Stability Analysis of Analytical and Numerical Solutions to Nonlinear Delay Differential Equations with Variable Impulses

Abstract

A stability theory of nonlinear impulsive delay differential equations (IDDEs) is established. Existing algorithm may not converge when the impulses are variable. A convergent numerical scheme is established for nonlinear delay differential equations with variable impulses. Some stability conditions of analytical and numerical solutions to IDDEs are given by the properties of delay differential equations without impulsive perturbations.

Modeling Corruption as a Contagious Disease

“…The small perturbation that was introduced to the equilibrium resulted into a slight change in the evolution of some classes, but they eventually returned back to the same equilibrium. This happens because the model is stable (Liu and Zeng, 2017). The stability analysis of the model is presented in another paper (Pabico, 2018b).…”

Stability of Differential Systems with Impulsive Effects

“…In [25], the authors focused on the linear form of IDDS (2), where f = Ax(t) + Bx(t − τ); they presented several asymptotic stability and exponential stability criteria by using Lyapunov functions and the comparison principle method. In [26][27][28], the authors investigated linear and nonlinear IDDS (2), respectively, where ∆x | t=t k = r k x(t k ), r k ∈ R and t k+1 − t k = τ. Some theorems are established such that the stability of IDDS can be transformed into the stability of the corresponding delay differential equations without impulsive effects, and the convergent numerical processes are also proposed to calculate numerical solutions of IDDS.…”

“…This 'equivalent method' has attracted the attention of many researchers, such as [4,5,[26][27][28]30,31]. Nowadays, impulsive delay differential systems have been widely studied, and a variety of applications have been found.…”

“…In [33], time-delay is state-dependent, i.e., τ = τ(t, x(t)), in addition to providing some Lyapunov-based sufficient conditions for the exponential stability, these results are also applied to the stability analysis of impulsive gene regulatory networks by using the linear matrix inequalities technique. More information can be found in the references cited in [24][25][26][27][28][29]32,33] and the studies in which they are cited.…”

“…[4,30,31] developed the 'equivalent method' in [29], established the relationship between SIDDS and a corresponding non-impulsive stochastic delay differential system, and obtained some stability results. Moreover, the 'equivalent method' has also been effectively applied to the stability of numerical solutions, such as [5,[26][27][28].…”