Multiple stability switches and Hopf bifurcation in a damped harmonic oscillator with delayed feedback

This paper takes into consideration a damped harmonic oscillator model with delayed feedback. After transforming the model into a system of first-order delayed differential equations with a single discrete delay, the single stability switch and multiple stability switches phenomena as well as the existence of Hopf bifurcation of the zero equilibrium of the system are explored by taking the delay as the bifurcation parameter and analyzing in detail the associated characteristic equation. Particularly, in view of the normal form method and the center manifold reduction for retarded functional differential equations, the explicit formula determining the properties of Hopf bifurcation including the direction of the bifurcation and the stability of the bifurcating periodic solutions are given. In order to check the rationality of our theoretical results, numerical simulations for some specific examples are also carried out by means of the MATLAB software package.

     

Bifurcation and chaos of a delayed predator-prey model with dormancy of predators

In this paper, a delayed predator-prey model with dormancy of predators is investigated. It shows that time delay in the prey-species growth can lead to the occurrence of Hopf bifurcation with stability switches at a coexistence equilibrium. The computing formulas of stability and direction of the Hopf bifurcating periodic solutions are given. Under appropriate conditions, the uniform persistence of this model with time delay is proved. In this simple model, multiple periodic solutions coexist. Through numerical simulation, it is shown that different values of time delay can generate or eliminate chaos. Biologically, our results imply that dynamical behaviors of this system with time delay strongly depend on the initial density of this model and the time delay of the growth of the prey.     

Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system

In this paper, we address the problem of the bifurcation control of a delayed fractional-order dual model of congestion control algorithms. A fractional-order proportional–derivative (PD) feedback controller is designed to control the bifurcation generated by the delayed fractional-order congestion control model. By choosing the communication delay as the bifurcation parameter, the issues of the stability and bifurcations for the controlled fractional-order model are studied. Applying the stability theorem of fractional-order systems, we obtain some conditions for the stability of the equilibrium and the Hopf bifurcation. Additionally, the critical value of time delay is figured out, where a Hopf bifurcation occurs and a family of oscillations bifurcate from the equilibrium. It is also shown that the onset of the bifurcation can be postponed or advanced by selecting proper control parameters in the fractional-order PD controller. Finally, numerical simulations are given to validate the main results and the effectiveness of the control strategy.

     

Stability Analysis of a Reaction–Diffusion Equation with Spatiotemporal Delay and Dirichlet Boundary Condition

In this paper, we concentrate on the study of a reaction–diffusion equation with spatiotemporal delay and homogeneous Dirichlet boundary condition. It is shown that a positive spatially nonhomogeneous equilibrium can bifurcate from the trivial equilibrium. Moreover, the stability of the bifurcated positive equilibrium is investigated. And we prove that, for the given spatiotemporal delay, the bifurcated equilibrium is stable under some conditions, and Hopf bifurcation cannot occur.     

Bifurcation analysis of 4-axle rail vehicle models in a curved track

The dynamics of a diffusive predator–prey model with time delay and Michaelis–Menten-type harvesting subject to Neumann boundary condition is considered. Turing instability and Hopf bifurcation at positive equilibrium for the system without delay are investigated. Time delay-induced instability and Hopf bifurcation are also discussed. By the theory of normal form and center manifold, conditions for determining the bifurcation direction and the stability of bifurcating periodic solution are derived. Some numerical simulations are carried out for illustrating the theoretical results.     

Hopf bifurcations in a predator-prey system of population allelopathy with a discrete delay and a distributed delay

A delayed Lotka?CVolterra predator-prey system of population allelopathy with discrete delay and distributed maturation delay for the predator population described by an integral with a strong delay kernel is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.     

Hopf bifurcation analysis in a delayed oncolytic virus dynamics with continuous control

A delayed oncolytic virus dynamics with continuous control is investigated. The local stability of the infected equilibrium is discussed by analyzing the associated characteristic transcendental equation. By choosing the delay ?? as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay ?? crosses some critical values. Using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to support the theoretical results.     

Stability and bifurcation analysis in the cross-coupled laser model with delay

The dynamics of the cross-coupled laser model with delay has been investigated. The investigation confirms that a Hopf bifurcation occurs due to the existence of stability switches when the product of the coupling strengths varies. An algorithm for determining the stability and direction of the Hopf bifurcation is derived by applying the normal form theory and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the analytic results.     

Nonlinear dynamics of a delayed Leslie predator–prey model

The present paper is concerned with a delayed Leslie predator–prey model. The conditions of boundedness of the solutions of the system, existence, and stability of the equilibrium of the system are investigated. Meanwhile, we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the delay crosses through a sequence of critical values. The extensive simulations carried out show that the bifurcations arise around the positive equilibrium.     

反馈时滞对van der Pol振子张弛振荡的影响

研究反馈控制环节时滞对van derPol振子张弛振荡的影响. 首先, 通过稳定性切换分析, 得到了系统的慢变流形的稳定性和分岔点分布图, 结果表明, 当时滞大于某临界值时, 系统慢变流形的结构发生本质的变化.其次, 基于几何奇异摄动理论, 分析了慢变流形附近解轨线的形状, 发现时滞反馈会引起张弛振荡中的慢速运动过程中存在微幅振荡, 其中微幅振荡来自于内部层引起的振荡和Hopf分岔产生的振荡两个方面; 同时, 时滞对张弛振荡的周期也具有显著的影响. 实例分析表明理论分析结果与数值结果相吻合.      

Stability and bifurcation in a generalized delay prey–predator model

The present paper considers a generalized prey–predator model with time delay. It studies the stability of the nontrivial positive equilibrium and the existence of Hopf bifurcation for this system by choosing delay as a bifurcation parameter and analyzes the associated characteristic equation. The researcher investigates the direction of this bifurcation by using an explicit algorithm. Eventually, some numerical simulations are carried out to support the analytical results.     

Complex dynamics of a four neuron network model having a pair of short-cut connections with multiple delays

This paper deals with dynamic behaviors on Hopfield type of ring neural network of four neurons having a pair of short-cut connections with multiple time delays. By suitable transformation and under certain assumptions on multiple time delays, the model is reduced to four dimensional nonlinear delay differential equations with three delays. Regarding these time delays as parameters, delay independent sufficient conditions for no stability switches of the trivial equilibrium of the linearized system are derived. Conditions for stability switching with respect to one delay parameter which is not associated with short-cut connection are obtained. Hopf bifurcations with respect to two other delays which are associated with short-cut connection are also obtained. Using the normal form method and center manifold theory, the direction of the Hopf bifurcation, stability and the properties of Hopf-bifurcating periodic solutions are determined. Using numerical simulations of the nonlinear model, different rich dynamical behaviors such as quasiperiodicity, torus attractor and chaotic-bands are also observed for suitable range of three delay parameters. Lyapunov exponents are also calculated using the AnT 4.669 tool for verification of chaotic dynamics.     

Hopf bifurcation and spatio-temporal patterns in?delay-coupled van der Pol oscillators

In this paper, the dynamics of a pair of van der Pol oscillators with delayed velocity coupling is studied by taking the time delay as a bifurcation parameter. We first investigate the stability of the zero equilibrium and the existence of Hopf bifurcations induced by delay, and then study the direction and stability of the Hopf bifurcations. Then by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups, we investigate the spatio-temporal patterns of Hopf bifurcating periodic oscillations. We find that there are different in-phase and anti-phase patterns as the coupling time delay is increased. The analytical theory is supported by numerical simulations, which show good agreement with the theory.     

Stability and bifurcation analysis in tri-neuron model with time delay

A simple delayed neural network model with three neurons is considered. By constructing suitable Lyapunov functions, we obtain sufficient delay-dependent criteria to ensure global asymptotical stability of the equilibrium of a tri-neuron network with single time delay. Local stability of the model is investigated by analyzing the associated characteristic equation. It is found that Hopf bifurcation occurs when the time delay varies and passes a sequence of critical values. The stability and direction of bifurcating periodic solution are determined by applying the normal form theory and the center manifold theorem. If the associated characteristic equation of linearized system evaluated at a critical point involves a repeated pair of pure imaginary eigenvalues, then the double Hopf bifurcation is also found to occur in this model. Our main attention will be paid to the double Hopf bifurcation associated with resonance. Some Numerical examples are finally given for justifying the theoretical results.     

Delayed feedback control and bifurcation analysis of Rossler chaotic system

In this paper, from the view of stability and chaos control, we investigate the Rossler chaotic system with delayed feedback. At first, we consider the stability of one of the fixed points, verifying that Hopf bifurcation occurs as delay crosses some critical values. Then, for determining the stability and direction of Hopf bifurcation we derive explicit formulae by using the normal-form theory and center manifold theorem. By designing appropriate feedback strength and delay, one of the unstable equilibria of the Rossler chaotic system can be controlled to be stable, or stable bifurcating periodic solutions occur at the neighborhood of the equilibrium. Finally, some numerical simulations are carried out to support the analytic results.     

Stability and bifurcation analysis of delay coupled Van der Pol–Duffing oscillators

In this paper, we aim to investigate the dynamics of a system of Van der Pol–Duffing oscillators with delay coupling. First, taking the time delay as a bifurcation parameter, the stability of the equilibrium, and the existence of Hopf bifurcation are investigated. Then using the center manifold reduction technique and normal form theory, we give the direction of the Hopf bifurcation. And then by means of the symmetric bifurcation theory for delay differential equations and the representation theory of groups, we claim the bifurcation periodic solution induced by time delay is antiphase locked oscillation. Finally, at the end of the paper, numerical simulations are carried out to support our theoretical analysis.     

Hopf bifurcation of an oscillator with quadratic and cubic nonlinearities and with delayed velocity feedback

This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms, and with linear delayed velocity feedback. The analysis indicates that for a sufficiently large velocity feedback gain, the equilibrium of the system may undergo a number of stability switches with an increase of time delay, and then becomes unstable forever. At each critical value of time delay for which the system changes its stability, a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay. The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability. It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions. The project supported by the National Natural Science Foundation of China (19972025)     

van der Pol-Duffing时滞系统的稳定性和Hopf分岔

研究了具有三次项的ｖａｎ ｄｅｒ Ｐｏｌ－Ｄｕｆｆｉｎｇ非线性时滞系统的稳定性和Ｈｏｐｆ分岔,分析了当线性化特征方程随两参数（时滞量和增益系数）变化时特征根的分布;证明了Ｈｏｐｆ分岔的存在性,通过构造中心流形并且使用范式方法给出的Ｈｏｐｆ分岔的方向以及周期解的稳定性,讨论时滞量对该系统的Ｈｏｐｆ分岔的影响。     

Analysis of the stability and Hopf bifurcation of money supply delay in complex macroeconomic models

Considering the macroeconomic model of money supply, this paper carries out the corresponding extension of the complex dynamics to macroeconomic model with time delays. By setting the parameters, we discuss the effect of delay variation on system stability and Hopf bifurcation. Results of analysis show that the stability of time-delay systems has important significance with the length of time delay. When time delay is short, the stable point of the system is still in a stable region; when time delay is long, the equilibrium point of the system will go into chaos, and the Hopf bifurcation will appear in certain conditions. In this paper, using the normal form theory and center manifold theorem, the periodic solutions of the system are obtained, and the related numerical analysis are also given; this paper has important innovation-theoretical value and acts as important actual application in macroeconomic system.     

Dynamics analysis of a non-smooth Filippov pest-natural enemy system with time delay

In this paper, we propose a non-smooth Filippov system that describes the interaction of the pest and natural enemy with considering time delay, which represents the change in the growth rate of natural enemies before it is released to prey on pests. When the number of the pest is below the threshold, no control is applied; otherwise, control measures will be adopted. We discuss the stability of the equilibria and the existence of Hopf bifurcation. The results show that the Hopf bifurcation occurs when the time delay passes through some critical values. By applying the Filippov convex method, we obtain the dynamics of the sliding mode. The solutions of the system eventually tend toward the regular equilibrium, the pseudo-equilibrium or a standard periodic solution. Numerical simulations show that time delay plays an important role in local and global sliding bifurcations. We can obtain boundary focus bifurcations from boundary node bifurcations by varying time delay. Furthermore, touching, buckling and crossing bifurcations can be obtained frequently by increasing time delay. The results can provide some insights in pest control.