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.     

Periodic solution of a pest management Gompertz model with impulsive state feedback control

In this paper, a new model with two state impulses is proposed for pest management. According to different thresholds, an integrated strategy of pest management is considered, that is to say if the density of the pest population reaches the lower threshold \(h_1\) at which pests cause slight damage to the forest, biological control (releasing natural enemy) will be taken to control pests; while if the density of the pest population reaches the higher threshold \(h_2\) at which pests cause serious damage to the forest, both chemical control (spraying pesticide) and biological control (releasing natural enemy) will be taken at the same time. For the model, firstly, we qualitatively analyse its singularity. Then, we investigate the existence of periodic solution by successor functions and Poincaré-Bendixson theorem and the stability of periodic solution by the stability theorem for periodic solutions of impulsive differential equations. Lastly, we use numerical simulations to illustrate our theoretical results.     

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.     

Periodic solutions of an impulsive system with two discontinuity points

We establish conditions for the existence of a 2π-periodic solution of an impulsive system with two discontinuity points at nonfixed times.     

非线性时滞反馈控制系统超谐共振分析

采用增量谐波平衡法求解了非线性时滞微分方程的超谐共振解,研究了时滞、反馈控制增益、激励幅值、非线性项系数等系统参数对系统超谐共振响应的影响,分析了超谐共振响应随系统参数变化的规律。结果表明:三次谐波与一次谐波振幅的比值随时滞量呈周期性变化;反馈控制增益对系统超谐共振的影响与非线性项系数和激励幅值有关;随着非线性项系数和激励幅值的不断增大,三次谐波项与一次谐波项振幅的比值都是先增大后减小,而且减小的趋势逐渐减弱;一次谐波成份在振幅中占主导地位。     

Complex dynamic behavior in a viral model with state feedback control strategies

With the consideration of mechanism of prevention and control for the spread of viral diseases, in this paper, we propose two novel virus dynamics models where state feedback control strategies are introduced. The first model incorporates the density of infected cells (or free virus) as control threshold value; we analytically show the existence and orbit stability of positive periodic solution. Theoretical results imply that the density of infected cells (or free virus) can be controlled within an adequate level. The other model determines the control strategies by monitoring the density of uninfected cells when it reaches a risk threshold value. We analytically prove the existence and orbit stability of semi-trivial periodic solution, which show that the viral disease dies out. Numerical simulations are carried out to illustrate the main results.     

Bifurcation analysis of a linear Hamiltonian system with two kinds of impulsive control

In this paper, the dynamical behavior of a linear Hamiltonian system under two kinds of impulsive control is discussed by means of both theoretical and numerical ways. The existence and stability of the periodic solution are investigated. Moreover, the conditions of existence for a Neimark?CSacker bifurcation are derived by using a discrete map. Numerical results for phase portraits, periodic solutions, and bifurcation diagrams are in good agreement with the theoretical analysis.     

分段线性耦合动力系统的周期解及稳定性分析

以某货车的主副钢板弹簧悬架为模型建立两自由度分段线性和轮胎非线性耦合动力学方程,采用打靶法求耦舍系统的周期解;将所得结果与近似解析法分析的结果进行比较,并利用Floquet理论判断周期解的稳定性。研究结果表明：当激励频率在等效固有频率附近时系统的周期解不稳定,其Floquet乘数的模大于1,系统振动剧烈,KBM法得到的周期解将产生较大误差;当路面工况突然发生改变或路面工况不变而载重发生较大变化时都有可能发生舅毡跃现象,造成系统的不稳定;周期解的长期时域图和Floquet理论验证了这一现象;     

非线性转子-轴承系统的周期解及近似解析表达式

通过对普通打靶方法进行改造提出一种确定非线性系统周期轨道及周期的新型打靶算法。首先通过改变系统的时间尺度,将非线性系统周期轨道的周期显式地出现在非线性系统的系统方程中,然后对传统打靶法进行改造,将周期也作为一个参数一起参与打靶法的迭代过程,迭代过程包含对周期轨道和周期的求解,迭代过程中的增量通过优化方法选择,从而能迅速确定出系统的周期轨道及其周期。应用所求的结果结合谐波平衡方法求得了非线性系统的周期轨道的近似解析表达式,理论上通过增加谐波的阶数任何精度的周期解都可以得到。最后将该方法应用于非线性转子轴承系统,求出了在某些参数下转子的周期解及其近似解析表达式,通过与四阶Runge-Kutta数值积分结果比较,验证了方法的有效性,计算结果对于转子系统运动的定量控制有重要理论指导意义。     

Impulsive control and synchronization of a new unified hyperchaotic system with varying control gains and impulsive intervals

In this paper, we investigate the impulsive control and synchronization of a new unified hyperchaotic system. This new system unifies both the hyperchaotic Lorenz system and the hyperchaotic Chen system. Some conditions are given to guarantee the global asymptotic stability of the controlled and synchronized system. The control gains and impulsive intervals are both variable. Moreover, we estimate the upper bound of impulsive interval for stable control and synchronization. Simulations are included to show the effectiveness of the theoretical results.     

Periodic solution of a second order, autonomous, nonlinear system

There exist a sufficient condition for the existence of at least one periodic solution for a type of second order autonomous ordinary differential equations. The correctness of the condition has been pointed out by Schauder's fixed point theorem. In order to indicate the validity of the assumptions made, two illustrative examples, showing its application in the nonlinear vibration and relaxation oscillation are presented.     

Discrete-in-time feedback control of near-grazing dynamics in the two-degree-of-freedom vibro-impact system with a clearance

Nonlinear Dynamics - Grazing bifurcation might bring dramatic and abrupt qualitative changes of system responses in non-smooth systems. The control problem of near-grazing dynamics in a...     

Response of Duffing system with delayed feedback control under bounded noise excitation

This paper presents a procedure for predicting the response of Duffing system with time-delayed feedback control under bounded noise excitation by using stochastic averaging method. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged It? stochastic differential equations for the system are derived by using the stochastic averaging method. Finally, the response of the system is obtained by solving the Fokker?CPlank?CKolmogorov equation associated with the averaged It? equations. It is shown that the time delay in feedback control will deteriorate the control effectiveness and cause bifurcation of stochastic jump of Duffing system. The validity of the proposed method is confirmed by digital simulation.     

ROBUST CONTROL VIA STATE FEEDBACK FOR A CLASS OF UNCERTAIN BILINEAR SYSTEMS

ＲＯＢＵＳＴＣＯＮＴＲＯＬＶＩＡＳＴＡＴＥＦＥＥＤＢＡＣＫＦＯＲＡＣＬＡＳＳＯＦＵＮＣＥＲＴＡＩＮＢＩＬＩＮＥＡＲＳＹＳＴＥＭＳ￥ＣｈｅｎＳｏｎｇｌｉｎ（陈松林）；ＺｈｕＺｕｃｈｉ（朱祖慈）（ＥａｓｔＣｈｉｎａＩｎｓｔｉｔｕｔｅｏｆＭｅｔａｌｌｕｒｇ...     

Hopf bifurcation control for stochastic dynamical system with nonlinear random feedback method

Hopf bifurcation control in nonlinear stochastic dynamical system with nonlinear random feedback method is studied in this paper. Firstly, orthogonal polynomial approximation is applied to reduce the controlled stochastic nonlinear dynamical system with nonlinear random controller to the deterministic equivalent system, solvable by suitable numerical methods. Then, Hopf bifurcation control with nonlinear random feedback controller is discussed in detail. Numerical simulations show that the method provided in this paper is not only available to control the stochastic Hopf bifurcation in nonlinear stochastic dynamical system, but is also superior to the deterministic nonlinear feedback controller.