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 t...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.展开更多
With noncritical eigenvalues assumed to be campletely controllable stability and linear feedback stabiliz-ablity probems are attacked by the center manifold method, and a procedure had been established for the constru...With noncritical eigenvalues assumed to be campletely controllable stability and linear feedback stabiliz-ablity probems are attacked by the center manifold method, and a procedure had been established for the construction of linear feedback stabilizing law on the basis of noncritical eigenvalue assignment.展开更多
The problem of state feedback stabilization of control systems is tackled in this paper.The stabilizability of the plane control systems with critical spectrum is considered,By using therecently developed center manif...The problem of state feedback stabilization of control systems is tackled in this paper.The stabilizability of the plane control systems with critical spectrum is considered,By using therecently developed center manifold method combined with Liapunov second method,it is shownthat in all critical cases the above mentioned systems are state feedback stabilizable.展开更多
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 ...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 reduction 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 bifurcations 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.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.10532050,10702024)the Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20070287029)
文摘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.
文摘With noncritical eigenvalues assumed to be campletely controllable stability and linear feedback stabiliz-ablity probems are attacked by the center manifold method, and a procedure had been established for the construction of linear feedback stabilizing law on the basis of noncritical eigenvalue assignment.
文摘The problem of state feedback stabilization of control systems is tackled in this paper.The stabilizability of the plane control systems with critical spectrum is considered,By using therecently developed center manifold method combined with Liapunov second method,it is shownthat in all critical cases the above mentioned systems are state feedback stabilizable.
基金Supported by the National Natural Science Foundation of China (Grant Nos.10532050)Supported by the National Natural Science Foundation of China (Grant Nos.10702024)the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No.20070287029)
文摘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 reduction 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 bifurcations 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.
