Hopf Bifurcation Theory Research Articles

Global dynamics of an immunosuppressive infection model with stage structure.

In this paper, we propose an immunosuppressive infection model incorporating natural mortality of immune cells during the time lag needed for the expansion of immune cells. Starting from a stage structure model for the immune cells with various ages, we use the method of characteristic lines to derive a delay differential equation for the population of mature immune cells. Then, we use Lyapunov functional techniques to obtain the global dynamics of the model system. Specifically, we show that the virus dominant equilibrium is globally asymptotically stable when the delay is large. Next, we conduct local and global Hopf bifurcation analysis for the proposed model via Hopf bifurcation theory of delay differential equations. We choose the delay as a bifurcation parameter and examine the onset and termination of Hopf bifurcations of periodic solutions from the immune control equilibrium. We also prove that the model has only a finite number of Hopf bifurcation values, and the periodic solutions with specific oscillation frequencies occur only in bounded delay intervals. Under some technical conditions, we show that two global Hopf branches bifurcated from different Hopf bifurcation values may connect to each other and thus be bounded. However, unlike the global Hopf bifurcation results in the existing literature, the Hopf branches for our model system are not necessarily bounded, though the delay components are always bounded. Numerical simulation suggests that bounded and unbounded Hopf branches may co-exist in the bifurcation diagram. Moreover, we observe a new interesting phenomenon that a global Hopf branch may have uniformly bounded periodic solutions, bounded delays, but unbounded periods.

Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system

In this paper, we consider a diffusive Leslie-Gower predator-prey system with prey subject to Allee effect. First, taking into account the diffusion of both species, we obtain the existence of traveling wave solution connecting predator-free constant steady state and coexistence steady state by using the upper and lower solutions method. However, due to the singularity in the predator equation, we need construct a positive suitable lower solution for the prey density. Such a traveling wave solution can model the spatial-temporal process where the predator invades the territory of the prey and they eventually coexist. Second, taking into account two cases: the diffusion of both species and the diffusion of prey-only, we prove the existence of small amplitude periodic traveling wave train solutions by using the Hopf bifurcation theory. Such traveling wave solutions show that the predator invasion leads to the periodic population densities in the coexistence domain.

Hopf Bifurcation for a Susceptible-Infective Model with Infection-Age Structure

An SIS model is investigated in which the infective individuals are assumed to have an infection-age structure. The model is formulated as an abstract non-densely defined Cauchy problem. We study some dynamical properties of the model by using the theory of integrated semigroups, the Hopf bifurcation theory and the normal form theory for semilinear equations with non-dense domain. Qualitative analysis indicates that there exist some parameter values such that this SIS model has a non-trivial periodic solution which bifurcates from the positive equilibrium. Furthermore, the explicit formulae are given to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Numerical simulations are also carried out to support our theoretical results.

Stability and dynamic characteristics of the nonlinear coupling system of hydropower station and power grid

This paper aims to study the stability and dynamic characteristics of the nonlinear hydro-turbine governing system (HTGS)–power grid (PG) coupling system. Firstly, for the coupling system, the nonlinear mathematical model considering the nonlinear characteristic of head loss in the penstock is established. Then, based on the nonlinear mathematical model and Hopf bifurcation theory, the stability is studied by using stable domain and verified by numerical simulation. Finally, by investigating the dynamic characteristics, the generation mechanism of two time scales oscillation of nonlinear HTGS–PG coupling system is revealed. The effect of HTGS and PG on the stability and dynamic characteristics of system is analyzed. Sensitivity analysis of nonlinear HTGS–PG coupling system to noise and variability of system parameters is conducted. The results show that, for nonlinear HTGS–PG coupling system, the emerged bifurcation is supercritical and the area at the bottom side of bifurcation line is stable domain. The nonlinear HTGS–PG coupling system possesses two time scales oscillation, i.e. subwave-1 with a low frequency and subwave-2 with a high frequency. Subwave-1 and subwave-2 are generated by PG and HTGS, respectively. The HTGS and PG have coupling effect on the stability and dynamic characteristics of the two time scales. The stability and dynamic characteristics of nonlinear HTGS–PG coupling system can be significantly improved by the reasonable determination of system parameters. The nonlinear HTGS–PG coupling system is the most sensitive to the variability of HTGS parameters, especially to ey and eqy.

Spatiotemporal Patterns Induced by Hopf Bifurcations in a Homogeneous Diffusive Predator-Prey System

In this paper, we consider a diffusive predator-prey system where the prey exhibits the herd behavior in terms of the square root of the prey population. The model is supposed to impose on homogeneous Neumann boundary conditions in the bounded spatial domain. By using the abstract Hopf bifurcation theory in infinite dimensional dynamical system, we are capable of proving the existence of both spatial homogeneous and nonhomogeneous periodic solutions driven by Hopf bifurcations bifurcating from the positive constant steady state solutions. Our results allow for the clearer understanding of the mechanism of the spatiotemporal pattern formations of the predator-prey interactions in ecology.

Setting condition of surge tank based on stability of hydro-turbine governing system considering nonlinear penstock head loss

In order to provide a basis for the setting of surge tank to achieve the safe and stable operation of hydropower station, this paper studies the setting condition of surge tank (SCST) based on stability of hydro-turbine governing system (HTGS) considering nonlinear penstock head loss. Firstly, the mathematical model of the HTGS considering nonlinear penstock head loss is established. Secondly, the SCST based on stability of HTGS considering nonlinear penstock head loss is derived by using Hopf bifurcation theory. Finally, the characteristics effects of SCST are analyzed. The effect mechanism of nonlinear penstock head loss on SCST is revealed by the comparative analysis of different SCSTs based on stability. The results indicate that the HTGS considering nonlinear penstock head loss is a third order nonlinear dynamic system. There are three inequality criterions for the SCST, and the third one is the controlling inequality criterion with the highest requirement and acts as a dominant role. The flow inertia time constant of penstock Twt0 corresponding to the endpoint of inequality interval is denoted as [Twt0]. [Twt0] is the allowable value of Twt0. When Twt0 is less than [Twt0], it is not necessary to set surge tank. To obtain a more accurate SCST, the penstock head loss and its nonlinear characteristic should be considered.

Local and global Hopf bifurcation in a neutral population model with age structure

In this paper, an age‐structured population model with the form of neutral functional differential equation is studied. We discuss the stability of the positive equilibrium by analyzing the characteristic equation. Local Hopf bifurcation results are also obtained by choosing the mature delay as bifurcation parameter. On the center manifold, the normal form of the Hopf bifurcation is derived, and explicit formulae for determining the criticality of bifurcation are theoretically given. Moreover, the global continuation of Hopf bifurcating periodic solutions is investigated by using the global Hopf bifurcation theory of neutral equations. Finally, some numerical examples are carried out to support the main results.

Hopf bifurcation of an infection-age structured eco-epidemiological model with saturation incidence

A novel infection-age structured eco-epidemiological model with saturation incidence is constructed. The model is regarded as an abstract non-densely defined Cauchy problem, and the condition of the existence of the interior equilibrium is derived. By employing the method of integrated semigroup and the Hopf bifurcation theory for semilinear equations with non-dense domain, we obtain that the model undergoes a Hopf bifurcation around the interior equilibrium which manifests that this model has a non-trivial periodic orbit that bifurcates from the interior equilibrium when bifurcation parameter τ crosses the bifurcation critical value τ0. In other words, the sustained periodic oscillation phenomenon appears. To support and extend our theoretically analytic results, numerical simulations are performed.

Hopf bifurcation of an age-structured prey–predator model with Holling type II functional response incorporating a prey refuge

In this paper, an age-structured prey–predator model with Holling type II functional response incorporating a prey refuge is constructed. Through applying the method of integrated semigroup and the Hopf bifurcation theory for semilinear equations with non-dense domain, we obtain that the model undergoes a Hopf bifurcation at the interior equilibrium which shows that this model has a non-trivial periodic orbit that bifurcates from the interior equilibrium when bifurcation parameter τ crosses the bifurcation critical value τ0. Numerical simulations are given to verify the theoretical analysis. The results manifest that the prey refuge has a stabilizing effect, namely, the prey refuge is a significant factor to maintain the balance between the prey population and the predator population.

Stability Analysis of Pump Turbine Regulating System Based On Hopf Bifurcation Theory

In order to study the stability of the pumped storage power station system, a model of the pump turbine regulating system under turbine operating conditions is established in this paper. Hopf bifurcation theory is applied to analyze the dynamic characteristics of pump turbine regulating system with different loads, and the results of theoretical analysis are verified by simulation calculation. The results show that the range of the regulator parameters increases significantly with the increase of the load disturbances of the regulating system, but the shape and direction of the bifurcation curves are basically the same. The change of the PID parameters will cause the change of the topology of the nonlinear system.

Multiple Limit Cycles Shimmy of the Dual-Front Axle Steering Heavy Truck Based on Bisectional Road

The shimmy problem causes considerable harm to vehicles and is difficult to solve, especially multiple limit cycle shimmy. Moreover, the dynamic behavior of the multiple limit cycle shimmy of vehicles based on a bisectional road is more complex. Shimmy is practically observed in trucks of cooperative factories during utilization. Thus, we take a heavy truck of a cooperative factory as the prototype and establish a dynamic model of the vehicle-road coupling shimmy system, considering the road adhesion coefficient and dry friction between the suspension and steering system. Based on the dynamic model, the Hopf bifurcation theory is used to qualitatively analyze the existence of the limit cycle for the vehicle shimmy system, and the multiple limit cycle shimmy phenomenon is successfully reproduced using a numerical method. Moreover, the effect of the road adhesion coefficient on the multiple limit cycle shimmy characteristic is studied. Results show that the speed interval and amplitude of the multiple limit cycle shimmy decrease with the road adhesion coefficient; when the coefficient is reduced to a certain extent, the multiple limit cycle shimmy phenomenon is not observed. In addition, the adhesion coefficient of the second axle has a stronger effect on the shimmy characteristic than that of the first axle.

Effect of throttling orifice head loss on dynamic behavior of hydro-turbine governing system with air cushion surge chamber

This paper aims to study the effect of throttling orifice head loss on the dynamic behavior of the hydro-turbine governing system with air cushion surge chamber. Firstly, a nonlinear mathematical model of hydro-turbine governing system considering the nonlinear head loss of throttling orifice is established. Then, the nonlinear dynamic behavior of the hydro-turbine governing system is investigated by using the Hopf bifurcation theory and direct numerical integration. The stability and dynamic response of the system are detailed described. Finally, the effect mechanism of throttling orifice head loss on the dynamic behavior of system is revealed based on the comparison between the cases with nonlinear head loss and no head loss. The results indicate that the mathematical model considering the nonlinear head loss of throttling orifice for the hydro-turbine governing system is a fifth-order nonlinear state equation. The system goes through Hopf bifurcation at the bifurcation points and the type of Hopf bifurcation is supercritical. The domain at the bottom of the bifurcation line is the stable domain. The throttling orifice cannot affect the system stability through its head loss. The throttling orifice can affect the system dynamic response through its head loss. The throttling orifice head loss is favorable for the damping of the water level oscillation in the surge chamber.

Bifurcation Analysis for Simplified Five-Neuron Bidirectional Associative Memory Neural Networks with Four Delays

The paper deals with the stability and bifurcation analysis of a class of simplified five-neuron bidirectional associative memory neural networks with four delays. By discussing the characteristic transcendental equation and applying Hopf bifurcation theory, some sufficient conditions which guarantee the local stability and the existence of Hopf bifurcation of the neural networks are established. With the aid of the normal form theory and center manifold theory, we obtain some specific formulae to determine the stability and the direction of the Hopf bifurcation. Computer simulations are implemented to explain the key mathematical predictions. The paper ends with a brief conclusion.

Exploring the dynamics of a tritrophic food chain model with multiple gestation periods.

This work is mainly focused on the series of dynamical analysis of tritrophic food chain model with Sokol-Howell functional response, incorporating the multiple gestation time delays for more realistic formulation. Basic properties of the proposed model are studied with the help of boundedness, stability analysis, and Hopf-bifurcation theory. By choosing the fixed parameter set and varying the value of time delay, the stability of the model has been studied. There is a critical value for delay parameter. Steady state is stable when the value of delay is less than the critical value and further increase the value of delay beyond the critical value makes the system oscillatory through Hopf-bifurcation. Whereas, another delay parameter has a stabilizing effect on the system dynamics. Chaotic dynamics has been explored in the model with the help of phase portrait and sensitivity on initial condition test. Numerical simulations are performed to validate the effectiveness of the derived theoretical results and to explore the various dynamical structures such as Hopf-bifurcation, periodic solutions and chaotic dynamics.

Periodic oscillations in age-structured ratio-dependent predator–prey model with Michaelis–Menten type functional response

A novel Michaelis–Menten type ratio-dependent predator–prey model with predator-age structure is investigated. The predator fertility function is considered to be the piecewise function related to predator maturation period τ. Our methods rely upon integrated semigroup theory and Hopf bifurcation theory for semilinear equations with non-dense domain. By local analysis and bifurcation analysis of the model, a non-trivial periodic oscillation phenomenon through Hopf bifurcation appears when parameter τ passes through some critical values. Numerical simulations and sensitivity analysis on several important parameters are also performed to illustrate the conclusions.

Stability analysis of fluid film bearings under laminar and turbulent regimes

Stability of short journal bearings is investigated in bith the linear and nonlinear regimes under laminar and turbulent operating conditions with consideration of drag force effects. Exerted forces on fluid film bearings comprises of two different components. (1) Pressure force components in radial and tangential directions, and (2) Drag force (shear/friction force) components in radial and tangential directions. In most existing literature, related to rotor bearing system supported on journal bearing stability analysis, the effect of fluid film drag force has been neglected. Utilizing Ng-Pan-Elrod model and the modified Reynold’s equation, detailed calculation of the turbulent pressure distribution is carried out. Local stability of periodic solutions was studied using Hopf bifurcation theory. The shaft stiffness and inclusion of drag force and turbulent effects were found to play an important role in stability regions and bifurcation profiles of a flexible rotor bearing system.It was found that for shafts supported on short journal bearings, consideration of drag force effect could lead to expansion of un-stable region at high Sommerfeld numbers (low shaft static loads). Dynamic coefficients of short length journal bearing were analytically calculated under laminar and turbulent regimes based on Ng–Pan–Elrod model. Linear stability charts of a flexible rotor supported on laminar and turbulent journal bearings are found by calculating the threshold speed of instability associated to the start of instable oil whirl phenomenon. Local journal trajectories of the rotor-bearing system were found at different operating conditions solely based on the calculated dynamic coefficients in laminar and turbulent flow. Results show discrepancy between laminar and turbulent models, when shaft stiffness is great than 10, for the entire range of Sommerfeld number by increasing the Reynolds number in turbulent models.

Hopf bifurcation control of a Pan-like chaotic system**Project supported by the National Natural Science Foundation of China (Grant No. 11372102).

This paper is concerned with the Hopf bifurcation control of a modified Pan-like chaotic system. Based on the Routh–Hurwtiz theory and high-dimensional Hopf bifurcation theory, the existence and stability of the Hopf bifurcation depending on selected values of the system parameters are studied. The region of the stability for the Hopf bifurcation is investigated. By the hybrid control method, a nonlinear controller is designed for changing the Hopf bifurcation point and expanding the range of the stability. Discussions show that with the change of parameters of the controller, the Hopf bifurcation emerges at an expected location with predicted properties and the range of the Hopf bifurcation stability is expanded. Finally, numerical simulation is provided to confirm the analytic results.

Bifurcation Analysis of an Age Structured HIV Infection Model with Both Virus-to-Cell and Cell-to-Cell Transmissions

We make a mathematical analysis of an age structured HIV infection model with both virus-to-cell and cell-to-cell transmissions to understand the dynamical behavior of HIV infection in vivo. In the model, we consider the proliferation of uninfected CD[Formula: see text] T cells by a logistic function and the infected CD[Formula: see text] T cells are assumed to have an infection-age structure. Our main results concern the Hopf bifurcation of the model by using the theory of integrated semigroup and the Hopf bifurcation theory for semilinear equations with nondense domain. Bifurcation analysis indicates that there exist some parameter values such that this HIV infection model has a nontrivial periodic solution which bifurcates from the positive equilibrium. The numerical simulations are also carried out.

Dynamic Analysis of a Bistable Bi-Local Active Memristor and Its Associated Oscillator System

This paper proposes a new type of memristor with two distinct stable pinched hysteresis loops and twin symmetrical local activity domains, named as a bistable bi-local active memristor. A detailed and comprehensive analysis of the memristor and its associated oscillator system is carried out to verify its dynamic behaviors based on nonlinear circuit theory and Hopf bifurcation theory. The local-activity domains and the edge-of-chaos domains of the memristor, which are both symmetric with respect to the origin, are confirmed by utilizing the mathematical cogent theory. Finally, the subcritical Hopf bifurcation phenomenon is identified in the subcritical Hopf bifurcation region of the memristor.

Delay Feedback Control of the Lorenz-Like System

We choose the delay as a variable parameter and investigate the Lorentz-like system with delayed feedback by using Hopf bifurcation theory and functional differential equations. The local stability of the positive equilibrium and the existence of Hopf bifurcations are obtained. After that the direction of Hopf bifurcation and stability of periodic solutions bifurcating from equilibrium is determined by using the normal form theory and center manifold theorem. In the end, some numerical simulations are employed to validate the theoretical analysis. The results show that the purpose of controlling chaos can be achieved by adjusting appropriate feedback effect strength and delay parameters. The applied delay feedback control method in this paper is general and can be applied to other nonlinear chaotic systems.