Theory Of Impulsive Differential Equations Research Articles

Analysis of an Impulsive Predator‐Prey System with Monod‐Haldane Functional Response and Seasonal Effects

For a class of impulsive predator‐prey systems with Monod‐Haldane functional response and seasonal effects, we investigate conditions for the local and global stabilities of prey‐free solutions and for the permanence of the system by using the Flquet theory of impulsive differential equations and comparison techniques. In addition, we numerically analyze the phenomena caused by seasonal effects and impulsive perturbation. It will be applicable to the controllability for the population of prey and predator.

Dynamic Complexities of a Three-Species Beddington-DeAngelis System with Impulsive Control Strategy

By introducing the impulsive control strategy, the dynamical properties of the three-species food chain systems with Beddington-DeAngelis functional response are investigated. Using the Floquet theory of impulsive differential equations and small amplitude perturbation techniques, we prove that there exists an asymptotically locally stable prey-free periodic solution under some conditions, and permanence conditions are established via the comparison theorem. Furthermore, from numerical examples, we figure out that the system has complex dynamic behaviors.

A practical approach to robust impulsive lag synchronization between different chaotic systems

In this paper, a practical impulsive lag synchronization scheme for different chaotic systems with parametric uncertainties is proposed. By virtue of the new definition of synchronization and the theory of impulsive differential equations, some new and less conservative sufficient conditions are established to guarantee that the error dynamics can converge to a predetermined level. The idea and approach developed in this paper can provide a more practical framework for the synchronization between identical and different chaotic systems in parameter perturbation circumstances. Simulation results finally demonstrate the effectiveness of the method.

Dynamic analysis of mathematical model of ethanol fermentation with gas stripping

In this paper, a mathematical model for ethanol fermentation with gas stripping is investigated. Firstly, the model with continuous substrate input is taken. We study the existence and local stability of two equilibrium points. According to Poincare–Bendixson Theorem, the sufficient condition for the globally asymptotical stability of positive equilibrium point is obtained, which implies that we can get stable ethanol product. Secondly, we study the model with impulsive substrate input and obtain the sufficient condition for the local stability of cell-free periodic solution by using the Floquet’s theory of impulsive differential equation and small-amplitude perturbation skills. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical (subcritical) bifurcation. Finally, our results are confirmed by means of numerical simulation.

Predicting the potential impact of a cytotoxic T-lymphocyte HIV vaccine: How often should you vaccinate and how strong should the vaccine be?

To stimulate the immune system’s natural defenses, a post-infection HIV vaccination program to regularly boost cytotoxic T-lymphocytes has been proposed. We develop a mathematical model to describe such a vaccination program, where the strength of the vaccine and the vaccination intervals are constant. We apply the theory of impulsive differential equations to show that the model has an orbitally asymptotically stable periodic orbit, with the property of asymptotic phase. We show that, on this orbit, the vaccination frequency can be chosen so that the average number of infected CD4 + T cells can be made arbitrarily low. We illustrate the results with numerical simulations and show that the model is robust with respect to both the parameter choices and the formulation of the model as a system of impulsive differential equations.

Impulsive control of projective synchronization in chaotic systems

Scaling factor of projective synchronization in coupled partially linear chaotic systems is hardly predictable. To control projective synchronization of chaotic systems in a preferred way, an impulsive control scheme is introduced to direct the scaling factor onto a desired value. The control approach is derived from the impulsive differential equation theory. Numerical simulations on the chaotic Lorenz system are illustrated to verify the theoretical results. Furthermore, some interesting and surprising numerical results are discussed.

Trajectory tracking for a particle in elliptical billiards

An infinitely rigid unitary mass (particle) is considered, moving on a planar region delimited by a rigid elliptical barrier (elliptical billiards) under the action of proper control forces. A class of periodic trajectories, involving an infinite sequence of non-smooth impacts between the mass and the barrier at fixed times, is found by using an LMIs based procedure. The jumps in the velocities at the impact times render difficult (if not impossible) to obtain the classical stability and attractivity properties for the dynamic system describing the tracking error behaviour. Hence, the tracking control problem is properly stated using notions similar to the quasi stability concept in V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, 6, World Scientific, 1989. A controller (whose state is subject to discontinuities) based on the internal model principle is shown to solve the proposed tracking problem, giving rise to control forces that are piecewise continuous function of time, with discontinuities at the desired impact times and at the impact times of the particle with the barrier.

Some modern aspects of the theory of impulsive differential equations

We give a brief survey of the main results obtained in recent years in the theory of impulsive differential equations.

Extinction and Permanence of a Three‐Species Lotka‐Volterra System with Impulsive Control Strategies

A three‐species Lotka‐Volterra system with impulsive control strategies containing the biological control (the constant impulse) and the chemical control (the proportional impulse) with the same period, but not simultaneously, is investigated. By applying the Floquet theory of impulsive differential equation and small amplitude perturbation techniques to the system, we find conditions for local and global stabilities of a lower‐level prey and top‐predator free periodic solution of the system. In addition, it is shown that the system is permanent under some conditions by using comparison results of impulsive differential inequalities. We also give a numerical example that seems to indicate the existence of chaotic behavior.

Robust Impulsive Synchronization for a Class of Unified Chaotic Systems with Parameter Uncertainty

For a class of unified chaotic systems with parameter uncertainty, a robust impulsive synchronization scheme is proposed. Based on the theory of impulsive differential equations, some new and less conservative sufficient conditions are established in order to guarantee the robust synchronization of the chaotic systems. In particular, some simple and practical conditions are derived in synchronizing the chaotic systems by equal impulsive distances and control gains. Simulation results finally demonstrate the effectiveness of the method.

Permanence in a food chain system with impulsive perturbations

In this article, we investigate three species food chain system with Holling II functional responses and periodic constant impulsive perturbations of predator at different fixed time. The conditions for extinction of prey and top predator are given. By using the Floquet theory of impulsive differential equation and small amplitude perturbation skills, we consider the local stability of prey and top predator eradication periodic solution. Further, we obtain the conditions of permanence of the system. The complex dynamics behavior of the system (1.3) is found by using numerical method.

THE BEST DOSAGE SCHEME ON MULTIPLE RAPID VEIN INJECTION

In this paper, we mainly consider the compartment models of the multiple rapid vein injection. Applying the theory of impulsive differential equation, we study the steady drug concentration in the blood, its maximum, minimum and mean value, dosing interval, etc. to design the best dosage schemes on these models. This paper is useful in instructing the use of pharmaceuticals in clinical practice and enriches the research of pharmacokinetics.

Impulsive control of chaotic systems with exogenous perturbations

The impulsive control of chaotic systems, which are subjected to unbounded exogenous perturbations, is considered. By using the theory of impulsive differential equation together with the fuzzy control technique, the authors propose an impulsive robust chaos controlling criterion expressed as linear matrix inequalities (LMIs). Based on the proposed control criterion, the procedure for designing impulsive controllers of common (perturbed) chaotic systems is provided. Finally, a numerical example is given to demonstrate the obtained theoretical result.

Permanence of a general periodic single-species system with periodic impulsive perturbations

Sufficient conditions for permanence of a general periodic single-species system with periodic impulsive perturbations are obtained via comparison theory of impulsive differential equations. An application is given to the periodic impulsive logistic system.

Impulsive synchronization for unified chaotic systems with channel time-delay and parameter uncertainty

In this paper, an impulsive control for synchronization of a class of unified chaotic systems with channel time-delay and parameter uncertainty is proposed. The practical stability of impulsive synchronization between two nonautonomous chaotic systems is studied. This practical stability is equivalent to that of the synchronization error system at the origin. Based on the theory of impulsive differential equation, the criterion for the practical stability of the synchronization error system at the origin is presented. Numerical simulations show the effectiveness of the method.

Impulsive Intergo-differential Equations and Stability of Moving Invariant Manifolds

Abstract. This paper study the stability of moving invariant manifolds of nonlinear impulsiveintegro-differential equations. The obtain results are based on the method of piecewise continuousLyapunov’s functions and the comparison principle.Key words. Uncertain impulsive integro-differential system, moving invariant manifold, stabil-ity theoryAMS subject classifications. 34A37 1. Preliminary notes. Impulsive integro-differential equations arise naturallyfrom a wide variety of applications such as aircraft control, inspection process inoperations research, drug administration, and threshold theory in biology. There hasbeen a significant development in the theory of impulsive differential equations in thelast years [1–3].Now there also exist a well developed qualitative theory for impulsive integro-differential equations [7, 8].The efficient applications of impulsive integro-differential equations to mathemat-ical simulation request the finding of criteria for stability of their solutions.In this paper we use piecewise continuous Lyapunov’s functions to study thestability of moving invariant manifolds for general class of uncertain impulsive integro-differential equations. In the few publications dedicated to the subject of movinginvariant manifold for differential equations without impulses, earlier works were doneby [5, 6-7, 10].Our results are obtained by means of the comparison principle which permitsus to reduce the study of impulsive integro-differential equations to the study of ascalar differential equation.Let R

A FOOD CHAIN SYSTEM WITH DENSITY-DEPENDENT BIRTH RATE AND IMPULSIVE PERTURBATIONS

We investigate a three-species food chain system with density-dependent birth rate and impulsive effect concerning biological and chemical control strategy — periodic releasing of natural enemies or spraying pesticide at different fixed times. Conditions for the extinction of the prey and top predator are given. By using the Floquet theory of impulsive differential equations and small amplitude perturbation skills, we consider the local stability of the prey and top predator eradication periodic solution. Further, we obtain the conditions of permanence of the system.

Impulsive set differential equations with delay

The basic theory of set impulsive set differential equations with delay is initiated by combining the theories of impulsive differential equations, set differential equations in metric spaces and delay differential equations.

Some impulsive synchronization criterions for coupled chaotic systems via unidirectional linear error feedback approach

Based on stability theory of impulsive differential equation and new comparison theory of impulsive differential system, we study the chaos impulsive synchronization of two coupled chaotic systems using the unidirectional linear error feedback scheme. Some generic conditions of chaos impulsive synchronization of two coupled chaotic systems are derived, and to apply the conditions to typical chaotic system––the original Chua’s circuit. The example illustrates the effectiveness of the proposed result.

Impulsive Synchronization of Nonautonomous Chaotic Systems

In this paper-the practical stability of impulsive synchronization between two-nonautonomous chaotic systems is studied. This practical stability is equivalent to that of the origin of the synchronization error system. We present the crite ria for the practical stability of the origin of the synchronization error syste m based on the theory of impulsive differential equation. We also provide the theoretical results on the range of synchronization errors. Numerical experimental-results are given to compare with theoretical results.