Hopf Bifurcation Phenomenon Research Articles

Randomness suppress backward bifurcation in an epidemic model with limited medical resources

This paper delves into the dynamic features of a stochastic SIR epidemic model featuring a perturbed transmission rate influenced by white noise. Our primary aim is to unravel the intricate interplay between restricted medical resources, their supply efficiency, and environmental stochasticity, shedding light on their collective impact on the transmission dynamics of infectious diseases. Our findings bring to light a notable distinction from the deterministic counterpart of the model. Specifically, under varying scenarios of medical resource availability and supply efficiency, the stochastic model exhibits a departure from bifurcation phenomena. This stands in contrast to the deterministic model, which is characterized by the presence of both backward bifurcation and Hopf bifurcation phenomena. To complement and validate our theoretical findings, numerical simulations are employed, providing concrete illustrations of the dynamical phenomena discussed in the paper. This research contributes to a nuanced understanding of the intricate interplay between stochastic environmental factors, medical resource constraints, and disease transmission dynamics, offering valuable insights for public health management and epidemic control strategies.

A novel approaches to 6th-order delay differential equations in toxic plant interactions and soil impact: beyond newton-raphson

This paper focuses on investigating a 6th-order delay differential equation root within the context of toxic interactions between competing plant populations and their impact on soil dynamics. The study introduces a novel approach for approximating solutions to nonlinear delay differential equations, drawing inspiration from the fundamental principles of Newton-Raphson’s method. This technique leverages the complex root theorem to ensure stability, enabling it to effectively handle widely dispersed roots within dynamic systems. Consequently, this approach holds considerable potential for a diverse array of applications. The analysis introduces time delay into a nonlinear dynamical system and explores the system’s threshold value. At this threshold, the dynamical system’s stability undergoes fluctuations, and observations of hopf bifurcation phenomena are made. The study also successfully identifies both real and complex roots of the dynamical system. Visualization of the dynamic system is accomplished using MATLAB-generated graphical representations. Moreover, this research’s implications extend to the realm of climate action and terrestrial ecosystems, underscoring its significance for promoting a sustainable environment and fostering healthy life on land.

A novel fractional-order stochastic epidemic model to analyze the role of media awareness in the spread of conjunctivitis

This study introduces a novel fractional-order stochastic epidemic model to analyze the spread of conjunctivitis, a prevalent ocular infection, while accounting for the influence of media awareness on disease transmission. The model incorporates fractional derivatives to capture memory effects and non-local interactions inherent in epidemic processes, allowing for a more accurate representation of disease dynamics. The stability analysis of equilibrium points is carried out based on the basic reproduction number ℛ0 and fractional-order α. Further, the Hopf bifurcation phenomenon is discussed in this paper. Stochasticity accounts for the randomness in transmission events. The findings of this study provide insights into the complex interrelationship between disease dynamics and media influence, shedding light on the role of public awareness in mitigating or exacerbating conjunctivitis outbreaks. The implications of this work extend to public health policy formulation, highlighting the importance of targeted communication strategies in controlling and preventing the spread of conjunctivitis and similar infectious diseases.

Tanh-like models for analysis and prediction of time-dependent flow around a circular cylinder at low Reynolds numbers

When employing traditional low-order approximation equations to forecast the Hopf bifurcation phenomenon in the wake of a circular cylinder at low Reynolds numbers, inaccuracies may arise in estimating the phase. This is due to the fact that, in this transition process, the frequency varies with time. In this paper, we propose a method for analyzing and predicting the vortex shedding behind a cylinder at low Reynolds numbers. The proposed method is based on coordinate transformation and description function and is demonstrated using data from computational fluid dynamics simulation of flow around a cylinder at Reynolds number 100. The resulting governing equations explicitly contain the flow amplitude and implicitly contain the flow frequency. The proposed method is found to have higher accuracy compared to other methods for nonlinear identification and order reduction. Finally, the method is extended to predict nonlinear vortex shedding in the Reynolds number range of 80–200.

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.

Hidden Dynamics, Multistability and Synchronization of a Memristive Hindmarsh–Rose Model

Reliable neuron models play an essential role in identifying the electrical activities, global bifurcation patterns, and dynamic mechanisms of neurons in electromagnetic environments. Considering that memristive autapse can characterize the self-induced effect of neurons, a five-dimensional Hindmarsh–Rose (HR) neuron model involving electric and magnetic fields is established. The detailed existence and stability analyses for equilibrium points are performed, and the complex time-varying stability, saddle-node bifurcation, and Hopf bifurcation behaviors are demonstrated. Interestingly, the bistable structures consisting of quiescent state and periodic bursting modes near the subcritical Hopf bifurcation and counterintuitive dynamic phenomena can be induced via appropriately adjusting the memristive current. Accordingly, the mechanism of positive feedback autaptic current decreases its firing frequency, while negative feedback autaptic current promotes its excitability and is revealed by the fast–slow dynamic analysis. Generally, the system possesses period-adding bifurcation patterns and comb-shaped chaotic structures as demonstrated by the numerical results. Importantly, it can be confirmed that the electrical activities and multistability of the system can be accurately predicted by analyzing the global dynamic behaviors of the Hamilton energy. Furthermore, it is verified that the unidirectional coupling controller involving energy is far more efficient and consumes lower energy than electrical synaptic coupling in achieving complete synchronization with mismatched parameters. These results provide potential guidance and help for further research in computational neuroscience and the design and control of intelligent sensors.

The dynamics of delayed models for interactive wild and sterile mosquito populations

The sterile insect technique (SIT) has been applied as an alternative method to reduce or eradicate mosquito-borne diseases. To explore the impact of the sterile mosquitoes on controlling the wild mosquito populations, in this paper, we further extend the work in [J. Li, New revised simple models for interactive wild and sterile mosquito populations and their dynamics, J. Biol. Dyn. 11(S2) (2017) 316–333] and formulate delayed models for interactive wild and sterile mosquitoes, which can depict wild mosquito population undergoing distinct stages of development during a lifetime. By performing mathematical analysis, the threshold dynamics of the proposed models are explored, respectively. In particular, Hopf bifurcation phenomena are observed as the delay [Formula: see text] is varying. Numerical examples illustrate our findings.

Kinematics and dynamics simulations of cold orbital forging machines based on ADAMS

The vibration of cold orbital forging (COF) machines is detrimental to the quality of the forged part. Therefore, it is necessary to carry out kinematics and dynamics analyses, study the vibration of the COF machine, and provide effective methods for reducing the vibration. In terms of kinematics and dynamics analyses, the mathematical modeling and model solving processes are complicated, and it takes a long time to obtain the kinetic and vibration performances of the COF machine. In this study, the kinematics and dynamics models of the COF machine were investigated. Subsequently, automatic dynamic analysis of mechanical systems (ADAMS) was applied to conduct kinematics and dynamics simulations. The simulation process was convenient and time-saving, and the simulation of the COF machine was effectively verified by conducting an experiment. Based on the simulation results, the kinetic and vibration performances of the COF machine were studied. In addition, a structure optimization method for the COF machine is proposed. Based on the singularity stability theory, the Hopf bifurcation and chaos phenomena are less likely to occur on the optimized COF machine, and therefore the vibration of the optimized COF machine is stable. Meanwhile, the structure optimization method was proven to be effective in reducing the vibration of the COF machine, which improves the quality of the forged part.

A bidirectional Hopf bifurcation analysis of Parkinson’s oscillation in a simplified basal ganglia model

In this paper, we study the parkinson oscillation mechanism in a computational model by bifurcation analysis and numerical simulation. Oscillatory activities can be induced by abnormal coupling weights and delays. The bidirectional Hopf bifurcation phenomena are found in simulations, which can uniformly explain the oscillation mechanism in this model. The Hopf1 represents the transition between the low firing rate stable state (SS) and oscillation state (OS), the Hopf2 represents the transition between the high firing rate stable state (HSS) and the OS, the mechanisms of them are different. The Hopf1 and Hopf2 bifurcations both show that when the state transfers from the stable region to the oscillation region, oscillatory activities originate from the beta frequency band or the gamma frequency band. We find that the changing trends of the frequency (DF) and oscillation amplitude (OSAM) are contrary in many cases. The effect of the delay in inhibitory pathways is greater than that of in excitatory pathways, and appropriate delays improve the discharge activation level (DAL) of the system. In all, we infer that oscillations can be induced by the follow factors: 1. Improvement of the DAL of the globus pallidus externa (GPe); 2. Reduce the DAL of the GPe from the HSS or the discharge saturation state; 3. The GPe can also resonate with the subthalamic nucleus (STN).

Fear Effect in a Tri-Trophic Food Chain Model with Holling Type IV Functional Response

Of concern the present study deals with an updated food chain model comprising one prey and two predator species in a natural environment with the inclusion of fear effect in both the prey and the median predator population through Holling type IV functional response. The present model is affluent with intra-specific competition among the hunter species having specific mortality. The model system emphasizes its characteristics in the proximity of the probable equilibrium position in the realm of biological dynamics. The response of the system is explored further for its stability analysis based on prerequisites and Hopf-bifurcation phenomena as well with respect to some significant model parameters. A quantitative analysis based on numerical simulation is also carried out in order to validate the analytical results and thereby the applicability of the model is established.

Nonlinear vibration and control of maglev vehicle-switch beam coupling system

The electro-magnetic suspension (EMS)-based Maglev train, referred as one of the new transport mode, the two important issues are the loss of stability and the nonlinear coupled vibration because of it including suspension controller and control command. In this study, a Maglev train-controller coupled dynamic model is developed to investigate the stability of suspension controller and the coupling vibration, wherein, the tuned mass damper (TMD) control method is applied to avoid Hopf bifurcation phenomenon of the train-switch vibration. The eigenvalue and time integration methods are used to analysis the instability domain. The results demonstrate the proposed dynamic model is able to represent the loss of stability of the coupled vibration, which is comparable to those obtained via the experiments. Using this dynamic model, the dynamic performance and characteristic of the vehicle/switch coupled vibration is investigated, which further contributes to the TMD-based vibration control method for the vibration of the switch beam and the stability of the coupling system.

HOPF bifurcation and stability conditions for a class of nonlinear mass regulation systems with delay

The underwater glider platform is equipped with a mass block adjustment system, which can directly control the roll angle and pitch angle through mass block adjustment, and indirectly control the heading angle through the coupling of profile motion. As the underwater glider energy and space limitations drive the low power and low frequency control, it will inevitably lead to the underwater glider attitude adjustment process is subject to the delay problem of mass motion state signal feedback, the delay time and the nonlinear and coupling characteristics fused together more likely to cause the instability of the control system. In this paper, the time delay problem in the attitude control of underwater glider is summarized as a class of nonlinear stability problems with time delay parameters. The HOPF bifurcation phenomenon caused by the time delay parameter is discussed, and the critical time delay conditions for HOPF bifurcation generation are given. The effect of the time delay parameter on the bifurcation direction and the stability of the periodic solution is analyzed using the central flow pattern method. Finally, the correctness of the theoretical analysis is verified by numerical simulation, which provides a theoretical basis for the platform stability control and drive system design of the underwater glider.

Global $5.1 bn photocatalysts technologies and markets, 2020-2021 & 2021-2026

The underwater glider platform is equipped with a mass block adjustment system, which can directly control the roll angle and pitch angle through mass block adjustment, and indirectly control the heading angle through the coupling of profile motion. As the underwater glider energy and space limitations drive the low power and low frequency control, it will inevitably lead to the underwater glider attitude adjustment process is subject to the delay problem of mass motion state signal feedback, the delay time and the nonlinear and coupling characteristics fused together more likely to cause the instability of the control system. In this paper, the time delay problem in the attitude control of underwater glider is summarized as a class of nonlinear stability problems with time delay parameters. The HOPF bifurcation phenomenon caused by the time delay parameter is discussed, and the critical time delay conditions for HOPF bifurcation generation are given. The effect of the time delay parameter on the bifurcation direction and the stability of the periodic solution is analyzed using the central flow pattern method. Finally, the correctness of the theoretical analysis is verified by numerical simulation, which provides a theoretical basis for the platform stability control and drive system design of the underwater glider.

Spatial dynamics and optimization method for a rumor propagation model in both homogeneous and heterogeneous environment.

Considering the influence of environmental capacity and forgetting on rumor spreading, we improve the traditional SIR (susceptible–infected–removed) rumor propagation model and give two dynamic models of rumor propagation in heterogeneous environment and homogeneous environment, respectively. The main purpose of this paper is to make a dynamic analysis of rumor propagation models. In the spatial heterogeneous environment, we have analyzed the uniform persistence of the rumor propagation model and the asymptotic behavior of the positive equilibrium point when the diffusion rate of rumor–susceptible tends to zero. In the spatial homogeneous environment, we discuss the stability of rumor propagation model. Further, optimal control and the necessary optimality conditions are obtained by using the maximum principle. Finally, we study the Hopf bifurcation phenomenon through inducing time delay in the reaction–diffusion model. In addition, the existence of Hopf bifurcation is verified and the influence of diffusion coefficients is studied by numerical simulations.

Stability and bifurcation in a two-species reaction–diffusion–advection competition model with time delay

In this paper, we are concerned with the dynamics of a class of two-species reaction–diffusion–advection competition models with time delay subject to the homogeneous Dirichlet boundary condition or no-flux boundary condition in a bounded domain. The existence of steady state solution is investigated by means of the Lyapunov–Schmidt reduction method. The stability and Hopf bifurcation at the spatially nonhomogeneous steady-state are obtained by analyzing the distribution of the associated eigenvalues. Finally, the effect of advection on Hopf bifurcation is explored, which shows that with the increase of convection rate, the Hopf bifurcation phenomenon is more likely to emerge.

Numerical study of equilibrium radial positions of neutrally buoyant balls in circular Poiseuille flows

In this article, we have studied, via direct numerical simulations, the equilibrium radial positions of neutrally buoyant balls moving in circular Poiseuille flows. For the one ball case, the Segre–Silberberg effect takes place at low Reynolds numbers (Re) as expected. However, at higher Re, the ball moves to one of two equilibrium positions. At even higher Re, the ball is pinched to a radial position closer to the central axis. For the case of two neutrally buoyant balls placed on a line parallel with the central axis initially, this two-ball train is stable at low Re and its mass center moves to the outer Segre–Silberberg equilibrium position like the migration of a single neutrally buoyant ball. Moreover, for Re values greater than the critical value, Rec = 435, the two-ball train is unstable. The two balls interact periodically, suggesting a (kind of) Hopf bifurcation phenomenon. Nevertheless, the averaged mass center of the two balls is located at the inner equilibrium radial position.

Chemical and biological control of parasite-borne disease Schistosomiasis: An impulsive optimal control approach

This article deals in a continuous Schistosomiasis transmission model involving hosts human and freshwater snail, and two free living Schistosoma larva forms Miracidia and Cercariae. Along with derivation of basic reproduction number and standard equilibrium analysis, Hopf bifurcation phenomenon is demonstrated analytically and numerically. Parameter sensitivity analysis of system solution has been used to identify significant parameters for the change of disease dynamics. Impulsive optimal control of the same system, adding combined biological and chemical control of snail population to minimize human infection is undertaken. Snail natural predator crayfish is used for biological control and molluscicide is used for chemical control, where both controls are released in water periodically for finite number of times. Mathematically, the continuous system is converted to a discrete-continuous hybrid control problem applying Pontryagin’s Maximum Principle. The optimum solution revealed the quantity of each control to be released at each time point. The control analysis identified parameters crucial to success of combined control.

Delayed Dynamics of SIR Model for COVID-19

This paper presents a new modified SIR model which incorporates appropriate delay parameters leading to a more precise prediction of COVID-19 real time data. The efficacy of the newly developed SIR model is proven by comparing its predictions to real data obtained from four counties namely Germany, Italy, Kuwait, and Oman. Two included delay periods for incubation and recovery within the SIR model produce a sensible and more accurate representation of the real time data. In the absence of the two-delay period () the dynamical behavior of the model will not correspond to today’s picture and lag the detection of the epidemic peak. The reproductive number R0 is defined for the model for values of recovery time delay of the infective case. The effect of recovery time may produce second wave, and/or an oscillation which could destabilize the behavior of the system and a periodic oscillation can arise due to Hopf bifurcation phenomenon.

Modeling and analysis of a predator–prey type eco-epidemic system with time delay

In this research work, a delay-induced eco-epidemic model using a reconstructed Leslie–Gower-type growth rate is formulated and analyzed. An extended qualitative nature of the solutions of the model system like boundedness, strong uniform persistence, and permanence is examined to secure the longstanding viability of the system. The stability of the system is investigated at different stationary points, and sufficient conditions are obtained for the local as well as global stability. The dynamics of the delay-induced model system, including the Hopf bifurcation phenomenon, is rigorously studied around the coexisting equilibrium using the normal form method and center manifold theorem. Also, the length of the delay to preserve the stability of the coexisting equilibrium is evaluated. It is observed that the effect of infection on the total harvest is negligible, but the effort to harvest can reduce the infection and preserve the system’s stability. The results may help to determine the point of reference for disease persistence and extinction. Based on our analytical results, several numerical simulations are also performed.

Bifurcation Analysis for a Food Chain Model with Nonmonotonic Nutrition Conversion Rate of Predator to Top Predator

In this paper, a prey–predator-top predator food chain model with nonmonotonic functional response in the predators is studied. With an emphasis on the nutrition conversion rate of predator to top predator, one can get two important thresholds: the top predator extinction threshold and the coexistence threshold. The top predator will die out if the nutrition conversion rate of predator to top predator is less than the top predator extinction threshold; the prey, predator and top predator will coexist if the rate is larger than the coexistence threshold. While between the two thresholds is a bistable interval. When the nutrition conversion rate of predator to top predator is in the bistable interval, the system will see the emergence of bistability. The bifurcation analysis of the system depending on parameters indicates that it exhibits saddle-node bifurcation and Hopf bifurcation phenomena.