Hopf bifurcation control for delayed complex networks

In this paper, we consider the problem of Hopf bifurcation control for a complex network model with time delays. We know that for the system without control, as the positive gain parameter of the system passes a critical point, Hopf bifurcation occurs. To control the Hopf bifurcation, a time-delayed feedback controller is proposed to delay the onset of an inherent bifurcation when such bifurcation is undesired. Furthermore, we can also change the stability and direction of bifurcating periodic solutions by choosing appropriate control parameters. Numerical simulation results confirm that the new feedback controller using time delay is efficient in controlling Hopf bifurcation.     

Bifurcation and control of a planktonic ecological system with double delays by delayed feedback control

In this paper, a delayed feedback controller with the delay-dependent coefficient is introduced into a multiple delay phytoplankton-zooplankton system. For uncontrolled system, choosing delays as the bifurcation parameters, we prove that Hopf bifurcation can occur when the delays change and cross some values. Then, the delays are still chosen as the bifurcation parameters to research the dynamic behaviors of the controlled system. Under this control mechanism, the onset of Hopf bifurcation can be delayed by selecting the appropriate control parameters and the stability domain can be extended as feedback gain (the decay rate) decreases (increases), and the influence of the decay rate cannot be ignored. Furthermore, using the crossing curve methods, the stable changes of equilibrium in two delay plane can be obtained. Some numerical simulations are given to verify the correctness and validity of the delayed feedback controller in the bifurcation control.     

Dynamical behavior in a harvested differential-algebraic prey-predator model with discrete time delay and stage structure

A differential-algebraic model system which considers a prey-predator system with stage structure for prey and harvest effort on predator is proposed. By using the differential-algebraic system theory and bifurcation theory, the dynamic behaviors of the proposed model system with and without discrete time delay are investigated. Local stability analysis of the model system without discrete time delay reveals that there is a phenomenon of singularity induced bifurcation due to variation of the economic interest of harvesting, and a state feedback controller is designed to stabilize the proposed model system at the interior equilibrium; on the other hand, the local stability of the model system with discrete time delay is also studied. The theoretical analysis shows that the discrete time delay has a destabilizing effect in the model of population dynamics, and a phenomenon of Hopf bifurcation occurs as the discrete time delay increases through a certain threshold. Numerical simulations are carried out to show the consistency with theoretical analysis.     

Stability and Hopf bifurcation analysis of an eco-epidemiological model with delay

In this paper, an eco-epidemiological model with time delay is considered. The asymptotical stability of the three equilibria, the existence of stability switches about both the disease-free planar equilibrium and the positive equilibrium are investigated. It is found that Hopf bifurcation occurs when the delay τ passes through a critical value. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations at the positive equilibrium are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.     

Stability and bifurcation of delay-coupled genetic regulatory networks with hub structure

In this paper, we study the local stability and bifurcation of a delay-coupled genetic regulatory networks consisting of two modes with the hub structure. By analyzing the equilibrium equation, the number of the positive equilibria is discussed in both the cases that there are inhibition coupling and activation coupling in the networks. It is revealed that multiple equilibria could exist in the developed genetic networks and the number of the equilibria could be distinct under the two cases of delayed-coupling. For the equilibrium, the conditions of the coupling-delay-independent stability and the saddle-node bifurcation are derived with respect to the biochemical parameters. The coupling-delay-dependent stability and the Hopf bifurcation criteria on the biological parameters and the coupling delay are also given. Moreover, the complexity of the algorithm used in this paper is analyzed. The numerical simulations are made to certify the obtained results. The multistability of the developed genetic regulatory networks is displayed. The different effects of the coupling delay on the stability of the genetic networks under different biochemical parameters are shown.     

Output stabilization of a class of second-order linear time-delay systems: An eigenvalue approach

This paper studies linear time-invariant systems with an input delay and two repeated or distinct real poles. The closed-loop system eigenvalue-loci with respect to the output feedback controller gain are investigated by using the Lambert function and root-locus construction techniques. Output feedback stabilization conditions and stability robustness with respect to the delay time uncertainty are established. Also, the response performance is discussed. Three examples and related simulations are presented to illustrate the analysis results.     

Dynamical behavior in a hybrid stochastic triple delayed prey predator bioeconomic system with Lévy jumps

In this paper, a hybrid triple delayed prey predator bioeconomic system with prey refuge and Lévy jumps is established, where both maturation delay for prey and predator population and gestation delay for predator population are considered. For deterministic system, positivity and uniform permanence of solution are discussed. Local stability of deterministic system around interior equilibrium is investigated due to variations of triple time delays. For stochastic system without time delay, sufficient conditions for stochastically ultimate boundedness and stochastic permanence are discussed. Existence of stochastic Hopf bifurcation and stochastic stability are investigated. For stochastic system with triple time delays, existence and uniqueness of global positive solution are studied. Finally, combined dynamic effects of triple time delays and Lévy jumps on the hybrid stochastic system are discussed by constructing appropriate Lyapunov functions. Numerical simulations are supported to illustrate theoretical analysis.     

Hopf bifurcation and synchronization of a five-dimensional self-exciting homopolar disc dynamo using a new fuzzy disturbance-observer-based terminal sliding mode control

Having found hidden hyperchaos in a 5D self-exciting homopolar disc dynamo, we study the existence of a Hopf bifurcation, which leads to unstable limit cycles bifurcating from a stable equilibrium. Hidden chaos with only stable equilibria can be observed from the Hopf bifurcation: a typical way to enable hidden attractors to be located. We then provide a new fuzzy controller, and a fast fuzzy disturbance observer, based on terminal sliding mode control for synchronization of the hyperchaotic system. Fuzzy inference is considered to weaken the chattering phenomena. Using Lyapunov stability theory, the stability of the closed-loop system is proved. Finally, simulations of synchronization are illustrated to show the efficient performance of the designed control method via external disturbances and dynamic uncertainties.     

Bifurcation with regard to combined interaction parameter in a life energy system dynamic model of two components with multiple delays

Bifurcation theory is commonly used to study the dynamical behavior of ecosystems. It involves the analysis of points in the parameter space where the stability of the system changes qualitatively. The type of bifurcation that associates equilibria with periodic solution is called Hopf bifurcation. In this paper, a life energy system dynamic model of two components with multiple delays is presented. It is shown that the interaction parameters of the delayed ecosystem play a fundamental role in classifying the rich dynamics and bifurcation phenomena. Regarding the combined interaction parameter as a bifurcation parameter, the bifurcation values in the parameter plane are displayed. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by applying the normal form theory and the center manifold theorem. Moreover, the amplitudes of oscillations always increase as the interaction parameters increase, while the robustness of periods occurs as the interaction parameters vary. From an ecological point of view, the existence of Hopf bifurcation expresses periodic oscillatory behavior of the life energy system.     

Bifurcation analysis of a competitive system with general toxic production and delayed toxic effects

Population interaction may release poisonous chemicals to inhibit other species’ growth in the ecosystem, especially for the competitive populations. The negative effect of toxic chemical substances may not display immediately and appear with time lag during the species’ growth. In this work, we investigate a competitive system with the delayed toxic effects of the chemicals from species interaction. Theoretical results obtained in this work help us reveal the delayed toxic factors on species’ growth. We first consider the existence and the stability of the equilibria. The influence of delay terms on the positive steady state is validated. The delayed toxic effects here will contribute to the oscillation for the concentration of species when the value of time delay passes through a critical point. Besides, the stability of periodic solutions from the Hopf bifurcation and the direction of the Hopf bifurcation are also determined. Finally, several numerical examples are provided to validate the theoretical conclusions.     

Spatiotemporal dynamics of a predator-prey system with fear effect

Recent field experiments on vertebrates show that though mere presence of a predator causes a dramatic change in prey demography, the fear of predators increases the survival probability of prey leading to a cost of prey production. Based on the experimental findings, we proposed and analyzed a mathematical model that incorporates the fear-induced birth reduction in the prey population due to presence of predator. A modified and more realistic fear function is proposed in this study. Qualitative behavior of the model is performed including positivity and boundedness of solutions, existence of critical points and their local stability analysis, existence of transcritical and Hopf bifurcation. We analyzed Hopf bifurcation with respect to the prey growth rate and the level of fear. Transcritical bifurcation is analyzed by varying the prey growth rate. Distribution of the population of interacting species in a large scale natural system is heterogeneous and subject to alter for different reasons. Thus, we investigate how behavioral modification in prey population due to fear for predators and mutual interference among predator species can create various spatiotemporal pattern formation in population distribution. In the spatially extended system, we provide a detailed stability analysis and obtain the conditions for Turing instability. Numerical simulations are performed to validate analytical results for both non-spatial and spatial models. Warm spot patterns are obtained by considering three different types of initial data and discussed the biological significance of these patterns for the two-dimensional spatial model. Our numerical simulation demonstrates that the fear effect in a diffusive predator-prey system with mutual interference between predators may exhibit more complicated dynamics.     

H∞ tracking control for a class of asynchronous switched nonlinear systems with uncertain input delay

This paper studies the problems of stability and H∞ model reference tracking performance for a class of asynchronous switched nonlinear systems with uncertain input delay. First, it is assumed switched controller and corresponding piecewise Lyapunov function are unknown but the derivative of piecewise Lyapunov function has a condition; this condition implies that the nominal system (system without input delay and disturbance) is exponentially stable by any switched controller which satisfies this condition. With this assumption, a proper Lyapunov–Krasovskii functional is constructed. By employing this new functional and average dwell time technique, the delay-dependent input-to-state stability criteria are derived under a certain delay bound; in addition, a mechanism which finds the upper bound of input delay is proposed. Finally, a kind of state feedback control law which fulfils condition of aforesaid piecewise Lyapunov function is introduced to guarantee the input-to-state stability and H∞ model reference tracking performance. Simulation examples are presented to demonstrate the efficacy of results.     

利用脉冲非线性状态反馈控制混沌

研究了脉冲非线性反馈控制方法控制Lorenz系统的混沌问题，首先从理论上论证了控制方法的正确性．然后设计出了三种控制器并给出了控制器应满足的条件，理论分析和数值模拟结果表明混沌Lorenz系统中的不稳定不动点能被稳定控制，而且Hopf分岔也能产生，给出了相应的数值模拟结果，例如不动点、极限环(IP)轨道，采用脉冲控制方法的优点是控制代价小，工程上易于实现，最后指出，对于其他具有平衡点对称的混沌系统如蔡氏电路系统的混沌控制，此控制策略同样有效。     

经济系统中几个微分方程模型

建立了市场经济中供求关系的两类数学模型。根据商品数量的不同,对供给函数和需求函数的假设不同,建立了几个微分方程模型。研究了其中一个模型的Hopf分支问题,给出了均衡价格的局部稳定性条件和出现Hopf分支的条件     

The effect of delay interval on the feedback control for a turbidostat model

In this paper, we propose a turbidostat model with delay interval on its output using a feedback control law, aiming to investigate how the delay interval affects the feedback control of the model. The delay interval is represented by two parameters, which describe the time delay distributed in a past sub-interval. We first prove the positivity and boundedness of solutions and the permanence of the model. Then, using the input flow rate as a feedback control variable, we discuss the asymptotical stabilization of a given state (i.e., the positive equilibrium) employing the method of Lyapunov functionals. Moreover, we further study the Hopf bifurcations induced by the two delay parameters. Our theoretical and numerical results all show that the delay interval can have a significantly different effect on the dynamics of a turbidostat model from other delay types.     

Hopf bifurcation for a predator-prey biological economic system with Holling type II functional response

In this paper, a biological economic system which considers a prey-predator system with Holling type II functional response and harvest effort on prey is proposed. By using the differential-algebraic system theory and Hopf bifurcation theory, Hopf bifurcation of the proposed system is investigated. Different from previous researches on the dynamic behaviors of predator-prey systems, our model is described by differential-algebraic equations due to the economic factor. The economic profit is chosen as a positive bifurcation parameter here. It is found that a phenomenon of Hopf bifurcation occurs as the economic profit increases beyond a certain threshold. Lastly, with the help of Matlab software, numerical simulations are carried out to demonstrate the effectiveness of our results.     

Modeling and stabilization of a wireless network control system with packet loss and time delay

The problem of modeling and stabilization of a wireless network control system (NCS) is considered in this paper, where packet loss and time delay exist simultaneously in the wireless network. A discrete-time switched system with time-varying delay model is first proposed to describe the system closed by a static state feedback controller. A sufficient criteria for the discrete-time switched system with time-varying delay to be stable is proposed, based on which, the corresponding state feedback controller is obtained by solving a set of linear matrix inequalities (LMIs). Numerical examples show the effectiveness of the proposed method.     

Complex dynamics and stability analysis in a discrete hybrid bioeconomic system with double time delays

In this paper, a discrete hybrid three-species food chain system is proposed, where commercial harvesting on top predator is considered. Two time delays are introduced to represent gestation delay for prey and predator population, respectively. In absence of time delay, sufficient conditions associated with economic interest and step size are derived to show system undergoes flip bifurcation. In presence of double time delays, existence of Neimark–Sacker bifurcation and local stability switch are discussed due to variations of time delays. Furthermore, by utilizing new normal form of delayed discrete hybrid system and center manifold theorem, direction and stability of Neimark–Sacker bifurcation are studied. Numerical simulations are performed not only to validate theoretical analysis, but also exhibit cascades of period-doubling bifurcation, chaotic behavior and stable closed invariant curve.     

Bifurcation analysis and control of a discrete harvested prey-predator system with Beddington-DeAngelis functional response

In this paper, we study a discrete prey-predator system with harvesting of both species and Beddington-DeAngelis functional response. By using the center manifold theorem and bifurcation theory, we establish that the system undergoes flip bifurcation and Hopf bifurcation when the harvesting effort of prey population passes some critical values. Numerical simulations exhibit period-6, 10, 12, 14, 20 orbits, cascade of period-doubling bifurcation in period-2, 4, 8, 16 orbits and chaotic sets. At the same time, the numerically computed Lyapunov exponents confirm the complex dynamical behaviors. Moreover, a state delayed feedback control method, which can be implemented only by adjusting the harvesting effort for the prey population, is proposed to drive the discrete prey-predator system to a steady state.     

Bifurcation analysis in a predator-prey system with stage-structure and harvesting

In this paper, we consider a predator-prey model with stage-structure and harvesting. This model is the same as the one developed by Kar and Pahari (2007) [9], but we make bifurcation analysis more general than their work. In particular, using the approach of Beretta and Kuang (2002) [4], we show that the positive steady state can be destabilized through a Hopf bifurcation. We also investigate the stability and direction of periodic solutions bifurcating from Hopf bifurcation by using the normal form theory and the center manifold theorem presented in Hassard et al. (1981) [8]. Numerical simulations are then carried out as supporting evidences of our analytical results.