This paper presents both analytical and numerical studies on the global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback.Based on a detailed analysis of the stability switches of the trivial equilibrium of the system,the stability charts are given in a parameter space consisting of the time delay and the feedback gains.The center manifold reduc-tion and the normal form method are used to study Hopf bifurcations with respect to the time delay.To gain an insight into the persistence of a Hopf bifurcation as the time delay varies farther away from its critical value,the method of multiple scales is used to obtain the global view of Hopf bifurcations with respect to the time delay.Both the analytical results of Hopf bifurca-tions and global view of those bifurcations are validated via a collocation scheme implemented on DDE-Biftool.The most important discovery in this paper is the well-structured global view of Hopf bifurcations for the system of concern,showing the generality of the persistence of Hopf bifurcations.
ZHANG Li1,WANG HuaiLei1 & HU HaiYan1,2 1 MOE Key Lab of Mechanics and Control of Aerospace Structures,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China
Time delays in the feedback control often dete-riorate the control performance or even cause the instabilityof a dynamic system.This paper presents a control strategyfor the dynamic system with a constant or a slowly time-varying input delay based on a transformation,which sim-plifies the time-delay system into a delay-free one.Firstly,the relation is discussed for two existing reduction-basedlinear quadratic controls.One is continuous and the otheris discrete.By extending the relation,a new reduction-basedcontrol is then developed with a numerical algorithm pre-sented for practical control implementation.The controllersuggested by the proposed method has such a promisingproperty that it can be used for the cases of different values ofan input time delay without redesign of controller.This prop-erty provides the potential for stabilizing the dynamic systemwith a time-varying input delay.Consequently,the applica-tion of the proposed method to the dynamic system with aslowly time-varying delay is discussed.Finally,numericalsimulations are given to show the efficacy and the applica-bility of the method.
