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.     

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.     

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.     

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.     

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 control of a delayed fractional-order eco-epidemiological model with incommensurate orders

In this paper, a delayed fractional eco-epidemiological model with incommensurate orders is proposed, and a control strategy of this model is discussed. Firstly, for the system with no controller, the stability and Hopf bifurcation with respect to time delay are investigated. Secondly, under the influence of the controller, the stability and Hopf bifurcation of the system is discussed, and it is indicated that the stability of the system can be changed by increasing the feedback control delay. In particular, a separate study is carried out on the bifurcation with respect to the extended feedback delay, and the bifurcation point is calculated. At last, to support the theoretical results, some numerical simulations are depicted.     

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.     

The impact of hospital resources and environmental perturbations to the dynamics of SIRS model

Incorporating the environmental perturbations and available resources of the public health system, we construct both deterministic and stochastic models of SIRS type. The deterministic model exhibits very rich dynamics, such as Hopf bifurcation and backward bifurcation which leads to the co-existence of the stable disease-free state and a stable endemic equilibrium. For the stochastic model, we show that under mild extra conditions, if the basic reproduction number is less than one, then the disease will be eradicated almost surely, and if the basic reproduction number is greater than one, the stochastic model will admit a unique ergodic stationary distribution, which implies that the disease persists almost surely. Part of our numerical simulations indicate that: (i) The introduction of environmental perturbations may drift the endemic equilibrium to the disease-free equilibrium, or vice versa; (ii) Increasing available resources is necessary in order to mitigate the infections.     

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.     

Interaction between prey and mutually interfering predator in prey reserve habitat: Pattern formation and the Turing–Hopf bifurcation

In this paper, we propose a diffusive prey-predator system with mutually interfering predator (Crowley–Martin functional response) and prey reserve. In particular, we develop and analyze both spatially homogeneous model based on ordinary differential equations and reaction-diffusion model. We mainly investigate the global existence and boundedness of positive solution, stability properties of homogeneous steady state, non-existence of non-constant positive steady state, conditions for Turing instability and Hopf bifurcation of the diffusive system analytically. Conventional stability properties of the non-spatial counterpart of the system are also studied. The analysis ensures that the prey reserve leaves stabilizing effect on the stability of temporal system. The prey reserve and diffusive parameters leave parallel impact on the stability of the spatio-temporal system. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the model dynamics exhibits diffusion controlled pattern formation by different interesting patterns.     

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.     

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     

Nonlinear dynamic analysis for a Francis hydro-turbine governing system and its control

In this paper, we introduce a novel model of a hydro-turbine system with the effect of surge tank based on state-space equations to study the nonlinear dynamical behaviors of the hydro-turbine system. The critical points of Hopf bifurcation and the relationship of the stability satisfying with the adjustment coefficients are obtained from direct algebraic criterion. Furthermore, the bifurcation diagrams and Lyapunov exponents are presented and analyzed. The dynamical behaviors of the points with representative characteristics are identified and studied in detail. Both theoretical analysis and numerical simulations show that chaotic oscillations, which cannot stabilize the system, may occur with the changes of adjustment coefficients. To control the undesirable chaotic behaviors in this system, fuzzy sliding mode governor based on the sliding mode control (SMC) and the fuzzy logic are designed, and considering the bounded disturbance. Finally, series of numerical simulations are presented to verify the effectiveness of the proposed governor, which prove that the hydro-turbine governing system can maintain a better operation station under the designed governor.     

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.     

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.     

Simple and double Hopf bifurcations in aeroelastic oscillators with tuned mass dampers

The effects of an added mass on the oscillations of a SDOF bluff body, elastically supported, exposed to a steady flow and undergoing galloping oscillations, are investigated. The stability boundaries of the trivial equilibrium position of the 2DOF system are determined in a four parameters space. The occurrence of different types of bifurcation on these boundaries is highlighted, namely, simple Hopf, non-resonant double Hopf and 1 : 1 resonant double Hopf. The perturbation multiple scale method is employed to analyze the system postcritical behavior around the codimension-1 and codimension-2 critical manifolds. The analytical results are compared with numerical solutions obtained through direct integration of the equations of motion. Finally, the effects of the closeness of the critical frequencies on the non-resonant double Hopf manifold, are discussed by using a quasi-resonant asymptotic solutions.     

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.     

Detecting collusive spammers with heterogeneous graph attention network

Detecting collusive spammers who collaboratively post fake reviews is extremely important to guarantee the reliability of review information on e-commerce platforms. In this research, we formulate the collusive spammer detection as an anomaly detection problem and propose a novel detection approach based on heterogeneous graph attention network. First, we analyze the review dataset from different perspectives and use the statistical distribution to model each user's review behavior. By introducing the Bhattacharyya distance, we calculate the user-user and product-product correlation degrees to construct a multi-relation heterogeneous graph. Second, we combine the biased random walk strategy and multi-head self-attention mechanism to propose a model of heterogeneous graph attention network to learn the node embeddings from the multi-relation heterogeneous graph. Finally, we propose an improved community detection algorithm to acquire candidate spamming groups and employ an anomaly detection model based on the autoencoder to identify collusive spammers. Experiments show that the average improvements of precision@k and recall@k of the proposed approach over the best baseline method on the Amazon, Yelp_Miami, Yelp_New York, Yelp_San Francisco, and YelpChi datasets are [13%, 3%], [32%, 12%], [37%, 7%], [42%, 10%], and [18%, 1%], respectively.     

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.     

Characterization of two-dimensional cellular automata over ternary fields

The set of papers [3], [4], [6] and [7] (Chattopadhyay et al., 1999; Dihidar and Choudhury, 2004; Khan et al., 1997, 1999) deals with the behavior of the uniform two-dimensional cellular automata over binary fields (Z₂). Some structural properties and precise mathematical models using matrix algebra over the field Z₂ are reported for characterizing the behavior of two-dimensional nearest neighborhood linear cellular automata with null and periodic boundary conditions [3], [4], [6] and [7] (Chattopadhyay et al., 1999; Dihidar and Choudhury, 2004; Khan et al., 1997, 1999). In this paper, we characterize two-dimensional linear cellular automata transformations by using matrix algebra built on Z₃. We analyze some results for two-dimensional CA with rule numbers 2460N and 2460P. Finally, we investigate the dimension of the kernel of two-dimensional cellular automata defined by the rule number 2460N.