Ivanka M. Stamova

Lyapunov—Razumikhin method for impulsive functional differential equations and applications to the population dynamics

The present paper deals with the investigation of the stability of the zero solution of impulsive functional differential equations. By means of piecewise continuous functions coupled with the Razumikhin technique sufficient conditions for stability, uniform stability and asymptotic stability of the zero solution of such equations are found.

On global exponential stability for impulsive cellular neural networks with time-varying delays

In this paper, the problem of global exponential stability for cellular neural networks (CNNs) with time-varying delays and fixed moments of impulsive effect is studied. A new sufficient condition has been presented ensuring the global exponential stability of the equilibrium points by using piecewise continuous Lyapunov functions and the Razumikhin technique combined with Youngs inequality. The results established here extend those given previously in the literature. Compared with the method of Lyapunov functionals as in most previous studies, our method is simpler and more effective for stability analysis.

Almost necessary and sufficient conditions for survival of species

Abstract We consider a nonautonomous competitive Lotka–Volterra system of two species, satisfying two inequalities involving averages of the growth rates and the interaction coefficients, which imply persistence. We introduce a third species and give a third inequality, involving the average of the growth rate of the third species and solutions of a linear algebraic system, which guarantees persistence of the system. It is also shown that reversing this inequality implies non-persistence; more specifically, extinction of the third species with small positive initial values, in the autonomous case. Our conditions are simple and computable.

Global stability of impulsive fractional differential equations

Abstract In the present paper the global stability of systems of impulsive differential equations of fractional order with impulse effect at fixed moments of time is considered. Using piecewise continuous functions of the type of Lyapunov’s functions and a new fractional comparison principle sufficient conditions for various types of global stability are proved.

Impulsive control on global asymptotic stability for a class of impulsive bidirectional associative memory neural networks with distributed delays

In this paper, we study the problem of global asymptotic stability for a class of bidirectional associative memory neural networks with distributed delays and nonlinear impulsive operators. We establish stability criteria by employing Lyapunov functions and the Razumikhin technique. These results can easily be used to design and verify globally stable networks. An illustrative example is given to demonstrate the effectiveness of the obtained results.

Lipschitz stability criteria for functional differential systems of fractional order

In this paper, a class of fractional functional differential equations is investigated. Using differential inequalities and Lyapunov-like functions, Lipschitz stability, uniform Lipschitz stability, and global uniform Lipschitz stability criteria are proved. Since the problem of Lipschitz stability of dynamic systems is relevant in various contexts, including many inverse and control problem, our results can be applied in the qualitative investigations of many practical problems of diverse interest.

Impulsive control for a class of neural networks with bounded and unbounded delays

In this paper, we study the problem of global exponential stability for a class of impulsive neural networks with bounded and unbounded delays and fixed moments of impulsive effect. We establish stability criteria by employing Lyapunov functions and Razumikhin technique. An illustrative example is given to demonstrate the effectiveness of the obtained results.

Almost periodic solutions for impulsive fractional differential equations

In this paper, we consider the existence of almost periodic solutions for impulsive fractional evolution equations involving Caputo fractional derivative. The main results are obtained by means of the theory of operators semi-group, probability density functions, fixed point theorems and the techniques based on fractional calculus. An example is also discussed to illustrate the theory. Some known results are improved and generalized.

Lyapunov method for boundedness of solutions of nonlinear impulsive functional differential equations

Sufficient conditions are investigated for boundedness of solutions of nonlinear impulsive functional differential equations with impulses at fixed moments. The main tools are Lyapunov functions and Gronwall type of inequality.

Applied impulsive mathematical models

Introduction.-Basic Theory.- Impulsive Biological Models.- Impulsive Models in Population Dynamics.- Impulsive Neural Networks.- Impulsive Models in Economics.- References.- Index.