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.

     

One-to-three resonant Hopf bifurcations of a maglev system

This paper studies the dynamics of a maglev system around 1:3 resonant Hopf–Hopf bifurcations. When two pairs of purely imaginary roots exist for the corresponding characteristic equation, the maglev system has an interaction of Hopf–Hopf bifurcations at the intersection of two bifurcation curves in the feedback control parameter and time delay space. The method of multiple time scales is employed to drive the bifurcation equations for the maglev system by expressing complex amplitudes in a combined polar-Cartesian representation. The dynamics behavior in the vicinity of 1:3 resonant Hopf–Hopf bifurcations is studied in terms of the controller’s parameters (time delay and two feedback control gains). Finally, numerical simulations are presented to support the analytical results and demonstrate some interesting phenomena for the maglev system.     

Periodic Delay Effects on Cutting Dynamics

We consider a delay equation whose delay is perturbed by a small periodic fluctuation. In particular, it is assumed that the delay equation exhibits a Hopf bifurcation when its delay is unperturbed. The periodically perturbed system exhibits more delicate bifurcations than a Hopf bifurcation. We show that these bifurcations are well explained by the Bogdanov-Takens bifurcation when the ratio between the frequencies of the periodic solution of the unperturbed system (Hopf bifurcation) and the external periodic perturbation is 1:2. Our analysis is based on center manifold reduction theory.     

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.     

Multiple scales for two-parameter bifurcations in a neutral equation

Van der Pol??s equation with extended delay feedback is investigated as a neutral differential-difference equation. Normal forms near codimension two bifurcations, including Hopf?Cpitchfork and Hopf?CHopf bifurcation, are determined by the method of multiple scales. Through analyzing the associated amplitude equations, we obtain the detailed bifurcation sets and find some interesting phenomena such as quasi-periodic oscillations and strange attractor, which are confirmed by several numerical simulations.     

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 in an electromechanical system with time delays

The effect of time delays occurring in a proportional-integral-derivative feedback controller on the linear stability of a simple electromechanical system is investigated by analyzing the characteristic transcendental equation. It is found that the trivial fixed point of the system can lose its stability through Hopf bifurcations when the time delay crosses certain critical values. Codimension two bifurcations, which result from non-resonant and resonant Hopf–Hopf bifurcation interactions, are also found to exist in the system.     

索-梁耦合结构Hopf分岔的反控制

本文研究了索-梁耦合结构的Hopf分岔的反控制,动态窗口滤波反馈控制器在反控制领域有着很广泛的应用。本文通过使用这种控制器,可以使得受控系统在指定的平衡点处产生Hopf分岔。最后,根据庞加莱截面和级数展开法,证明了上述方法的有效性及可行性。     

Dynamics of microbubble oscillators with delay coupling

Two vibrating bubbles submerged in a fluid influence each others’ dynamics via sound waves in the fluid. Due to finite sound speed, there is a delay between one bubble’s oscillation and the other’s. This scenario is treated in the context of coupled nonlinear oscillators with a delay coupling term. It has previously been shown that with sufficient time delay, a supercritical Hopf bifurcation may occur for motions in which the two bubbles are in phase. In this work, we further examine the bifurcation structure of the coupled microbubble equations, including analyzing the sequence of Hopf bifurcations that occur as the time delay increases, as well as the stability of this motion for initial conditions which lie off the in-phase manifold. We show that in fact the synchronized, oscillating state resulting from a supercritical Hopf is attracting for such general initial conditions.     

Hopf and Bogdanov�CTakens bifurcations in a coupled FitzHugh�CNagumo neural system with delay

In this paper, Hopf bifurcation and Bogdanov?CTakens bifurcation with codimension 2 in a coupled FitzHugh?CNagumo neural system with gap junction are investigated. At first, a general bifurcation diagram on the plane of coupling strength and delay is derived. Then, explicit algorithms due to Hassard and Faria are applied to determine the normal forms of Hopf and Bogdanov?CTakens bifurcations, respectively. Next, we analyze the codimension-2 unfolding for Bogdanov?CTakens bifurcation, and give complete bifurcation diagrams and phase portraits. Furthermore, we also consider the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. By the results of theoretical analysis, we obtain that the values of coupled strength, which make the transmission and received signals be synchronous and anti-phase, are opposite. And universal unfolding of Bogdanov?CTakens bifurcation indicates that the neuron signals can transit between resting and spiking.     

Bifurcations in a predator-prey model with discrete and distributed time delay

In this paper, a class of predator-prey model with discrete and distributed time delay is considered. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. By using the normal form theory and center manifold theory, we derive some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are included.     

Dynamics in two nonsmooth predator–prey models with threshold harvesting

Considering a good pest control program should reduce the pest to levels acceptable to the public, we investigate the threshold harvesting policy on pests in two predator–prey models. Both models are nonsmooth and the aim of this paper is to provide how threshold harvesting affects the dynamics of the two systems. When the harvesting threshold is larger than some positive level, the harvesting does not affect the ecosystem; when the harvesting threshold is less than the level, the model has complex dynamics with multiple coexistence equilibria, limit cycle, bistability, homoclinic orbit, saddle-node bifurcation, transcritical bifurcation, subcritical and supercritical Hopf bifurcation, Bogdanov–Takens bifurcation, and discontinuous Hopf bifurcation. Firstly, we provide the complete stability analysis and bifurcation analysis for the two models. Furthermore, some numerical simulations are given to illustrate our results. Finally, it is found that harvesting lowers the level of both species for natural enemy–pest system while raises the densities of both species for the pest–crop system. It is seen that the threshold harvesting policy of the enemy system is more effective than the crop system.     

Stability and Hopf bifurcation in a three-neuron unidirectional ring with distributed delays

In this paper, we consider the effect of distributed delays in a three-neuron unidirectional ring. Sufficient conditions for existence of unique equilibrium, multiple equilibria and their local stability are derived. Taking the average delay as a bifurcation parameter, we find two critical values at which the system undergoes Hopf bifurcations. The orbital asymptotic stability of the Hopf bifurcating periodic solutions is investigated by using the method of multiple scales. The global Hopf bifurcation is also studied. Finally, the theoretical results are illustrated by some numerical simulations.     

Double Hopf bifurcation of time-delayed feedback control for maglev system

This paper undertakes an analysis of a double Hopf bifurcation of a maglev system with time-delayed feedback. At the intersection point of the Hopf bifurcation curves in velocity feedback control gain and time delay space, the maglev system has a codimension 2 double Hopf bifurcation. To gain insight into the periodic solution which arises from the double Hopf bifurcation and the unfolding, we calculate the normal form of double Hopf bifurcation using the method of multiple scales. Numerical simulations are carried out with two pairs of feedback control parameters, which show different unfoldings of the maglev system and we verify the theoretical analysis.     

Nonlinear analysis of time-delay position feedback control of container cranes

Time-delay feedback control of container cranes is robustly stable and insensitive to initial conditions for most of the linearly stable region. To better understand this robustness and any limitations of the technique, we undertake a nonlinear analysis of the system. To this end, we develop a nonlinear model of the crane system by modeling the crane-hoist-payload assembly as a double pendulum. Then, we derive a linear approximation specific to this model. Finally, we derive a cubic model of the dynamics for nonlinear analysis. Using linear analysis, we determine the gain and time delay factors for stabilizing controllers. Also, we show that the controller undergoes a Hopf bifurcation at the linear stability boundary. Using the method of multiple scales on the cubic model, we determine the normal form of the Hopf bifurcation. We then show that for practical operating ranges, the controller undergoes a supercritical bifurcation that helps explain the robustness of the controller.     

Bifurcation and Chaos in the Duffing Oscillator with a PID Controller

We discuss in this paper the bifurcation, stability and chaos of the non-linear Duffing oscillator with a PID controller. Hopf bifurcation can occur and we show that there is a global stable fixed point. The PID controller works well in some fields of the parameter space, but in other fields of the parameter space, or if the reference input is not equal to zero, chaos is common for hard spring type system and so is fractal basin boundary for soft spring system. The Melnikov method is used to obtain the criterion of fractal basin boundary.     

The Bifurcation Study of 1:2 Resonance in a Delayed System of Two Coupled Neurons

In this paper, we consider a delayed system of differential equations modeling two neurons: one is excitatory, the other is inhibitory. We study the stability and bifurcations of the trivial equilibrium. Using center manifold theory for delay differential equations, we develop the universal unfolding of the system when the trivial equilibrium point has a double zero eigenvalue. In particular, we show a universal unfolding may be obtained by perturbing any two of the parameters in the system. Our study shows that the dynamics on the center manifold are characterized by a planar system whose vector field has the property of 1:2 resonance, also frequently referred as the Bogdanov–Takens bifurcation with $Z_2$ symmetry. We show that the unfolding of the singularity exhibits Hopf bifurcation, pitchfork bifurcation, homoclinic bifurcation, and fold bifurcation of limit cycles. The symmetry gives rise to a “figure-eight” homoclinic orbit.     

Oscillatory dynamics in a gene regulatory network mediated by small RNA with time delay

A genetic regulatory network mediated by small RNA with two time delays is investigated. We show by mathematical analysis and simulation that time delays can provide a mechanism for the intracellular oscillator. 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.     

The new result on delayed finance system

In this paper, a finance system with time delay is considered. By linearizing the system at the unique equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the unique 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.     

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.