Dynamical behaviors of a trimolecular response model with impulsive input

In the paper, we consider a trimolecular response model with impulsive input. Sufficient conditions for globally stable positive equilibrium point of the trimolecular response model with constant input are obtained. By impulsive differential theory, we obtain conditions for permanence of the trimolecular response model with impulsive input. By numerical analysis, we demonstrate complex phenomena such as limit cycles, periodic solutions, and chaos.     

Periodic solution of a turbidostat model with impulsive state feedback control

In this paper, a turbidostat model with impulsive state feedback control is considered. We obtain sufficient conditions of the global asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control may have order one periodic solution, and the sufficient condition for existence and stability of order one periodic solution is gotten as well. For some special cases, it is shown that in the system an order two periodic solution may exist. Our results show that the control measure is effective and reliable.     

Nonlinear modelling of chemostat model with time delay and impulsive effect

In this paper, a chemostat model with periodically pulsed input and time delay is considered. We show that there exists a microorganism-free periodic solution, which is globally attractive when the period of impulsive effect is less than some critical value. Further, we give the sufficient conditions for the permanence of the model with time delay and pulsed input. We show that time delay, impulsive input can bring different effects on the dynamic behavior of the model by numerical analysis. We show that impulsive effect destroys the equilibria of the unforced continuous system and initiates periodic solution. Our results can be applied to culture the microorganism.     

Global qualitative analysis of new Monod type chemostat model with delayed growth response and pulsed input in polluted environment

In this paper, we consider a new Monod type chemostat model with time delay and impulsive input concentration of the nutrient in a polluted environment. Using the discrete dynamical system determined by the stroboscopic map, we obtain a ＂microorganism-extinction＂ periodic solution. Further, we establish the sufficient conditions for the global attractivity of the microorganism-extinction periodic solution. Using new computational techniques for impulsive and delayed differential equation, we prove that the system is permanent under appropriate conditions. Our results show that time delay is ＂profitless＂.     

A stage-structured single species model with pulse input in a polluted environment

In this paper, we study a stage-structured single species model with pulse input in a polluted environment. Using the discrete dynamical system determined by the stroboscopic map, we obtain the globally attractive condition for the species-extinction periodic solution of the investigated system. By the use of the theory of impulsive delayed differential equation, we also obtain sufficient condition of the permanence of the investigated system. Our results reveal that long mature period of the population in polluted environment can cause it to go extinct. Supported by National Natural Science Foundation of China (10647005), and the Nomarch Foundation of Guizhou Province.     

Dynamic stability of shallow arch with elastic supports—application in the dynamic stability analysis of inner winding of transformer during short circuit

In this paper, the dynamic stability of a shallow arch with elastic supports subjected to impulsive load is used as a theoretical model to investigate the dynamic stability problem of inner windings of power transformer under short-circuit condition. Firstly, the series solution representing the equilibrium configurations of a shallow arch is obtained by solving the corresponding non-linear integration-differential equation. The local stability of each equilibrium configuration is discussed, and the sufficient condition for stability of the shallow arch system as well as the critical load against snap-through is obtained. Secondly, the equivalent relation between short-circuit load and impulsive one, and the electrical forces transferred pattern between the coils of inner windings are assumed. Then the results of the shallow arch model are applied to the case of the inner winding of transformer and the formulas for computing critical electromagnetic force and the dynamic stability criterion of the inner windings are established. Finally, examples are offered and the theoretical results are shown to agree well with the experimental ones.     

Nonlinear analysis of a microbial pesticide model with?impulsive?state feedback control

In this paper, a new mathematical model for the entomopathogenic nematode attacking pests with impulsive state feedback control is considered. By using the Poincaré map, we obtain that the system with impulsive state feedback control has a periodic solution of order one. Sufficient conditions for existence and stability of the order one periodic solution are given. Specifically, the system has a singular order one periodic solution. In some cases, it is possible that the system may also have an order two periodic solution. Our results show that the control measure is effective and reliable.     

The study of a ratio-dependent predator–prey model with stage structure in the prey

In this paper, a ratio-dependent predator–prey model with stage structure in the prey is constructed and investigated. In the first part of this paper, some sufficient conditions for the existence and stability of three equilibriums are obtained. In the second part, we consider the effect of impulsive release of predator on the original system. A sufficient condition for the global asymptotical stability of the prey-eradication periodic solution is obtained. We also get the condition, under which the prey would never be eradicated, i.e., the impulsive system is permanent. At last, we give a brief discussion.

     

Dynamics of an inshore–offshore fishery model with impulsive pollutant input in inshore area

Concerning the polluted environment in inshore areas and the effects of unbalanced impulsive diffusion between inshore and offshore areas on the size of fish population, based on optimal harvesting policy, a logistic offshore fishery system with impulsive pollutant input in inshore areas and impulsive diffusion at different fixed time is formulated, and we investigate the dynamics of such system. The sufficient conditions are obtained for the existence of the positive periodic solution and the global asymptotic stability of both the trivial periodic solution and the positive periodic solution. Moreover, the maximum sustainable yield and the corresponding harvesting effort are obtained.     

Periodic solution of the system with impulsive state feedback control

The order-1 periodic solution of the system with impulsive state feedback control is investigated. We get the sufficient condition for the existence of the order-1 periodic solution by differential equation geometry theory and successor function. Further, we obtain a new judgement method for the stability of the order-1 periodic solution of the semi-continuous systems by referencing the stability analysis for limit cycles of continuous systems, which is different from the previous method of analog of Poincarè criterion. Finally, we analyze numerically the theoretical results obtained.     

The dynamics of a high-dimensional delayed pest management model with impulsive pesticide input and?harvesting prey at different fixed moments

In this paper, a delayed pest control model with stage-structure for pests by introducing a constant periodic pesticide input and harvesting prey (Crops) at two different fixed moments is proposed and analyzed. We assume only the pests are affected by pesticide. We prove that the conditions for global asymptotically attractive ??predator-extinction?? periodic solution and permanence of the population of the model depend on time delay, pulse pesticide input, and pulse harvesting prey. By numerical analysis, we also show that constant maturation time delay, pulse pesticide input, and pulse harvesting prey can bring obvious effects on the dynamics of system, which also corroborates our theoretical results. We believe that the results will provide reliable tactic basis for the practical pest management. One of the features of present paper is to investigate the high-dimensional delayed system with impulsive effects at different fixed impulsive moments.     

弹性支持扁拱动力稳定性分析和变压器内线圈短路动稳定分析

本文采取脉冲载荷激励下弹支扁拱弹性动力稳定性的理论模型分析电力变压器内线圈短路动稳定问题.首先导出了脉冲载荷激励下弹支扁拱非线性运动积分微分方程级数解,给出了各平衡位置(奇点)的局部稳定(不稳定)性证明,导出弹支扁拱抗跳跃失稳的稳定性充分条件、判别公式和临界载荷;其次导出了短路载荷与脉冲载荷的相当关系,考虑内线圈各线匝(铜条)短路电动力到内线匝的传递,给出了内线圈短路动稳定临界电动力计算公式和动稳定判别方法;最后作出与实验结果的比较和算例.     

Qualitative analysis of a predator–prey system with mutual interference and impulsive state feedback control

In this paper, a predator–prey system with mutual interference and impulsive state feedback control is constructed. Firstly, by using the stability theorem, we obtain the global asymptotically stability of the positive equilibrium for the system without impulse effects. Secondly, by using the geometric theory of differential equations, the method of successor functions and mathematical analysis, we obtain some sufficient conditions for the existence, uniqueness and orbital asymptotically stability of the order-1 periodic solution for the system with impulse effects. Finally, the main conclusions are verified by numerical simulation, and the pest prevention and control strategies on combining light trapping and water-gun spraying are proposed.     

Dynamic complexity of microbial pesticide model

In this paper, a mathematical model for the entomopathogenic nematode attacking the pest is investigated. This novel theoretical framework could result in an objective criterion on how to release the entomopathogenic nematode in order to control the pest population under the economic threshold (ET) which indicates the maximally admissible pest densities. Firstly, continuous release of the entomopathogenic nematode is taken. By using qualitative analysis method, the sufficient condition of the global stability of the positive equilibria and the existence and uniqueness of limit cycle of the system are obtained. Secondly, impulsive release of the entomopathogenic nematode is also considered. Using the Floquet’s theorem and small-amplitude perturbation, we obtain that the pest-free periodic solution is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations.     

The effect of pulsed harvesting policy on the inshore–offshore fishery model with the impulsive diffusion

In this paper, we propose the inshore–offshore fishing model with impulsive diffusion and pulsed harvesting at the different fixed time. The existence and globally asymptotical stability of both the trivial periodic solution and the positive periodic solution are obtained. We show that the pulsed harvesting has a strong impact on the persistence of the fish population. By the numerical simulation, we obtain that the best time of fishing is at the end of the period τ.     

Dynamical behavior of a class of prey-predator system with impulsive state feedback control and Beddington–DeAngelis functional response

This paper studies systematically a Bedd-ington?CDeAngelis prey?Cpredator system with harvesting and impulsive state feedback control. Conditions for existence and stability of predator-free periodic solution are obtained. When the predator-free periodic solution loses its stability, the existence and stability of nontrivial period solution are also established. Furthermore, computer simulations show that this impulsive system displays a series of complex phenomena, including period-doubling bifurcation and cascade, period window, and chaotic bands. Through numerical simulation, it is also observed that capture capability can influence the amount of predator released and the interval of the stability for nontrivial period-1 solution. Moreover, the superiority of impulsive state feedback control strategy is also exhibited over the impulsive fixed-time control.     

Input-to-state stability of impulsive reaction–diffusion neural networks with infinite distributed delays

In this paper, we focus on the global existence–uniqueness and input-to-state stability of the mild solution of impulsive reaction–diffusion neural networks with infinite distributed delays. First, the model of the impulsive reaction–diffusion neural networks with infinite distributed delays is reformulated in terms of an abstract impulsive functional differential equation in Hilbert space and the local existence–uniqueness of the mild solution on impulsive time interval is proven by the Picard sequence and semigroup theory. Then, the diffusion–dependent conditions for the global existence–uniqueness and input-to-state stability are established by the vector Lyapunov function and M-matrix where the infinite distributed delays are handled by a novel vector inequality. It shows that the ISS properties can be retained for the destabilizing impulses if there are no too short intervals between the impulses. Finally, three numerical examples verify the effectiveness of the theoretical results and that the reaction–diffusion benefits the input-to-state stability of the neural-network system.

     

Nonlinear incidence rate of a pest management SI model with biological and chemical control concern

A pest management SI model with impulsive releases of infective pests and spraying pesticides is proposed and investigated. We prove that all solutions of the model are uniformly ultimately bounded. We also obtain the sufficient conditions of globally asymptotic stability periodic solution of pest-extinction and permanence of the model. The approach of combining impulsive releasing infective pests with impulsive spraying pesticides provides reliable tactical basis for the practical pest management.     

Bifurcation and chaos of biochemical reaction model with impulsive perturbations

In this paper, a biochemical model with the impulsive perturbations is considered. By using the Floquet theorem for the impulsive equation and small-amplitude perturbation skills, we see that the boundary-periodic solution ([(x)\tilde](t),0)(\tilde{x}(t),0) is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. By numerical simulation, we can show that the system presents rich dynamics, including periodic solutions, quasi-periodic oscillations, period doubling cascades, periodic halving cascades, symmetry bifurcations, and chaos.     

A numerical method on estimation of stable regions of rotor systems supported on lubricated bearings

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