BIFURCATION CONTROL FOR A FRACTIONAL-ORDER DELAYED SEIR RUMOR SPREADING MODEL WITH INCOMMENSURATE ORDERS

A fractional-order delayed SEIR rumor spreading model with a nonlinear incidence function is established in this paper,and a novel strategy to control the bifurcation of this model is proposed.First,Hopf bifurcation is investigated by considering time delay as bifurcation parameter for the system without a feedback controller.Then,a state feedback controller is designed to control the occurrence of bifurcation in advance or to delay it by changing the parameters of the controller.Finally,in order to verify the theoretical results,some numerical simulations are given.

Hopf bifurcation control of a Pan-like chaotic system

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 control and eigenstructure assignment in continuous solution polymerization of vinyl acetate

The major difficulty in achieving good performance of industrial polymerization reactors lies in the lack of understanding of their nonlinear dynamics and the lack of well-developed techniques for the control of nonlinear processes, which are usually accompanied with bifurcation phenomenon. This work aims at investigating the nonlinear behavior of the parameterized nonlinear system of vinyl acetate polymerization and further modifying the bifurcation characteristics of this process via a washout filter-aid controller, with all the original steady state equilibria preserved. Advantages and possible extensions of the proposed methodology are discussed to provide scientific guide for further controller design and operation improvement.

Bifurcation control and chaos in a linear impulsive system

Bifurcation control and the existence of chaos in a class of linear impulsive systems are discussed by means of both theoretical and numerical ways. Chaotic behaviour in the sense of Marotto＇s definition is rigorously proven. A linear impulsive controller, which does not result in any change in one period-1 solution of the original system, is proposed to control and anti-control chaos. The numerical results for chaotic attractor, route leading to chaos, chaos control, and chaos anti-control, which are illustrated with two examples, are in good agreement with the theoretical analysis.

The Dynamics and Bifurcation Control of a Singular Biological Economic Model

The objective of this paper is to study systematically the dynamics and control strategy of a singular biological economic model that is described by a differential-algebraic equation. It is shown that when the economic profit passes through zero, this model exhibits the transcritical bifurcation, the Hopf bifurcation, and the limit cycle. In particular, the system undergoes the singularity induced bifurcation at the positive equilibrium, which can result in impulse. Then, state feedback controllers closer to the actual control strategies are designed to eliminate the unexpected singularity induced bifurcation and stabilize the positive equilibrium under the positive profit. Finally, numerical simulations verify the results and illustrate the effectiveness of the controllers. Also, the model with positive economic profit is shown numerically to have different dynamics.

Bifurcation analysis and control study of improved full-speed differential model in connected vehicle environment

In recent years, the traffic congestion problem has become more and more serious, and the research on traffic system control has become a new hot spot. Studying the bifurcation characteristics of traffic flow systems and designing control schemes for unstable pivots can alleviate the traffic congestion problem from a new perspective. In this work, the full-speed differential model considering the vehicle network environment is improved in order to adjust the traffic flow from the perspective of bifurcation control, the existence conditions of Hopf bifurcation and saddle-node bifurcation in the model are proved theoretically, and the stability mutation point for the stability of the transportation system is found. For the unstable bifurcation point, a nonlinear system feedback controller is designed by using Chebyshev polynomial approximation and stochastic feedback control method. The advancement, postponement, and elimination of Hopf bifurcation are achieved without changing the system equilibrium point, and the mutation behavior of the transportation system is controlled so as to alleviate the traffic congestion. The changes in the stability of complex traffic systems are explained through the bifurcation analysis, which can better capture the characteristics of the traffic flow. By adjusting the control parameters in the feedback controllers, the influence of the boundary conditions on the stability of the traffic system is adequately described, and the effects of the unstable focuses and saddle points on the system are suppressed to slow down the traffic flow. In addition, the unstable bifurcation points can be eliminated and the Hopf bifurcation can be controlled to advance, delay, and disappear,so as to realize the control of the stability behavior of the traffic system, which can help to alleviate the traffic congestion and describe the actual traffic phenomena as well.

Analysis of complex intermittency in Boost converter from a bifurcation control viewpoint

Due to undesirable interference via unintended coupling paths, switching converters may exhibit complex intermittency, which appears as a form of bifurcation undergoing regular operation, subharmonics, and chaos orderly and repeatedly for a long period of time. Such intermittent operation, being an unwanted operating state, should normally be avoided in power converters. This paper expounds the mechanism and conditions for the emergence of intermittency in a common current-mode controlled Boost converter. It is found that interference at frequencies near the switching frequency or its rational multiples may induce intermittent operation. The strengths and frequencies of the interfering signals determine the type and period of intermittency. The problem is analyzed by transforming the time-bifurcation analysis to a conventional parameter-bifurcation analysis. Based on this transformation, intermittency can be investigated from the bifurcation control viewpoint. Furthermore, the critical circuit parameter conditions for the emergence of intermittency can be predicted and compared with those from circuit simulation.

Bifurcation control of a fractional-order delayed competition and cooperation model of two enterprises

The competition and cooperation among enterprises has become a hot topic and focus issue in today’s world. How to manage the enterprise well so as to achieve the maximum output is an important problem for enterprise managers. Optimizing output of two enterprises plays a key role in operating enterprises. Many scholars pay much attention to this aspect. However, the reports on the stability and Hopf bifurcation for fractional-order delayed competition and cooperation model of two enterprises are very few. This paper is concerned with the stability, the existence of Hopf bifurcation and the bifurcation control issue of fractionalorder delayed competition and cooperation model of two enterprises. Firstly, some new sufficient conditions that guarantee the stability and the existence of Hopf bifurcation for fractional-order delayed competition and cooperation model of two enterprises are obtained by regarding the delay as bifurcation parameter. Then a suitable time delayed feedback controller is designed to control the Hopf bifurcation for involved model. The study shows that the delay and the fractional order have an important effect on the stability and Hopf bifurcation of involved model. Some simulations justifying the validity of the derived analytical results are given. At last, we end this paper with a concise conclusion. The obtained results of this article are innovative and are of great significance in handling the competition and cooperation among enterprises.

Control of Codimension-2 Bautin Bifurcation in Chaotic Lü System

In this paper, we design a feedback controller, and analytically determine a control criterion so as to control the codimension-2 Bautin bifurcation in the chaotic Lfi system. According to the control criterion, we determine a potential Bautin bifurcation region （denoted by P） of the controlled system. This region contains the Bautin bifurcation region （denoted by Q） of the uncontrolled system as its proper subregion. The controlled system can exhibit Bautin bifurcation in P or its proper subregion provided the control gains are properly chosen. Particularly, we can control the appearance of Bautin bifurcation at any appointed point in the region P. Due to the relationship between Bantin bifurcation and Hopf bifurcation, the control scheme thereby is also viable for controlling the creation and stability of the Hopf bifurcation. In the controller, there are two terms： a linear term and a nonlinear cubic term. We show that the former determines the location of the Hopf bifurcation while the latter regulates its criticality. We also carry out numerical studies, and the simulation results confirm our analyticai predictions.

ROBUST CONTROL OF PERIODIC BIFURCATION SOLUTIONS

The topological bifurcation diagrams and the coefficients of bifurcation equation were obtained by C_L method.According to obtained bifurcation diagrams and combining control theory,the method of robust control of periodic bifurcation was presented,which differs from generic methods of bifurcation control.It can make the existing motion pattern into the goal motion pattern.Because the method does not make strict requirement about parametric values of the controller,it is convenient to design and make it.Numerical simulations verify validity of the method.

Hopf Bifurcation Analysis and Control for Electrical System of Military Vehicle

Electrical system of military vehicle is a typical parameterized nonlinear system where complicated bifurcations may exist and threaten its safe and stable operation. An algebraic criterion for Hopf bifurcation is presented briefly and applied to find Hopf bifurcation point of the electrical system with automatic voltage regulator(AVR) dynamics in military vehicle. Hopf bifurcation controllers are designed for this electrical system by using wash-out filter,linear feedback,nonlinear feedback and their combination. The linear feedback control makes the system bring Hopf bifurcation at preferable parameter,the nonlinear feedback control modifies the type of the bifurcation,and the wash-out filter enhances the system damping,thus,the Hopf bifurcation is eliminated and the electrical system stability is ensured. Simulation results show the controller's validity.

Bifurcation suppression of nonlinear systems via dynamic output feedback and its applications to rotating stall control

This paper deals with the problems of bifurcation suppression and bifurcation suppression with stability of nonlinear systems. Necessary conditions and sufficient conditions for bifurcation suppression via dynamic output feedback are presented; Sufficient conditions for bifurcation suppression with stability via dynamic output feedback are obtained. As an application, a dynamic compensator,which guarantees that the bifurcation point of rotating stall in axial flow compressors is stably suppressed, is constructed.

BIFURCATION CONTROL OF A PREDATOR-PREY MODEL BASED ON NUTRITION KINETICS

The existence conditions of Hopf bifurcation for a predator prey model based on nutri- tion kinetics are given. The two results may appear as follows： one is that the model has a stable periodic trajectory from Hopf bifurcation, which shows the system is in an eco- logical balance; the other is that periodic trajectory from Hopf bifurcation is unstable, which indicates the system is in a sharp or catastrophic loss of stability. For the latter, a bifurcation controller is designed to make the periodic trajectory stable. Finally, some simulations are carried out to prove the results.

WAMS-based monitoring and control of Hopf bifurcations in multi-machine power systems

A method is proposed to monitor and control Hopf bifurcations in multi-machine power systems using the information from wide area measurement systems (WAMSs). The power method (PM) is adopted to compute the pair of conjugate eigenvalues with the algebraically largest real part and the corresponding eigenvectors of the Jacobian matrix of a power system. The distance between the current equilibrium point and the Hopf bifurcation set can be monitored dynamically by computing the pair of con- jugate eigenvalues. When the current equilibrium point is close to the Hopf bifurcation set, the approximate normal vector to the Hopf bifurcation set is computed and used as a direction to regulate control parameters to avoid a Hopf bifurcation in the power system described by differential algebraic equations (DAEs). The validity of the proposed method is demonstrated by regulating the reactive power loads in a 14-bus power system.

Amplitude control of limit cycle in a van der Pol Duffing system

This paper applies washout filter technology to amplitude control of limit cycles emerging from Hopf bifurcation of the van der Pol-Duffing system. The controlling parameters for the appearance of Hopf bifurcation are given by the Routh-Hurwitz criteria. Noticeably, numerical simulation indicates that the controllers control the amplitude of limit cycles not only of the weakly nonlinear van der Pol-Duffing system but also of the strongly nonlinear van der Pol-Duffing system. In particular, the emergence of Hopf bifurcation can be controlled by a suitable choice of controlling parameters. Gain-amplitude curves of controlled systems are also drawn.

Solution and transcritical bifurcation of Burgers equation

Burgers equation is reduced into a first-order ordinary differential equation by using travelling wave transformation and it has typical bifurcation characteristics. We can obtain many exact solutions of the Burgers equation, discuss its transcritical bifurcation and control dynamical behaviours by extending the stable region. The transcritical bifurcation exists in the （2 ＋ 1）-dimensional Burgers equation.

Hybrid Control of Delay Induced Hopf Bifurcation of Dynamical Small-World Network

In this paper, we focus on the Hopf bifurcation control of a small-world network model with time-delay. With emphasis on the relationship between the Hopf bifurcation and the time-delay, we investigate the effect of time-delay by choosing it as the bifurcation parameter. By using tools from control and bifurcation theory, it is proved that there exists a critical value of time-delay for the stability of the model. When the time-delay passes through the critical value, the model loses its stability and a Hopf bifurcation occurs. To enhance the stability of the model, we propose an improved hybrid control strategy in which state feedback and parameter perturbation are used. Through linear stability analysis, we show that by adjusting the control parameter properly, the onset of Hopf bifurcation of the controlled model can be delayed or eliminated without changing the equilibrium point of the model. Finally, numerical simulations are given to verify the theoretical analysis. © 2017, Shanghai Jiaotong University and Springer-Verlag Berlin Heidelberg.

Principle of designing slope compensation in PFC Boost converter

Due to wide input fluctuation with line frequency of 50 Hz, power-factor-correction （PFC） Boost converters tend to exhibit fast-scale instability over time domain. The traditional remedy is to impose slope compensation so as to weaken or eliminate this instability. A theoretical principle on the implementation of slope compensation signal is still lacking. Empirical design will induce over compensation frequently, resulting in a large decrease of power factor. In order to tackle this issue, by constructing the discrete-time iterative map of the PFC Boost converter from the viewpoint of bifurcation control theory of nonlinear systems, consequently, the criterion of critical stability for the PFC circuit can be established. Based on this stability criterion, appropriate design of slope compensation can be achieved. Our work indicates that 3 main circuit parameters （i.e. switching cycle, output reference voltage and inductor） determine the effective amplitude design of the slope compensation signal. The results, validated by a large quantity of analytical and numerical studies, show that appropriate slope compensation can be effective in weakening （or controlling） fast-scale bifurcation while maintaining a rather high input power factor.