State feedback control at Hopf bifurcation in an exponential RED algorithm model

In this paper, we show that a state feedback method, which has successfully been used to control unstable steady states or periodic orbits, provides a tool to control the Hopf bifurcation for a novel congestion control model, i.e., the exponential RED algorithm with a single link and single source. We choose the gain parameter as the bifurcation parameter. Without control, the bifurcation will occur early; meanwhile, the model can maintain a stationary sending rate only in a certain domain of the gain parameter. However, outside of this domain the model still possesses a stable sending rate that can be guaranteed by the state feedback control, and the onset of the undesirable Hopf bifurcation is postponed. Numerical simulations are given to justify the validity of the state feedback controller in the bifurcation control.     

Hopf and resonant double Hopf bifurcation in congestion control algorithm with heterogeneous delays

The congestion control algorithm, which has dynamic adaptations at both user ends and link ends, with heterogeneous delays is considered and analyzed. Some general stability criteria involving the delays and the system parameters are derived by generalized Nyquist criteria. Furthermore, by choosing one of the delays as the bifurcation parameter, and when the delay exceeds a critical value, a limit cycle emerges via a Hopf bifurcation. Resonant double Hopf bifurcation is also found to occur in this model. An efficient perturbation-incremental method is presented to study the delay-induced resonant double Hopf bifurcation. For the bifurcation parameter close to a double Hopf point, the approximate expressions of the periodic solutions are updated iteratively by use of the perturbation-incremental method. Simulation results have verified and demonstrated the correctness of the theoretical results.     

Global stability and Hopf bifurcation of a predator-prey model with stage structure and delayed predator response

A Holling type predator-prey model with stage structure for the predator and a time delay due to the gestation of the mature predator is investigated. By analyzing the characteristic equations, the local stability of a predator-extinction equilibrium and a coexistence equilibrium of the model is addressed and the existence of Hopf bifurcations at the coexistence equilibrium is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium is feasible. By using Lyapunov functionals and the LaSalle invariance principle, it is shown that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and sufficient conditions are derived for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.     

Effect of delay combinations on stability and Hopf bifurcation of an oscillator with acceleration-derivative feedback

This paper presents new observations of delayed AD (acceleration-derivative) controller in active vibration control and in bifurcation control of a Duffing oscillator. Based on the stability analysis of the linear delayed oscillator, it is found that combination of the two delays in acceleration feedback and velocity feedback has a significant influence on the stable region in the parameter plane of the gains. By calculating the real part of the rightmost characteristic roots of the controlled oscillator with fixed delays, it is shown that a delayed acceleration feedback with positive gain can work much better than the corresponding delayed negative acceleration feedback, which is used in classic control theory. For given feedback gains, by calculating the critical delay values, it is shown that a delayed positive acceleration feedback can result in a much larger stable delay interval than the corresponding delayed negative acceleration feedback does. As an application of these results to a delayed Duffing oscillator with acceleration-derivative feedback, a delayed positive acceleration feedback can be well used to postpone the occurrence of Hopf bifurcation in the delayed nonlinear oscillators.     

Stability switches and global Hopf bifurcation in a nutrient-plankton model

This paper is devoted to the analysis of a nutrient-plankton model with delayed nutrient cycling. Firstly, stability and Hopf bifurcation of the positive equilibrium are given, and the direction and stability of Hopf bifurcation are also studied. We show that delay, which is considered in the decomposition of dead phytoplankton, can induce stability switches, such that the positive equilibrium switches from stability to instability, to stability again and so on. One can observe that the influence of delay on the system dynamics is essential. Then, we prove that there exists at least one positive periodic solution as the time delay varies in some regions using the global Hopf bifurcation result of Wu (1998, Trans Am Math Soc 350:4799–4838) for functional differential equations. Furthermore, the impact of input rate of nutrient is discussed along with numerical results, and the role of delay in the nutrient cycling is interpreted ecologically. Finally, several groups of illustrations are performed to justify analytical findings.     

Hopf bifurcation in a delayed food-limited model with feedback control

In this paper, we consider a delayed food-limited model with feedback control. By regarding the delay as the bifurcation parameter and analyzing the corresponding characteristic equations, the linear stability of the system is discussed, and Hopf bifurcations are demonstrated. By the normal form and the center manifold theory, the explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some examples are presented to verify our main results.     

On stability, persistence, and Hopf bifurcation in fractional order dynamical systems

This is a preliminary study about the bifurcation phenomenon in fractional order dynamical systems. Persistence of some continuous time fractional order differential equations is studied. A numerical example for Hopf-type bifurcation in a fractional order system is given, hence we propose a modification of the conditions of Hopf bifurcation. Local stability of some biologically motivated functional equations is investigated.     

Stability and Hopf bifurcation analysis for an energy resource system

This paper investigates the dynamical behaviors for a four-dimensional energy resource system with time delay, especially in terms of equilibria analyses and Hopf bifurcation analysis. By setting the time delay as a bifurcation parameter, it is shown that Hopf bifurcation would occur when the time delay exceeds a sequence of critical values. Furthermore, the stability and direction of the Hopf bifurcation are determined via the normal form theory and the center manifold reduction theorem. Numerical examples are given in the end of the paper to verify the theoretical results.     

Criticality of Hopf bifurcation in state-dependent delay model of turning processes

In this paper the non-linear dynamics of a state-dependent delay model of the turning process is analyzed. The size of the regenerative delay is determined not only by the rotation of the workpiece, but also by the vibrations of the tool. A numerical continuation technique is developed that can be used to follow the periodic orbits of a system with implicitly defined state-dependent delays. The numerical analysis of the model reveals that the criticality of the Hopf bifurcation depends on the feed rate. This is in contrast to simpler constant delay models where the criticality does not change. For small feed rates, subcritical Hopf bifurcations are found, similar to the constant delay models. In this case, periodic orbits coexist with the stable stationary cutting state and so there is the potential for large amplitude chatter and bistability. For large feed rates, the Hopf bifurcation becomes supercritical for a range of spindle speeds. In this case, stable periodic orbits instead coexist with the unstable stationary cutting state, removing the possibility of large amplitude chatter. Thus, the state-dependent delay in the model has a kind of stabilizing effect, since the supercritical case is more favorable from a practical viewpoint than the subcritical one.     

Example of a non-smooth Hopf bifurcation in an aero-elastic system

We investigate a typical aerofoil section under dynamic stall conditions, the structural model is linear and the aerodynamic loading is represented by the Leishman–Beddoes semi-empirical dynamic stall model. The loads given by this model are non-linear and non-smooth, therefore we have integrated the equation of motion using a Runge–Kutta–Fehlberg algorithm equipped with event detection. The main focus of the paper is on the interaction between the Hopf bifurcation typical of aero-elastic systems, which causes flutter oscillations, and the discontinuous definition of the stall model. The paper shows how the non-smooth definition of the dynamic stall model can generate a non-smooth Hopf bifurcation. The mechanisms for the appearance of limit cycle attractors are described by using standard tools of the theory of dynamical systems such as phase plots and bifurcation diagrams.     

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.

     

Hopf bifurcation for a ecological mathematical model on microbe populations

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Hopf bifurcation from nonperiodic solutions of differential equations. I. Linear theory

In this and succeeding papers we consider some foundational elements of a theory of Hopf bifurcation from non-periodic solutions of ordinary differential equations.     

Hopf bifurcation phenomenon in catalytic reaction

In this paper, we intend to discuss Hopf bifurcation phenomenon under the effect of the periodic small perturbation on local temperature fluctuations (flikering) phenomenon of catalytic reaction. With the obtained results, we expect to provide basis for selecting reaction parameters.     

Stability and Hopf bifurcation in an inverted pendulum with delayed feedback control

In this paper, we investigate the dynamics of the inverted pendulum with delayed feedback control. The existence and stability of multiple equilibria depending on the control strengths are studied. Taking the time delay of the control terms as a parameter, periodic oscillations induced by delay are found. By using the method of multiple scales, the effect of the control gains and the relative mass of the pendulum on the stability and direction of Hopf bifurcations are discussed. Numerical simulations are employed to illustrate the obtained theoretical results.     

Stochastic Hopf bifurcation in quasi-integrable-Hamiltonian systems

A new procedure is developed to study the stochastic Hopf bifurcation in quasiintegrable-Hamiltonian systems under the Gaussian white noise excitation. Firstly, the singular boundaries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system‘s energy levels with respect to the stochastic averaging method. Secondly, the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones. Lastly, a quasi-integrable-Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure.Moreover, simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure. It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system‘s parameters. Therefore, one can see that the numerical results are consistent with the theoretical predictions.     

Hopf bifurcation in a three-dimensional system

In this paper,Liapunor-Schmidl reduction and singularity theory are employed to discuss Hopf and degenerate Hopf bifureations in global parametric region in a three-dimensional system x=-βx+y, y=-x-βy(1-kz), z=β[α(1-z)-ky²], The conditions on existence and stability are given.     

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.