Hopf Bifurcation Control Research Articles

Hopf bifurcation exploration and control technique in a predator-prey system incorporating delay

<abstract><p>Recently, delayed dynamical model has witnessed a great interest from many scholars in biological and mathematical areas due to its potential application in describing the interaction of different biological populations. In this article, relying the previous studies, we set up two new predator-prey systems incorporating delay. By virtue of fixed point theory, inequality tactics and an appropriate function, we explore well-posedness (includes existence and uniqueness, boundedness and non-negativeness) of the solution of the two formulated delayed predator-prey systems. With the aid of bifurcation theorem and stability theory of delayed differential equations, we gain the parameter conditions on the emergence of stability and bifurcation phenomenon of the two formulated delayed predator-prey systems. By applying two controllers (hybrid controller and extended delayed feedback controller) we can efficaciously regulate the region of stability and the time of occurrence of bifurcation phenomenon for the two delayed predator-prey systems. The effect of delay on stabilizing the system and adjusting bifurcation is investigated. Computer simulation plots are provided to sustain the acquired prime outcomes. The conclusions of this article are completely new and can provide some momentous instructions in dominating and balancing the densities of predator and prey.</p></abstract>

Stability and Hopf Bifurcation Control for Fractional-Order Two-Gene Regulatory Network With Multiple Delays

Stability and Hopf Bifurcation Control for Fractional-Order Two-Gene Regulatory Network With Multiple Delays

Exploration and Control of Bifurcation in a Fractional-Order Delayed Glycolytic Oscillator Model

Recently, establishing proper dynamical models to describe the relationship among different chemical substances has become a vital theme in chemistry. In this present article, we set up a new fractional-order delayed glycolytic oscillator model. Utilizing the contraction mapping theorem, we explore the existence and uniqueness of the solution to the involved fractional glycolytic oscillator model with delay. By virtue of some suitable analytical skills, we discuss the non-negativeness of the solution to the established fractional glycolytic oscillator system. Taking advantage of a suitable function, we investigate the boundedness of the fractional glycolytic oscillator system. Exploiting the stability and bifurcation theory of fractional dynamical system, we study the stability and the generation of Hopf bifurcation of the fractional glycolytic oscillator system with delay. Making use of delayed feedback controller and PDα controller, we deal with the Hopf bifurcation control of the fractional glycolytic oscillator system owing delay. Computer simulation results are displayed to support the obtained assertions. The acquired results of this article own great theoretical value in dominating the concentrations of different chemical compositions.

Further Insight into Bifurcation and Hybrid Control Tactics of a Chlorine Dioxide-Iodine-Malonic Acid Chemical Reaction Model Incorporating Delays

Delayed differential equation is an important tool to describe the interaction of different chemical substance in chemistry. In this present research, we set up a novel chlorine dioxide-iodine-malonic acid chemical reaction model incorporating delays. The peculiarity of solution and Hopf bifurcation of the formulated delayed chlorine dioxide-iodine-malonic acid chemical reaction model are explored. Firstly, the existence and uniqueness is investigated via fixed point theorem. Secondly, the non-negativeness of solution is studied via some proper mathematical inequality shills. Thirdly, the stability and bifurcation of the formulated delayed chlorine dioxide-iodine-malonic acid chemical reaction model are analyzed. The influence of delay on the delayed chlorine dioxide-iodine-malonic acid chemical reaction model is uncovered. Fourthly, Hopf bifurcation control issue of the formulated delayed chlorine dioxide-iodine-malonic acid chemical reaction model is studied via two hybrid controllers. To check the soundness of acquired key assertions, Matlab simulations are executed. The gained assertions of this research are completely novel and possess tremendous theoretical value in maintaining the balance of the concentrations of chlorine dioxide, iodine in chemistry.

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

EXPLORING BIFURCATION IN A FRACTIONAL-ORDER PREDATOR-PREY SYSTEM WITH MIXED DELAYS

This work chiefly develops and discusses a fractional-order predatorprey model with distributed delay and discrete delay. Applying skilly an appropriate variable substitution, a novel equivalent form of the fractional-order predator-prey model with distributed delay and discrete delay is derived. By virtue of the stability theorem and bifurcation principle of fractional-order dynamical system, we establish a delay-independent stability and bifurcation criterion ensuring the stability and the onset of Hopf bifurcation for the involved predator-prey system. The role of the time delay in stabilizing system and controlling Hopf bifurcation of the considered fractional-order predatorprey model is displayed. Software simulation results are presented to support the key theoretical fruits.

Stability and Hopf bifurcation analysis of a fractional-order ring-hub structure neural network with delays under parameters delay feedback control.

In this paper, a fractional-order two delays neural network with ring-hub structure is investigated. Firstly, the stability and the existence of Hopf bifurcation of proposed system are obtained by taking the sum of two delays as the bifurcation parameter. Furthermore, a parameters delay feedback controller is introduced to control successfully Hopf bifurcation. The novelty of this paper is that the characteristic equation corresponding to system has two time delays and the parameters depend on one of them. Selecting two time delays as the bifurcation parameters simultaneously, stability switching curves in $ (\tau_{1}, \tau_{2}) $ plane and crossing direction are obtained. Sufficient criteria for the stability and the existence of Hopf bifurcation of controlled system are given. Ultimately, numerical simulation shows that parameters delay feedback controller can effectively control Hopf bifurcation of system.

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.

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.

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.

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.

Theoretical analysis and computer simulations of a fractional order bank data model incorporating two unequal time delays

In order to reveal the change law of bank data and manage bank effectively, building mathematical models is a very effective approach. In this present study, we set up a novel fractional-order bank data model incorporating two unequal time delays. Firstly, we discuss the existence and uniqueness, non-negativeness, boundedness of the solution to the established bank data model by virtue of contraction mapping theorem, mathematical analysis technique, construct of an appropriate function, respectively. Secondly, the stability and the creation of Hopf bifurcation are investigated via the stability criterion and bifurcation principle of fractional-order differential equation, five new delay-independent stability conditions and bifurcation criteria ensuring the stability behavior and the onset of Hopf bifurcation of the involved bank data model are established. Furthermore, the role of time delay in stabilizing system and controlling the generation of Hopf bifurcation is sufficiently displayed. Thirdly, the global stability of the considered fractional-order bank data model is systematically explored. Fourthly, the Hopf bifurcation control issue of fractional-order bank data model is studied via PDξ controller. Finally, computer simulations are executed to verify the established primary results. The derived conclusions of this study are absolutely innovative and possess important theoretical guiding significance in maintaining a good operation of banks.

Bifurcation Analysis of Xenon Oscillations in Large Pressurized Heavy Water Reactors with Spatial Control

Large nuclear reactors operating in the thermal spectrum are prone to both global and regional oscillations in power due to variation of 135Xe concentration. These power oscillations are self-stabilizing up to a certain operating power level, beyond which spatial power control becomes necessary for suppressing these oscillations. Especially for large pressurized heavy water reactors (PHWRs), which are natural uranium–fueled reactors using heavy water as coolant and moderator, the modes of xenon instabilities decide the extent and scheme for spatial power control. In this paper, the effect of spatial control on the bifurcation characteristics is demonstrated using a two-region model. The error signal for movement of the reactivity device has a global component for bulk power control and a local component for regional power control. The amount of regional power control determines the power level at which the spatial xenon oscillations stabilize. Using bifurcation analysis, it is found that in case of limited regional control, both supercritical and subcritical Hopf bifurcations exist, whereas in the case of increased regional control only supercritical Hopf bifurcations exist. However, these supercritical Hopf oscillations are due to time lag in control and have short timescales and lower amplitudes as compared to xenon oscillations. Hence, a proper choice of spatial control enables a PHWR to operate at rated full power capacity without any spatial Xenon instability.

Hopf bifurcation control of the ML neuron model with Hc bifurcation type

<abstract><p>It is shown that many neurological diseases are caused by the changes of firing patterns induced by bifurcations. Therefore, the bifurcation control may provide a potential therapeutic method of these neurodegenerative diseases. In this paper, we investigate the Hopf bifurcation control of the Morris-Lecar (ML) model with Homoclinic (Hc) bifurcation type by introducing a dynamic state-feedback control. The results indicate that the linear term can change the ML model from Hc bifurcation type to SNIC bifurcation type without changing the firing patterns. The cooperation of linear and cubic term can transform the ML model from the Hc bifurcation type to the Hopf bifurcation type, resulting in the transformation of firing patterns from type I to type II. Besides, we utilize the Poincare Birkhoff (PB) normal form method to derive the analytical expression of the bifurcation stability index for the controlled ML model with Hc bifurcation type, and the results show that the cubic term can regulate the criticality of the Hopf bifurcation. Numerical simulation results are consistent with the theoretical analysis.</p></abstract>

Further investigation on bifurcation and their control of fractional‐order bidirectional associative memory neural networks involving four neurons and multiple delays

This work principally probes into the stability, the existence, and control of Hopf bifurcation of new established fractional‐order delayed bidirectional associative memory (BAM) neural networks with four neurons. Firstly, based on the earlier study, we set up a class of new fractional‐order multiple delayed BAM neural networks with four neurons. Secondly, applying an appropriate variable transformation, we get a new equivalent fractional‐order single delayed BAM neural networks with four neurons. With the aid of the stability theory and bifurcation knowledge of fractional‐order differential dynamical systems, a novel sufficient criterion to guarantee the stability and the appearance of Hopf bifurcation of the addressed fractional‐order delayed BAM neural networks is set up. Thirdly, designing a suitable time‐delayed feedback controller, the stability region and the time of appearance of Hopf bifurcation for the involved neural networks have been adjusted. Finally, two simulation examples are presented to illustrate the rationality of the mathematical derivation results. The study vindicates that the time delay has great effect on the stability, bifurcation, and its control for involved network models. The obtained fruits of this work can be effectively applied to control and design neural networks. Also, some research ideas will play a key role in studying the related fractional‐order dynamical models in actual world.

Bifurcation control and sound intensities in musical art

Qualitative changes in sound intensities frequently occur in musical signals and Hopf bifurcation control is the most natural mathematical approach to study them. We propose a natural formulation for investigating and generating varieties of pitch simultaneity and sound intensity using Eulerian flows with n -tuple Hopf singularity for a sufficiently large n . Chord leaves and chord limit sets are defined as two critical flow-invariant manifolds. The reduction of flows on chord leaves facilitate the study and bifurcation control through a generic codimension-three (a three-parametric family) scalar bifurcation problem. There is a one-to-one correspondence between bifurcations of positive equilibria in the scalar equation and bifurcations of invariant chord limit sets. An ordered set of double saddle-node and pitchfork bifurcation control of subcritical and supercritical types can describe and generate the sound intensity bifurcations of a sheet music. These scalar bifurcations are interpreted in terms of appearances and disappearances of various chord leaf-stable and unstable flow-invariant chord limit sets. We apply our proposed method to the first three bars of a sheet music. This is implemented in Matlab so that one can hear the smooth dynamic effect of bifurcations using a computer through a classical sheet music. Then, one can hear how pitchfork (subcritical and supercritical) and double fold-bifurcations of invariant hypertori sound in music.

Feedback control of a nonlinear aeroelastic system with non-semi-simple eigenvalues at the critical point of Hopf bifurcation

Abstract The feedback control of Hopf bifurcation of nonlinear aeroelastic systems with asymmetric aerodynamic lift force and nonlinear elastic forces of the airfoil is discussed. For the Hopf bifurcation analysis, the eigenvalue problems of the state matrix and its adjoint matrix are defined. The Puiseux expansion is used to discuss the variations of the non-semi-simple eigenvalues, as the control parameter passes through the critical value to avoid the difficulty for computing the derivatives of the non-semi-simple eigenvalues with respect to the control parameter. The method of multiple scales and center-manifold reduction are used to deal with the feedback control design of a nonlinear system with non-semi-simple eigenvalues at the critical point of the Hopf bifurcation. The first order approximate solutions are developed, which include gain vector and input. The presented methods are based on the Jordan form which is the simplest one. Finally, an example of an airfoil model is given to show the feasibility and for verification of the present method.

Four-wing chaotic system and its Hopf bifurcation control

In order to meet the practical needs of Hopf bifurcation control and communication security, we propose a four-wing chaotic attractor and a bifurcation controller based on washout filter. We analyze the dynamic evolution of system by means of bifurcation diagram, Lyapunov exponent and attractor simulation to determine its chaotic characteristics. Based on the high-dimensional bifurcation theory, we discuss the existence of Hopf bifurcation, and design a Hopf bifurcation controller based on washout filter, which is composed of linear and nonlinear terms. Without changing the equilibrium point of system, the controller can generate the Hopf bifurcation of system at the expected position, and realize the control of amplitude and stability of the limit cycle. The simulation results show that the amplitude of the limit cycle and the critical value of bifurcation parameters can be controlled, which verifies the rationality and effectiveness of the controller. This study expands the diversity of Hopf bifurcation control in chaotic system and has potential applications in communication and image encryption.

Hybrid Control Synthesis for Turing Instability and Hopf Bifurcation of Marine Planktonic Ecosystems With Diffusion

Great progress has been made in bifurcation control of systems described by ordinary differential equations. However, the control of Hopf bifurcation and Turing patterns is seldom reported in reaction-diffusion systems, which is formed by partial differential equations. In this paper, a hybrid control synthesis combining state feedback is firstly devised in the reaction-diffusion marine planktonic ecosystem. The Turing instability condition and Hopf bifurcation criterion are derived through carrying out the eigenvalue analysis of the controlled system. The numerical simulations show that the hybrid control strategy can not only suppress the formation of Turing patterns, but also delay or advance the Hopf bifurcation point. Therefore, the desired spatial dynamics behaviors can be generated by manipulate the control gain parameters, so as to achieve the purpose of maintaining the marine ecological balance.

Bifurcation analysis of a special delayed predator-prey model with herd behavior and prey harvesting

<abstract> In this paper, we propose a predator-prey system with square root functional response, two delays and prey harvesting, in which an algebraic equation stands for the economic interest of the yield of the harvest effort. Firstly, the existence of the positive equilibrium is discussed. Then, by taking two delays as bifurcation parameters, the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained. Next, some explicit formulas determining the properties of Hopf bifurcation are analyzed based on the normal form method and center manifold theory. Furthermore, the control of Hopf bifurcation is discussed in theory. What's more, the optimal tax policy of system is given. Finally, simulations are given to check the theoretical results. </abstract>