The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with t...The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance, it is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node （SN） or Hopf bifurcation （HB） as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.展开更多
The dynamical behavior of the extended Duffing-Van der Pol oscillator is investigated numerically in detail. With the aid of some numerical simulation tools such as bifurcation diagrams and Poinearé maps, the dif...The dynamical behavior of the extended Duffing-Van der Pol oscillator is investigated numerically in detail. With the aid of some numerical simulation tools such as bifurcation diagrams and Poinearé maps, the different routes to chaos and various shapes of strange attractors are observed. To characterize chaotic behavior of this oscillator system, the spectrum of Lyapunov exponent and Lyapunov dimension are also employed.展开更多
基金Project supported by the Scientific Research Foundation for Returned Overseas Chinese Scholar of Ministry of Eduction, China (No.2006-331)
文摘The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance, it is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node （SN） or Hopf bifurcation （HB） as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.
基金supported by the National Natural Science Foundation of China under Grant No.10875078the Natural Science Foundation of Zhejiang Province of China under Grant No.Y7080455
文摘The dynamical behavior of the extended Duffing-Van der Pol oscillator is investigated numerically in detail. With the aid of some numerical simulation tools such as bifurcation diagrams and Poinearé maps, the different routes to chaos and various shapes of strange attractors are observed. To characterize chaotic behavior of this oscillator system, the spectrum of Lyapunov exponent and Lyapunov dimension are also employed.
