give conditions under which the bifurcations occur. We also give application examples and find that Hopf and saddle-node bifurcations often occur infinitely many times.     

Exponential boundedness of solutions for impulsive delay differential equations

For an impulsive delay differential equation

exponent estimates of solutions have been obtained.     

Parameter identification in periodic delay differential equations with distributed delay

In this study, a parameter identification approach for identifying the parameters of a periodic delayed system with distributed delay is introduced based on time series analysis and spectral element analysis. Using this approach the parameters of the distributed delayed system can be identified from the time series of the response of the system. The experimental or numerical data of the response is examined with Floquet theory and time series analysis techniques to estimate a reduced order dynamics, or truncated state space to identify the Floquet multipliers. Parameter identification is then completed using a dynamic map developed for the assumed model of the system which can relate the Floquet multipliers to the unknown parameters in the model. The parameter identification technique is validated numerically for first and second order delay differential equations with distributed delay.     

Nontrivial periodic solutions for second-order differential delay equations

In this paper, we consider the existence of periodic solutions for second-order differential delay equations. Some existence results are obtained using Malsov-type index and Morse theory, which extends and complements some existing results.     

Exponential stability of impulsive stochastic functional differential equations

In this paper, we investigate the pth moment and almost sure exponential stability of impulsive stochastic functional differential equations with finite delay by using Lyapunov method. Several stability theorems of impulsive stochastic functional differential equations with finite delay are derived. These new results are employed to impulsive stochastic equations with bounded time-varying delays and stochastically perturbed equations. Meanwhile, an example and simulations are given to show that impulses play an important role in pth moment and almost sure exponential stability of stochastic functional differential equations with finite delay.     

Existence and exponential stability of periodic solution for impulsive delay differential equations and applications

In this paper, we consider a class of nonlinear impulsive delay differential equations. By establishing an exponential estimate for delay differential inequality with impulsive initial condition and employing Banach fixed point theorem, we obtain several sufficient conditions ensuring the existence, uniqueness and global exponential stability of a periodic solution for nonlinear impulsive delay differential equations. Furthermore, the criteria are applied to analyze dynamical behavior of impulsive delay Hopfield neural networks and the results show different behavior of impulsive system originating from one continuous system.     

Multiplicity of periodic solutions to symmetric delay differential equations

By applying the method based on the usage of the equivariant gradient degree introduced by G?ba (1997) and the cohomological equivariant Conley index introduced by Izydorek (2001), we establish an abstract result for G-invariant strongly indefinite asymptotically linear functionals showing that the equivariant invariant ${\omega(\nabla \Phi)}$ , expressed as that difference of the G-gradient degrees at infinity and zero, contains rich numerical information indicating the existence of multiple critical points of ${\Phi}$ exhibiting various symmetric properties. The obtained results are applied to investigate an asymptotically linear delay differential equation $$x\prime = - \nabla f \big ({x \big (t - \frac{\pi}{2} \big )} \big ), \quad x \in V \qquad \quad (*)$$ (here ${f : V \rightarrow \mathbb{R}}$ is a continuously differentiable function satisfying additional assumptions) with Γ-symmetries (where Γ is a finite group) using a variational method introduced by Guo and Yu (2005). The equivariant invariant ${\omega(\nabla \Phi) = n_{1}({\bf H}_{1}) + n_{2}({\bf H}_{2}) + \cdots + n_{m}({\bf H}_{m})}$ in the case n _k ≠ 0 (for maximal twisted orbit types (H _k )) guarantees the existence of at least |n _k | different G-orbits of periodic solutions with symmetries at least (H _k). This result generalizes the result by Guo and Yu (2005) obtained in the case without symmetries. The existence of large number of nonconstant periodic solutions for (*) (classified according to their symmetric properties) is established for several groups Γ, with the exact value of ${\omega(\,\nabla \Phi)}$ evaluated.     

On stability of linear periodic differential difference equations

The paper considers the equation

where the operator-valued bounded functions a_j and b_j are 2π-periodic, and the operator-valued kernels m and n are 2π-periodic with respect to the first argument. The connection between the input-output stability of the equation and the invertibility of a family of operators acting on the space of periodic functions is investigated.