Bifurcation of a three molecular saturated reaction with impulsive input

Although impulsive differential equations have become a widely concerned subject and a lot of models with impulsive effect have been studied in recent years, biochemical reaction models with impulsive input are rarely studied. In this paper, we consider an irreversible three molecular reaction model with impulsive input. By using the Floquet theorem and the method for the small parameter of impulsive differential equations, we obtain sufficient conditions for asymptotical stability and global stability of the given system. The existence of a positive periodic solution is also studied by the bifurcation theory. Further, we also show that our given conditions are right by numerical simulations.     

Almost periodic solution of an impulsive differential equation model of plankton allelopathy

In this paper, we consider an impulsive differential equation model of plankton allelopathy. Sufficient conditions ensuring the existence of a unique almost periodic solution of the system are obtained, by the relation between the solutions of impulsive system and the corresponding non-impulsive system.     

Extinction and stability of an impulsive system with pure delays

In this paper, we consider an impulsive competitive system with infinite delays and investigate the extinction and stability of the system. For the corresponding impulsive logistic model, our stability conditions are weaker than those of Yang et al. (2011).     

Dynamics behaviors of a delayed stage-structured predator-prey model with impulsive effect

In this paper, we analyzed a delayed stage-structured predator-prey model with impulsive stocking on prey and continuous harvesting on predator. Sufficient conditions for the global attractivity of ‘predator-eradication’ periodic solution and permanence of the system are obtained. The results show that the behavior of impulsive stocking on prey plays an important role for the permanence of the system.     

Threshold dynamics of a stage-structured single population model with non-transient and transient impulsive effects

In this paper, we present a stage-structured single population model with non-transient and transient impulsive effects. By the stroboscopic map theories of impulsive differential equations, we obtain the sufficient conditions for the permanence of the investigated system. The results indicate that the thresholds of the transient impulsive harvesting amount and the non-transient impulsive harvesting interval play important roles on the permanence of population. Our results also provide reliable tactic basis for the practical biological resource management.     

Chatter dynamic analysis for a planing model with the effect of pulse

In this paper, the phenomena of chatter vibration in a typicalplaning process is discussed from impulsive point of view. Considering theinstantaneous vibration as an impulse, we present the planing model as asecond-order impulsive dynamical system, which is a specic discontinuousone. Then we investigate its chatter conditions via the method of ow theoryin discontinuous systems, analyze ow's dynamical behaviors on separationsurface and get general results on chatter criterion. Finally, such dynamicalanalysis and criterion are applied to a specic planing model of coal plough.     

Fuzzy Impulsive Control and Synchronization of General Chaotic System

In this paper, by utilizing impulsive control theory and T-S fuzzy model, the fuzzy impulsive control and synchronization of general chaotic system are proposed. Some less conservative and more general conditions are obtained to guarantee the globally asymptotical stability for the impulsive control and synchronization of general chaotic system based on T-S fuzzy model. Moreover, some criteria of globally exponential stability of chaotic system are also derived. Finally, some numerical simulations are given to demonstrate the effectiveness of the proposed control method.     

具有脉冲输入和资源循环的恒化器模型

研究了一类在污染环境下的具有脉冲输入和资源循环的Monod型恒化器模型,利用Floquet定理和脉冲微分方程解的比较定理,我们得出了系统的微生物灭绝周期解全局渐近稳定以及系统持久的充分条件.     

Exponential stability of nonlinear state-dependent delayed impulsive systems with applications

The exponential stability problem for impulsive systems subject to double state-dependent delays is studied in this paper, where state-dependent delay (SDD) is involved in both continuous dynamics and discrete dynamics and the boundedness of it with respect to states is prior unknown. According to impulsive control theory, we present some Lyapunov-based sufficient conditions for the exponential stability of the concerned system. It is shown that the stabilizing effect of SDD impulses on an unstable SDD system changes the stability and achieves desired performance. In addition, the destabilizing effect of SDD impulses is also fully considered and the corresponding sufficient conditions are derived, which reveals the fact that a stable SDD system can maintain its performance when it is subject to SDD impulsive disturbance. As an application, the proposed result can be employed to the stability analysis of impulsive genetic regulatory networks (GRNs) with SDD and the corresponding sufficient conditions are proposed in terms of the model transformation technique and the linear matrix inequalities (LMIs) technique. In order to demonstrate the effectiveness and applicability of the derived results, we give two examples including impulsive GRNs with SDD and the impulsive controller design for the nonlinear system with SDD.     

Approximate controllability of impulsive semilinear stochastic system with delay in state

Many practical systems in physical and biological sciences have impulsive dynamical behaviors during the evolution process that can be modeled by impulsive differential equations. This article studies the approximate controllability of impulsive semilinear stochastic system with delay in state in Hilbert spaces. Assuming the conditions for the approximate controllability of the corresponding deterministic linear system, we obtain the sufficient conditions for the approximate controllability of the impulsive semilinear stochastic system with delay in state. The results are obtained by using Banach fixed point theorem. Finally, two examples are given to illustrate the developed theory.     

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.     

Global dynamics analysis of a nonlinear impulsive stochastic chemostat system in a polluted environment

This paper intends to develop a new method to obtain the threshold of an impulsive stochastic chemostat model with saturated growth rate in a polluted environment. By using the theory of impulsive differential equations and stochastic differential equations, we obtain conditions for the extinction and the permanence of the microorganisms of the deterministic chemostat model and the stochastic chemostat model. We develop a new numerical computation method for impulsive stochastic differential system to simulate and illustrate our theoretical conclusions. The biological results show that a small stochastic disturbance can cause the microorganism to die out, that is, a permanent deterministic system can go to extinction under the white noise stochastic disturbance. The theoretical method can also be used to explore the threshold of some impulsive stochastic differential equations.     

Analysis of mathematics and dynamics in a food web system with impulsive perturbations and distributed time delay

In this paper, on the basis of the theories and methods of ecology and ordinary differential equation, a food web system with impulsive perturbations and distributed time delay is established. By using the theories of impulsive equation, small amplitude perturbation skills and comparison technique, we get the condition which guarantees the global asymptotical stability of the prey and intermediate predator eradication periodic solution. On this basis, we get that the food web system is permanent if some parameters are satisfied with certain conditions. In order to show that these conditions are effective, the influences of impulsive perturbations on the inherent oscillation and distributed time delay are studied numerically; these show rich dynamics, such as period-halving bifurcation, chaotic band, narrow or wide periodic window, chaotic crises. Moreover, the computation of the largest Lyapunov exponent shows the chaotic dynamic behavior of the model. Meanwhile, we investigate the qualitative nature of strange attractor by using Fourier spectra. All of these results may be useful in the study of the dynamic complexity of ecosystems.     

Evolutionary dynamics in a Lotka–Volterra competition model with impulsive periodic disturbance

In this paper, we develop a theoretical framework to investigate the influence of impulsive periodic disturbance on the evolutionary dynamics of a continuous trait, such as body size, in a general Lotka–Volterra‐type competition model. The model is formulated as a system of impulsive differential equations. First, we derive analytically the fitness function of a mutant invading the resident populations when rare in both monomorphic and dimorphic populations. Second, we apply the fitness function to a specific system of asymmetric competition under size‐selective harvesting and investigate the conditions for evolutionarily stable strategy and evolutionary branching by means of critical function analysis. Finally, we perform long‐term simulation of evolutionary dynamics to demonstrate the emergence of high‐level polymorphism. Our analytical results show that large harvesting effort or small impulsive harvesting period inhibits branching, while large impulsive harvesting period promotes branching. Copyright © 2015 John Wiley & Sons, Ltd.     

Feasibility of time-limited control of a competition system with impulsive harvest

A kind of time-limited control model on a competition system with impulsive harvest, described by impulsive differential equation with the initial and boundary value problem, is presented. The existence of solution of the model, corresponding to the feasibility of the short-term control, is discussed. By the comparison principle, the conditions under which the model has a solution are found by a series of the upper solutions, and the conditions under which the model has no solution are also given by a series of the lower solutions. Finally, the practical meanings of those conditions are explained. As an example, if other parameters are given, the times of the impulsive control is estimated and the theoretical results are verified by numerical simulations.     

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.     

Integral boundary value problems for nonlinear non-instantaneous impulsive differential equations

In this paper, we study integral boundary value problems for two classes of nonlinear non-instantaneous impulsive ordinary differential equations. More precisely, one is integer order impulsive model, the other is fractional order impulsive model. By using standard fixed point approach, a series of existence results are presented under different conditions.     

Periodic solution of a chemostat model with variable yield and impulsive state feedback control

In this paper, a chemostat model with variable yield and impulsive state feedback control is considered. We obtain sufficient conditions of the globally asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control has periodic solution of order one. Sufficient conditions for existence and stability of periodic solution of order one are given. In some cases, it is possible that the system exists periodic solution of order two. Our results show that the control measure is effective and reliable.     

Impulsive control and its application to Lü''s chaotic system

In this paper, we derive some less stringent conditions for the exponential and asymptotic stability of impulsive control systems with impulses at fixed times. These conditions are then used to design an impulsive control law for Lü system, which drives the chaotic state to a zero equilibrium.