Hopf Bifurcation Control Research Articles

Hopf bifurcation control of memristor-based fractional delayed tri-diagonal bidirectional associative memory neural networks under various controllers

In this paper, authors present a memristor-based fractional order system of tri-diagonal bidirectional associative memory neural networks (TdBAMNNs) incorporating leakage and communication delays. The existence and uniqueness theorem is established for the memristor-based fractional-order TdBAMNNs system. Analysis of Hopf bifurcation anti-control is conducted using various feedback controllers by exploring single-parameter and double-parameter approaches. A study of the stability switching curves is undertaken to determine the direction of stability transitions in the system. The theoretical results and graphical illustrations are validated through numerical simulations to exhibit their feasibility. Comparative analysis of the system’s dynamical behavior under different controllers is performed. This paper comprehensively explores the proposed memristor-based system, proving its theoretical underpinnings, and analyzing stability through bifurcation control strategies and comparative assessments under different control scenarios.

Bifurcation and Controller Design of 5D BAM Neural Networks With Time Delay

ABSTRACTAll the time delayed dynamical system plays a vital role in describing the dynamical phenomenon of neural networks. In the current article, we study a class of 5D delayed bidirectional associative memory (BAM) neural networks that conform to objective reality. First of all, we prove that the solution of the delayed 5D BAM neural networks exists and is unique by virtue of fixed point theorem and some inequality techniques. Secondly, the Hopf bifurcation and stability of the delayed 5D BAM neural networks are investigated by exploiting the stability criterion and bifurcation theory. Once more, Hopf bifurcation control strategy of the delayed 5D BAM neural networks is explored by virtue of two different hybrid controllers. By adjusting the parameters of the controllers, we can control the stability domain and Hopf bifurcation onset. Eventually, the correctness of the theoretical results was verified through numerical simulations. The conclusions obtained in this paper are new and have important theoretical value in neural network area.

Hopf bifurcation control for the traffic flow model considering the tail light effect

The research on traffic congestion control has seen rapid development in recent years. Investigating the bifurcation characteristics of traffic flow and designing control schemes for unstable bifurcation points can offer new methods for alleviating traffic congestion. This paper focuses on studying the bifurcation characteristics and nonlinear control of traffic flow based on the continuous model and the taillight effect. Firstly, the traffic flow model is transformed into a stability model suitable for branching analysis through the use of the traveling wave transform. This transformation facilitates the analysis of stability that reflects unstable traffic characteristics such as congestion. Based on this stability model, the existence condition of Hopf bifurcation is proved and some bifurcation points of the traffic system are identified. Secondly, the congestion and stability mutation behaviors near equilibrium and branching points are studied to understand the formation mechanism of traffic congestion. Finally, control schemes are designed using Chebyshev polynomial approximation and stochastic feedback control to delay or eliminate unstable bifurcation points and relieve traffic congestion. This improved traffic flow model helps explain changes in system stability through bifurcation analysis and identify unstable bifurcation points. It can also effectively manage these points by designing a feedback controller. It is beneficial for controlling sudden changes in traffic system stability behavior and mitigating traffic congestion, with important theoretical significance and practical application value.

Hopf bifurcation and optimal control of a delayed reaction–diffusion brucellosis disease model

Hopf bifurcation and optimal control of a delayed reaction–diffusion brucellosis disease model

Control of Hopf bifurcation for a four-dimensional fractional order hyperchaotic system with coexisting attractors

Control of Hopf bifurcation for a four-dimensional fractional order hyperchaotic system with coexisting attractors

Delay induced Hopf bifurcation and its control in a fractional p53 protein oscillator model

The p53 protein is essential for cancer prevention. Research on p53 mathematical models is popular since its oscillatory dynamics were observed. However, there is little research on fractional-order p53 oscillators, as well as on Hopf bifurcation control. We therefore introduce fractional-order into a p53 oscillator model and propose a delayed state feedback bifurcation controller. The stability of positive equilibrium and the existence of Hopf bifurcation are studied analytically regarding delay as the bifurcation parameter in both controlled and uncontrolled scenarios. Numerical simulations are consistent with the analytical results. We reveal that a smaller fractional-order facilitates the stability of fixed point and a larger feedback gain advances the Hopf bifurcation point. The work may inspire cancer therapy by artificially controlling the occurrence of p53 oscillations.

Novel Hopf Bifurcation Exploration and Control Strategies in the Fractional-Order FitzHugh–Nagumo Neural Model Incorporating Delay

In this article, we propose a new fractional-order delay-coupled FitzHugh–Nagumo neural model. Taking advantage of delay as a bifurcation parameter, we explore the stability and bifurcation of the formulated fractional-order delay-coupled FitzHugh–Nagumo neural model. A delay-independent stability and bifurcation conditions for the fractional-order delay-coupled FitzHugh–Nagumo neural model is acquired. By designing a proper PDp controller, we can efficaciously control the stability domain and the time of emergence of the bifurcation phenomenon of the considered fractional delay-coupled FitzHugh–Nagumo neural model. By exploiting a reasonable hybrid controller, we can successfully adjust the stability domain and the bifurcation onset time of the involved fractional delay-coupled FitzHugh–Nagumo neural model. This study shows that when the delay crosses a critical value, a Hopf bifurcation will arise. When we adjust the control parameter, we can find other critical values to enlarge or narrow the stability domain of the fractional-order delay-coupled FitzHugh–Nagumo neural model. In order to check the correctness of the acquired outcomes of this article, we present some simulation outcomes via Matlab 7.0 software. The obtained theoretical fruits in this article have momentous theoretical significance in running and constructing networks.

Hopf bifurcation control of a modified continuum traffic flow model

The stability of the transportation system refers to the structural stability of the system. When the system structure is unstable, local or global bifurcation phenomena will occur, which is one of the main reasons for nonlinear traffic phenomena such as congestion. To truly understand the internal mechanism of the formation of these phenomena, it is necessary to analyze the bifurcation of traffic flow. In this paper, the Hopf bifurcation control of a modified viscous macroscopic traffic flow model is studied by using the linear state feedback method, which changes the characteristics of the bifurcation phenomenon of the dynamic system and obtains the required dynamic behavior of the system. First, we can convert the original traffic model into the nonlinear ordinary differential form suitable for bifurcation analysis, solve the equilibrium point of the system, and carry out phase plane analysis. Then, the linear state feedback term is added and the corresponding controlled system is generated, the existence and type of Hopf bifurcation and the existence of saddle node bifurcation are proved. Numerical simulation results show that the analysis and control of Hopf bifurcation in the traffic model are well realized in this paper.

Stability and Hopf Bifurcation Analysis of A Fractional-Order BAM Neural Network with Two Delays Under Hybrid Control

In this paper, considering that fractional-order calculus can more accurately describe memory and genetic properties, we introduce fractional integral operators into neural networks and discuss the stability and Hopf bifurcation of a fractional-order bidirectional associate memory (BAM) neural network with two delays. In addition, the hybrid controller is proposed to achieve Hopf bifurcation control of the system. By taking two time delays as the bifurcation parameters and analyzing of the corresponding characteristic equation, stability switching curves of the controllable system for two delays are obtained. The direction of the characteristic root crossing the imaginary axis in stability switching curves is determined. Sufficient criteria are sequentially given to judge the local stability and the existence of Hopf bifurcation of a fractional-order BAM neural network system. The numerical simulation results show that the hybrid controller can effectively control Hopf bifurcation of a fractional-order BAM neural network system with two delays.

Hopf bifurcation analytical expression and control strategy in direct‐drive permanent magnet synchronous generator

SummaryThe unstable operation of a direct‐drive permanent magnet synchronous generator (DPMSG) is exacerbated by periodic oscillation, having a significant impact on the safe and stable functioning of wind energy generation systems. This paper proposes an approximate solution method for analyzing the periodic oscillation of the Hopf bifurcation and a method for suppressing its bifurcation through H∞ robust control. Firstly, a three‐dimensional nonlinear dynamic model of the DPMSG is constructed. Secondly, the Hopf bifurcation of the system with changes in internal parameters is solved using the time domain and frequency domain methods, and the harmonic balancing method is used to approximate the periodic solution at the point of Hopf bifurcation. The order of the Hopf bifurcation neighborhood is then decreased using the central manifold theorem. Finally, this study suggests a H∞ output feedback control employing bifurcation parameters. The Nyquist criteria are used for evaluating small signal stability and convergence speed of the response system in DPMSG before and after dimensionality reduction. The simulation results suggest that the proposed strategy helps to tackle the periodic solution expression and instability difficulties caused by Hopf bifurcation. This paper provides theoretical suggestions for the future reliable operation of new energy generation systems.

Further study on Hopf bifurcation and hybrid control strategy in BAM neural networks concerning time delay

<abstract><p>Delayed dynamical system plays a vital role in describing the dynamical phenomenon of neural networks. In this article, we proposed a class of new BAM neural networks involving time delay. The traits of solution and bifurcation behavior of the established BAM neural networks involving time delay were probed into. First, the existence and uniqueness is discussed using a fixed point theorem. Second, the boundedness of solution of the formulated BAM neural networks involving time delay was analyzed by applying an appropriate function and inequality techniques. Third, the stability peculiarity and bifurcation behavior of the addressed delayed BAM neural networks were investigated. Fourth, Hopf bifurcation control theme of the formulated delayed BAM neural networks was explored by virtue of a hybrid controller. By adjusting the parameters of the controller, we could control the stability domain and Hopf bifurcation onset, which was in favor of balancing the states of different neurons in engineering. To verify the correctness of gained major outcomes, computer simulations were performed. The acquired outcomes of this article were new and own enormous theoretical meaning in designing and dominating neural networks.</p></abstract>

Hopf bifurcation and hybrid control of a delayed diffusive semi-ratio-dependent predator-prey model

<p>A delayed diffusive predator-prey system with nonmonotonic functional response subject to Neumann boundary conditions is introduced in this paper. First, we analyze the associated characteristic equation to research the conditions for local stability of the positive equilibrium point and the occurrence of Turing instability induced by diffusion in the absence of delay. Second, we provide conditions for the existence of Hopf bifurcation driven by time delay. By utilizing the normal theory and center manifold theorem, we derive explicit formulas for Hopf bifurcation properties such as direction and stability from the positive equilibrium. Third, a hybrid controller is added to the system. By judiciously adjusting the control parameters, we effectively enhance the stability domain of the system, resulting in a modification of the position of the Hopf bifurcation periodic solutions. Numerical simulations demonstrate the presence of rich dynamical phenomena within the system. Moreover, sensitivity analysis was conducted using Latin hypercube sampling (LHS)/partial rank correlation coefficient (PRCC) to explore the impact of parameter variations on the output of prey and predator populations.</p>

Hopf bifurcation analysis and control of the continuum model considering the new energy vehicles effect

Hopf bifurcation analysis and control of the continuum model considering the new energy vehicles effect

Hopf bifurcation control of macroscopic traffic flow model considering vehicle braking effect.

Traffic congestion not only has a great impact on people's travel, but also increases energy consumption and air pollution. The control analysis of the macroscopic traffic flow model considering the vehicle braking effect is particularly important, reflecting the impact on the actual traffic flow density wave, so as to better solve the actual traffic problems. In this paper, based on a speed difference optimization speed model, the micro-macro-variables are transformed into a high-order continuous traffic flow model. Then, a random function considering the physical correlation of random components is added to the high-order continuous traffic flow model to establish a random traffic flow model that can reflect the uncertain behavior of traffic flow acceleration or deceleration. Based on this stochastic traffic model, the existence of Hopf bifurcation and bifurcation control of the traffic flow system model considering stochastic characteristics are derived by using Hopf bifurcation theorem. By Chebyshev polynomial approximation method, the stochastic problem of the system is transformed into the bifurcation control problem of its equivalent deterministic system. A feedback controller is designed to delay the occurrence of Hopf bifurcation and control the amplitude of the limit cycle. Without changing the equilibrium point of the system, the complete elimination of Hopf bifurcation can be achieved by controlling the amplitude of the limit cycle. That is, the feedback controller is used to modify the bifurcation characteristics of the system, such as the bifurcation appearing at the equilibrium point in the control system moves forward, moves backward or disappears, so as to achieve the effect of preventing or alleviating traffic congestion.

Hopf Bifurcation Control of a Fractional-Order Delayed Turbidostat Model via a Novel Extended Hybrid Controller

Building delayed dynamical models to describe the inherent laws of different chemical matters has become a hot theme in recent years. In this current study, we set up a new fractional-order delayed turbidostat model. By using laplace transform, we obtain the characteristic equation of established fractional-order delayed turbidostat model. By selecting the delay as bifurcation parameter and exploring the roots of the corresponding characteristic equation of the involved fractional-order delayed turbidostat model, a novel delay-dependent condition on stability and Hopf bifurcation is acquired. Taking advantage of a novel extended hybrid controller, the stability region and the time of Hopf bifurcation of the established fractional-order delayed turbidostat model are successfully controlled. The role of delay in stabilizing system and controlling Hopf bifurcation is revealed. Matlab experiments are carried out to check the rationality of the acquired key outcomes in this article. The acquired outcomes of this study are completely new and own great theoretical value in dominating concentrations of various chemical matters.

Hybrid control of Turing instability and Hopf bifurcation in CDK1-APC feedback systems with diffusion

For the key substances CDK1 and APC that affect cell cycle, we establish a new feedback model by introducing diffusion. In order to simulate the effects of CDK1 targeted inhibitor on system dynamics, for the first time, the hybrid controller is applied in microscopic systems involving cell cycle substances. Necessary and sufficient conditions for the occurrence of Turing modes and Hopf bifurcation of the controlled system are given. By using the formal theory and the central manifold theorem, the direction and stability of Hopf bifurcation are determined. Finally, the efficiency of the controller is verified by numerical simulation.

Bifurcation mechanism and hybrid control strategy of a finance model with delays

Establishing financial models or economic models to describe economic phenomena in real life has become a heated discussion in society at present. From a mathematical point of view, the exploration on dynamics of financial models or economic models is a valuable work. In this study, we build a new delayed finance model and explore the dynamical behavior containing existence and uniqueness, boundedness of solution, Hopf bifurcation, and Hopf bifurcation control of the considered delayed finance model. By virtue of fixed point theorem, we prove the existence and uniqueness of the solution to the considered delayed finance model. Applying a suitable function, we obtain the boundedness of the solutions for the considered delayed finance model. Taking advantage of the stability criterion and bifurcation argument of delayed differential equation, we establish a delay-independent condition ensuring the stability and generation of Hopf bifurcation of the involved delayed finance model. Exploiting hybrid controller including state feedback and parameter perturbation, we efficaciously adjust the stability region and the time of occurrence of Hopf bifurcation of the involved delayed finance model. The study manifests that time delay is a fundamental parameter in controlling stability region and the time of onset of Hopf bifurcation of the involved delayed finance model. To examine the soundness of established key results, computer simulation figures are concretely displayed. The derived conclusions of this study are perfectly new and has momentous theoretical value in economical operation.

Bifurcation control analysis based on continuum model with lateral offset compensation

The study of traffic flow bifurcation control analysis is of great significance for understanding the essential evolution law of traffic flow, controlling traffic flow with practical means and alleviating traffic congestion. This paper describes the influence of lateral distance compensation in the process of lane change, and adjusts the traffic flow from the perspective of bifurcation control, which can capture the characteristics of traffic flow better, so as to describe the practical significance of the actual traffic phenomenon better. For example, the green ratio of traffic lights in the road, the planning and design of traffic signs, and the guidance of traffic condition prediction in ITS. In addition, the traffic flow is introduced and applied to the traffic model from the perspective of random function, and the bifurcation control is carried out according to the bifurcation behavior in traffic. That is, the bifurcation characteristics of the system are modified by the designed feedback controller, and the appearance of the equilibrium point of the system is adjusted to make it move forward, backward or disappear, so as to prevent or alleviate traffic congestion. There are few researches on bifurcation control of traffic flow model with lateral distance compensation. This paper introduces stochastic function based on lateral compensation model to study Hopf bifurcation phenomenon and Hopf bifurcation control. Firstly, the existence condition of Hopf bifurcation at the equilibrium point in the model is proved theoretically. Then, a feedback controller is designed to control the amplitudes of the Hopf bifurcation and the limit cycles formed by the Hopf bifurcation. Finally, the theoretical results are verified by numerical simulation. The research shows that by adjusting the control parameters in the feedback controller, the influence of boundary conditions on the stability of the traffic system is fully described, the effect of unstable focus and saddle point on the system is inhibited, and the traffic flow is slowed down. In addition, the unstable bifurcation points can be eliminated, and the amplitude of the limit cycle formed by Hopf bifurcation can be adjusted, so as to achieve the stability of the control system.

Hopf bifurcation and optimal control of HCV/HIV co-infection dynamics within human: A theoretical study

Pathogen and human host interaction dynamics are often more complicated in the presence of co-pathogens. Co-infection can occur either as the presence of pre-existing pathogen which is accelerated by the new pathogen and more complications happened. In this study, we look at the burden of HIV/HCV viremia and the efficacy of treatment in reducing the severity of HIV/HCV co-infection patterns. Disease-free equilibrium and endemic equilibrium are two equilibrium states determined in the absence of drugs. The basic reproduction number is computed and the stability of the disease-free equilibrium of the model is analyzed using it. Here we have also incorporated the optimal drug therapy to control the co-infection disease progression. The efficacy of treatment has also been found to influence the natural progression of HCV in HIV/HCV co-infection. The numerical results suggest that the HIV viral load impacts the severity of the HCV infection impressively. This research is significant to develop antiviral therapy strategies to control HIV/HCV co-infection. The most effective way to control the co-infection with the minimum side effects is to take the combination of three medications with optimal dosing.

Mathematical study on bifurcation dynamics and control mechanism of tri‐neuron bidirectional associative memory neural networks including delay

Applying delayed dynamical models to characterize the dynamics of neural networks has attracted great interest from scientific community. In this current manuscript, a kind of new tri‐neuron bidirectional associative memory (BAM) neural networks including delay are formulated. The properties of solution and Hopf bifurcation issue of the established tri‐neuron BAM neural networks including delay are investigated. First, we check the existence and uniqueness of the solution to the formulated delayed tri‐neuron BAM neural networks by virtue of fixed point theorem. Second, we seek the boundedness of the solution of the formulated delayed tri‐neuron BAM neural networks in view of a suitable function and inequality skills. Third, the delay‐independent criteria on stability and bifurcation of the formulated delayed tri‐neuron BAM neural networks are acquired. Fourth, Hopf bifurcation control aspect of the formulated delayed tri‐neuron BAM neural networks is explored via designing two proper hybrid controllers. To prove the accuracy of gained primary assertions, software simulation experiments are put into practice.