Gani Tr. Stamov

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 the existence of almost periodic solutions for the impulsive Lasota–Wazewska model

Abstract By means of the Cauchy matrix we give sufficient conditions for the existence and exponential stability of almost periodic solutions for the delay impulsive Lasota–Wazewska model. The impulses are realized at fixed moments of time.

Positive almost periodic solutions for a delay logarithmic population model

By utilizing the continuation theorem of coincidence degree theory, we shall prove that a delay logarithmic population model has at least one positive almost periodic solution. An example is provided to illustrate the effectiveness of the proposed result.

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.

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.

On almost periodic processes in uncertain impulsive delay models of price fluctuations in commodity markets

We present an impulsive price model for a single commodity market with delays and uncertain terms. Impulsive perturbations are realized at fixed moments of time and are proposed to model price shocks in the case of continuous time representation. To do so, the paper resorts to the theory of impulsive differential equations. Uncertain terms are due to modeling errors, measurement inaccuracy, mutations in the fluctuation processes and so on. We investigate conditions under which the extended model is capable of generating a stable almost periodic process.

Impulsive Fractional Functional Differential Systems and Lyapunov Method for the Existence of Almost Periodic Solutions

In this paper, the existence of almost periodic solutions for a class of impulsive fractional functional differential systems is investigated. The investigations are carried out by using a new fractional comparison principle, coupled with the fractional Lyapunov method. The stability behaviour of the almost periodic solutions is also considered, extending the corresponding theory of impulsive functional differential equations.

Mittag-Leffler synchronization of fractional neural networks with time-varying delays and reactiondiffusion terms using impulsive and linear controllers

In this paper, we propose a fractional-order neural network system with time-varying delays and reaction-diffusion terms. We first develop a new Mittag-Leffler synchronization strategy for the controlled nodes via impulsive controllers. Using the fractional Lyapunov method sufficient conditions are given. We also study the global Mittag-Leffler synchronization of two identical fractional impulsive reaction-diffusion neural networks using linear controllers, which was an open problem even for integer-order models. Since the Mittag-Leffler stability notion is a generalization of the exponential stability concept for fractional-order systems, our results extend and improve the exponential impulsive control theory of neural network system with time-varying delays and reaction-diffusion terms to the fractional-order case. The fractional-order derivatives allow us to model the long-term memory in the neural networks, and thus the present research provides with a conceptually straightforward mathematical representation of rather complex processes. Illustrative examples are presented to show the validity of the obtained results. We show that by means of appropriate impulsive controllers we can realize the stability goal and to control the qualitative behavior of the states. An image encryption scheme is extended using fractional derivatives.

On the stability of sets for delayed Kolmogorov-type systems

In this paper we consider Kolmogorov-type delay systems. Criteria on the uniform global asymptotic stability of sets are established for the above systems using Lyapunov functions and the Razumikhin technique.