Bifurcation Control Research Articles

Hopf bifurcation control of a fractional-order delayed turbidostat model via a novel extended hybrid controller

Hopf bifurcation control of a fractional-order delayed turbidostat model via a novel extended hybrid controller

Bifurcation Phenomenon and Control Technique in Fractional BAM Neural Network Models Concerning Delays

In this current study, we formulate a kind of new fractional BAM neural network model concerning five neurons and time delays. First, we explore the existence and uniqueness of the solution of the formulated fractional delay BAM neural network models via the Lipschitz condition. Second, we study the boundedness of the solution to the formulated fractional delayed BAM neural network models using a proper function. Third, we set up a novel sufficient criterion on the onset of the Hopf bifurcation stability of the formulated fractional BAM neural network models by virtue of the stability criterion and bifurcation principle of fractional delayed dynamical systems. Fourth, a delayed feedback controller is applied to command the time of occurrence of the bifurcation and stability domain of the formulated fractional delayed BAM neural network models. Lastly, software simulation figures are provided to verify the key outcomes. The theoretical outcomes obtained through this exploration can play a vital role in controlling and devising networks.

Stability, Hopf Bifurcation and Optimal Control of Multilingual Rumor-Spreading Model with Isolation Mechanism

The propagation of rumors on online social networks (OSNs) brings an awful lot of trouble to people’s life and society. Aiming at combating rumors spreading on OSNs, two novel rumor-propagation models without and with time delays are proposed, which combine with the influence of the immune mechanism, isolation mechanism and network structure. Firstly, we analyze the existence of rumor equilibria and obtain some existence conditions of backward bifurcation. Secondly, the local stabilities of rumor-free and rumor equilibria are proved by using the Jacobian matrix method, and some critical conditions for the existence of Hopf bifurcation are acquired by selecting critical parameters and delays as bifurcation parameters. Furthermore, an optimal control method is proposed, which can prevent the spread of rumors within an expected time period and minimize the cost of control. Finally, some numerical simulations are provided to verify the effectiveness of the proposed theoretical results.

Bifurcation Control on the Un-Linearizable Dynamic System via Washout Filters

Information fusion integrates aspects of data and knowledge mostly on the basis that system information is accumulative/distributive, but a subtle case emerges for a system with bifurcations, which is always un-linearizable and exacerbates information acquisition and presents a control problem. In this paper, the problem of an un-linearizable system related to system observation and control is addressed, and Andronov-Hopf bifurcation is taken as a typical example of an un-linearizable system and detailed. Firstly, the properties of a linear/linearized system is upon commented. Then, nonlinear degeneracy for the normal form of Andronov-Hopf bifurcation is analyzed, and it is deduced that the cubic terms are an integral part of the system. Afterwards, the theoretical study on feedback stabilization is conducted between the normal-form Andronov-Hopf bifurcation and its linearized counterpart, where stabilization using washout-filter-aided feedback is compared, and it is found that by synergistic controller design, the dual-conjugate-unstable eigenvalues can be stabilized by single stable washout filter. Finally, the high-dimensional ethanol fermentation model is taken as a case study to verify the proposed bifurcation control method.

Dynamic analysis and bifurcation control of a fractional-order cassava mosaic disease model

Dynamic analysis and bifurcation control of a fractional-order cassava mosaic disease model

Long-Term Bifurcation and Stochastic Optimal Control of a Triple-Delayed Ebola Virus Model with Vaccination and Quarantine Strategies

Despite its high mortality rate of approximately 90%, the Ebola virus disease (EVD) has not received enough attention in terms of in-depth research. This illness has been responsible for over 40 years of epidemics throughout Central Africa. However, during 2014–2015, the Ebola-driven epidemic in West Africa became, and remains, the deadliest to date. Thus, Ebola has been declared one of the major public health issues. This paper aims at exploring the effects of external fluctuations on the prevalence of the Ebola virus. We begin by proposing a sophisticated biological system that takes into account vaccination and quarantine strategies as well as the effect of time lags. Due to some external perturbations, we extend our model to the probabilistic formulation with white noises. The perturbed model takes the form of a system of stochastic differential equations. Based on some non-standard analytical techniques, we demonstrate two main approach properties: intensity and elimination of Ebola virus. To better understand the impact of applied strategies, we deal with the stochastic control optimization approach by using some advanced theories. All of this theoretical arsenal has been numerically confirmed by employing some real statistical data of Ebola virus. Finally, we mention that this work could be a rich basis for further investigations aimed at understanding the complexity of Ebola virus propagation at pathophysiological and mathematics levels.

Assessing the Impact of Relapse, Reinfection and Recrudescence on Malaria Eradication Policy: A Bifurcation and Optimal Control Analysis.

In the present study, we propose and analyze an epidemic mathematical model for malaria dynamics, considering multiple recurrent phenomena: relapse, reinfection, and recrudescence. A limitation in hospital bed capacity, which can affect the treatment rate, is modeled using a saturated treatment function. The qualitative behavior of the model, covering the existence and stability criteria of the endemic equilibrium, is investigated rigorously. The concept of the basic reproduction number of the proposed model is obtained using the concept of the next-generation matrix. We find that the malaria-free equilibrium point is locally asymptotically stable if the basic reproduction number is less than one and unstable if it is larger than one. Our observation on the malaria-endemic equilibrium of the proposed model shows possible multiple endemic equilibria when the basic reproduction number is larger or smaller than one. Hence, we conclude that a condition of a basic reproduction number less than one is not sufficient to guarantee the extinction of malaria from the population. To test our model in a real-life situation, we fit our model parameters using the monthly incidence data from districts in Central Sumba, Indonesia called Wee Luri, which were collected from the Wee Luri Health Center. Using the first twenty months’ data from Wee Luri district, we show that our model can fit the data with a confidence interval of 95%. Both analytical and numerical experiments show that a limitation in hospital bed capacity and reinfection can trigger a more substantial possibility of the appearance of backward bifurcation. On the other hand, we find that an increase in relapse can reduce the chance of the appearance of backward bifurcation. A non-trivial result appears in that a higher probability of recrudescence (treatment failure) does not always result in the appearance of backward bifurcation. From the global sensitivity analysis using a combination of Latin hypercube sampling and partial rank correlation coefficient, we found that the initial infection rate in humans and the mosquito infection rate are the most influential parameters in determining the increase in total new human infections. We expand our model as an optimal control problem by including three types of malaria interventions, namely the use of bed net, hospitalization, and fumigation as a time-dependent variable. Using the Pontryagin maximum principle, we characterize our optimal control problem. Results from our cost-effectiveness analysis suggest that hospitalization only is the most cost-effective strategy required to control malaria disease.

Global Bifurcation Behaviors and Control in a Class of Bilateral MEMS Resonators

The investigation of global bifurcation behaviors the vibrating structures of micro-electromechanical systems (MEMS) has received substantial attention. This paper considers the vibrating system of a typical bilateral MEMS resonator containing fractional functions and multiple potential wells. By introducing new variations, the Melnikov method is applied to derive the critical conditions for global bifurcations. By engaging in the fractal erosion of safe basin to depict the phenomenon pull-in instability intuitively, the point-mapping approach is used to present numerical simulations which are in close agreement with the analytical prediction, showing the validity of the analysis. It is found that chaos and pull-in instability, two initial-sensitive phenomena of MEMS resonators, can be due to homoclinic bifurcation and heteroclinic bifurcation, respectively. On this basis, two types of delayed feedback are proposed to control the complex dynamics successively. Their control mechanisms and effect are then studied. It follows that under a positive gain coefficient, delayed position feedback and delayed velocity feedback can both reduce pull-in instability; nevertheless, to suppress chaos, only the former can be effective. The results may have some potential value in broadening the application fields of global bifurcation theory and improving the performance reliability of capacitive MEMS devices.

Hopf bifurcation and optimal control of a delayed malware propagation model on mobile wireless sensor networks

In present paper, a delayed SEIR model of malware propagation on mobile wireless sensor networks is formulated. Basic reproduction number of the model is calculated by employing the next generation matrix method and impact of the time delay due to incubation period of malware is discussed with the help of eigenvalue method. What is more, properties of the Hopf bifurcation are derived by using center manifold method. To minimize number of the infectious nodes and the cost related with detoxication, an optimal control strategy is presented by constructing a suitable Hamiltonian function. Numerical calculations are performed to justify the obtained theoretical findings.

Study of Torsional Vibration Bifurcation Characteristics of Direct-Drive Wind Turbine Shaft System

This paper set out to establish the dynamics model of shaft torsional vibration for direct-drive wind turbine with the phenomenon of unstable shaft system torsional vibration. The stability of the equilibrium point of the dynamical model is investigated, and the Routh–Hurwitz stability criterion is used to obtain a range of values for the bifurcation control parameters. For the stable equilibrium point, the stability domain of the system is calculated by constructing the Lyapunov function. The sensitivity analysis of system parameters is carried out to obtain the law of the effect of system parameters on system stability of the torsional vibration system. The results are substituted for example calculations, and the results verify the correctness of the theoretical analysis conclusions. It is proved that it is feasible to analyze the torsional vibration characteristics of the direct-drive wind turbine shaft system by using the principle of Routh–Hurwitz stability, etc., which provides a reference for the structural design of direct-drive wind turbine.

Stability, Bifurcation, and Chaos Control of Two-Sided Market Competition

Benefitting from the popular uses of internet technologies, two-sided market has been playing an increasing prominent role in modern times. Users and developers can interact with each other through two-sided platforms. The two-sided market structure has been investigated profoundly. Through building a dynamics two-sided market model with bounded rational, stability conditions of the two-sided market competition system are presented. With the help of bifurcation diagram, Lyapunov exponent, and strange attractor, the stability of the two-sided market competition model is simulated. At last, we use the time-delayed feedback control (TDFC) method to control the chaos. Our main results are as follows: (1) when the adjustment speed of two-sided increases, the system becomes bifurcation, and chaos state happens finally. When the system is stable, the consumer fee is positive while developer fee is negative. (2) When the user externality increases, the stable area of the system increases, and the difference in user externality leads the whole system more stable. When the system is stable, the developer fee decreases. (3) The stable area becomes larger when developer externality increases; when the system is stable, the user fee becomes lower and developer fee becomes higher when developer externality increases. (4) The TDFC method is presented for controlling the chaos; we find that the system becomes more stable under the TDFC method.

Bifurcation Dynamics and Control Mechanism of a Fractional-Order Delayed Brusselator Chemical Reaction Model

Building differential dynamical systems to describe the changing relationship among chemical components is a vital aspect in chemistry. In this present manuscript, we put forward a new fractional-order delayed Brusselator chemical reaction model. By virtue of contraction mapping principle, we investigate the existence and uniqueness of the solution of fractional-order delayed Brusselator chemical reaction model. With the aid of mathematical analysis technique, we consider the non-negativeness of the solution of the fractional-order delayed Brusselator chemical reaction model. Making use of the theory of fractional-order dynamical system, we explore the stability and Hopf bifurcation issue of the fractional-order delayed Brusselator chemical reaction model. By designing a reasonable controller, we have availably controlled the time of emergence of Hopf bifurcation of the fractional-order delayed Brusselator chemical reaction model. A sufficient criterion guaranteeing the stability and the onset of Hopf bifurcation of the fractional-order controlled delayed Brusselator chemical reaction model is set up. Computer simulations are implemented to validate the theoretical findings. The derived fruits of this manuscript have great theoretical significance in controlling the concentrations of chemical substances.

Stability Analysis, Hopf Bifurcation and Drug Therapy Control of an HIV Viral Infection Model with Logistic Growth Rate and Cell-to-Cell and Cell-Free Transmissions

It is well known that dynamical systems are very useful tools to study viral diseases such as HIV, HBV, HCV, Ebola and Influenza. This paper focuses on a mathematical model of the cell-to-cell and the cell-free spread of HIV with both linear and nonlinear functional responses and logistic target cell growth. The reproduction number of each mode of transmission has been calculated and their sum has been considered as the basic reproduction number. Based on the values of the reproduction number, the local and global stabilities of the rest points are investigated. Choosing a suitable bifurcation parameter, some conditions for the occurrence of Hopf bifurcation are also obtained. Moreover, numerical simulations are presented to support the analytical results. Finally, to study the effect of the drug on the disease process, some control conditions are determined. Since two modes of transmission and both linear and nonlinear functional responses have been included in this manuscript, our obtained results are a generalization of those in the literature. Moreover, the results are obtained with weaker assumptions in comparison with the previous ones.

Bifurcation Anti-Control Tactics of a Fractional-Order Stable Chemical Reaction System

Establishing suitable differential dynamical models to describe the real natural phenomenon in chemistry and physics has become a very hot topic in nowadays society. In this present research, we deal with a fractional-order chemical reaction system. Taking advantage of the fixed point theorem, we prove the existence and uniqueness of the fractional-order chemical reaction system. Using the inequality skill, we prove the non-negativeness of the fractional-order chemical reaction system. By applying a suitable function, we prove the uniform boundedness of the solution to the fractional-order chemical reaction system. With the aid of a hybrid controller including state feedback and parameter perturbation, we discuss the Hopf bifurcation anti-control issue of the fractional-order stable chemical reaction system. A novel delay-independent condition ensuring the stability and the onset of Hopf bifurcation of the involved fractional-order stable chemical reaction system is set up. The study manifests that the delay in the hybrid controller plays a vital role in stabilizing the system and controlling the occurrence of Hopf bifurcation of the fractional-order stable chemical reaction system. In order to validate the derived key conclusions, MATLAB simulations are executed and bifurcation plots are given. The obtained results of this article have momentous theoretical guiding value in controlling the chemical compositions. The exploration idea can also be utilized to investigate the bifurcation control and bifurcation anti-control problems in lots of other fractional-order differential systems in numerous disciplines.

Hopf bifurcation and optimal control of a delayed SLBPS virus-patch model

This paper formulates a Susceptible–Latent–Breaking out–Patched–Susceptible (SLBPS) virus propagation model with delay of temporary immunity. The influence of time delay on stability of the model is examined by analyzing the distribution of eigenvalues of the characteristic equation. Furthermore, we also demonstrate the direction and stability of the Hopf bifurcation. Lastly, optimal control strategies in terms of the rate at which susceptible, latent, break out nodes acquiring patches to improve performance of the model are presented. For the sake of the correctness of the theoretical analysis, some numerical simulations are also presented.

Analysis of Digital Operation of Rural Sewage Treatment System in Big Data Environment by Discrete Dynamic Modeling Technology

Sewage treatment is a process with high energy consumption and pollution. In the current situation of energy shortage and fierce competition in the paper industry market, how to reduce the energy cost is related to the survival and development of every sewage treatment plant. With the rise of big data environment in the internet, discrete dynamic modeling technology has developed very rapidly. Discrete dynamic modeling technology is a dynamic model system that can connect the information of different things, so as to form a dynamic model system that can control all devices or data. The object attributes processed by the model system also determine the nature of the dynamic model system. As we all know, the big data environment is a large database with complex data types, so the dynamic model system in the big data environment is nonlinear. This paper mainly studies the historical development trend of big data environment and discrete dynamic modeling technology, as well as the stability method of five level H-bridge when discrete dynamic modeling technology controls rural sewage treatment system. Finally, the control of chaotic bifurcation in discrete dynamic model system under digital background is studied. It includes the rectifier modeling of the current data required by large devices in the sewage treatment system and the result analysis of the generated values.

In the presence of fear and refuge: Permanence, bifurcation and chaos control of a discrete-time ecological system

In this paper, we aim to study the effects of fear of predation in the presence of a competitor species and availability of refuge for the prey population. A discrete-time model of prey–predator–competitor type is formulated to study the dynamical relationship between the interacting species by deriving the conditions for local stability, permanence and existence of Neimark–Sacker bifurcation. A hybrid control technique based on state feedback strategy is implemented to control the chaos in the system. Finally, some numerical simulations are carried out to illustrate the theoretical discussions.

Sliding Mode Bifurcation Control Based on Acceleration Feedback Correction Adaptive Compensation for Maglev Train Suspension System With Time-Varying Disturbance

In order to ensure the suspension stability of maglev train, the active control problems of static suspension and dynamic suspension are studied. First, the mathematical model of dynamic suspension characteristics of medium- and low-speed maglev train is established. Second, the PID controller is studied. The results show that the controller is very sensitive to time-varying disturbances such as nonuniformity and load, which leads to the decrease in system robustness. In order to suppress the disturbance effectively, a suspension control method based on the sliding mode reaching law is proposed. Based on the bifurcation phenomenon, the bifurcation point is solved to determine the main control parameters. On the basis of this, an acceleration feedback correction and adaptive control module based on the radial basis function (RBF) network is constructed to effectively suppress the vibration of the electromagnet. The controller consists of a sliding mode bifurcation control law, an acceleration feedback correction module, and an adaptive compensation loop. The simulation and experimental results show that the proposed control algorithm can significantly reduce the variation range of suspension gap with a smoother control current in the presence of complex disturbances and significantly suppress the coupled vibration.

Bifurcation control of a delayed fractional-order prey-predator model with cannibalism and disease

In this paper, the bifurcation control for a fractional-order delayed prey-predator system with disease and cannibalism is investigated. Firstly, the existence, uniqueness and non-negativity of the solutions are studies, and the stability of equilibrium points are discussed. Next, taking time delay as the bifurcation parameter, the conditions of creation for Hopf bifurcation are confirmed. Then, two feedback controllers are introduced, which successfully control the Hopf bifurcation, and bifurcation control of the two controllers are simply compared theoretically. Finally, numerical simulations show that cannibalism has an significant influence in controlling the stability of the model, and when the parameters are appropriate, the disease can be removed. Meanwhile, the feedback controllers can control the bifurcation well. In general, the control effect of time-delay feedback controller should be better than that of traditional feedback controller. However, we found that the control effect of the traditional feedback controller is better than the time-delay feedback controller, the control effect completely depends on the selection of parameters.

Global dynamics, Neimark-Sacker bifurcation and hybrid control in a Leslie’s prey-predator model

In the present study, we explore the topological classifications at fixed points, global dynamics, Neimark-Sacker bifurcation and hybrid control in the two-dimensional discrete-time Leslie’s prey-predator model. It is proved that for all involved parameters a,b,c,d,h and α, discrete-time Leslie’s prey-predator model has boundary and interior fixed points: Ex0(ab,0),Exy+aαcd+bα,adcd+bα respectively. Then by linear stability theory, local dynamics with different topological classifications are investigated at fixed points: Ex0(ab,0),Exy+aαcd+bα,adcd+bα. Further for the discrete-time Leslie’s prey-predator model, existence of periodic points are also investigated. By bifurcation theory, it is also proved that if (a,b,c,d,h,α)∈NSBExy+aαcd+bα,adcd+bα then at interior fixed point: Exy+aαcd+bα,adcd+bα, discrete Leslie’s prey-predator model undergo Neimark-Sacker bifurcation and no other bifurcation occurs at it. Moreover, hybrid control strategy is applied to control Neimark-Sacker bifurcation. Boundedness and global dynamics of the discrete-time Leslie’s prey-predator model are also investigated. Finally, obtained results are numerically verified.