Stochastic Hopf bifurcation of quasi-integrable Hamiltonian systems with multi-time-delayed feedback control and wide-band noise excitations

2012 ◽  
Vol 69 (3) ◽  
pp. 935-947 ◽  
Author(s):  
Z. H. Liu ◽  
W. Q. Zhu
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Hamiltonian Systems ◽  
Wide Band ◽  
Delayed Feedback Control ◽  
Integrable Hamiltonian Systems ◽  
Band Noise ◽  
Stochastic Hopf Bifurcation ◽  
Wide Band Noise ◽  
Time Delayed Feedback

Stationary Response of Duffing-van der Pol Oscillator with Delayed Feedback Control under Wide-Band Noise Excitation

2013 ◽  
Vol 300-301 ◽  
pp. 1518-1524
Author(s):  
Peng Lin ◽  
Chang Shui Feng ◽  
Qiao Yi Wang
Keyword(s):  
Feedback Control ◽  
Wide Band ◽  
Delayed Feedback ◽  
Van Der Pol Oscillator ◽  
Delayed Feedback Control ◽  
Control Force ◽  
Van Der Pol ◽  
Stationary Response ◽  
Wide Band Noise ◽  
Time Delayed Feedback

The stationary response of Duffing-van der Pol oscillator with time-delayed feedback control under wide-band noise excitations is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method. Finally, the stationary response of the system is obtained by solving the Fokker-Plank-Kolmogorov equation associated with the averaged Itô equation. The effect of time delay in feedback control force on the response is analyzed. The theoretical results are well verified through digital simulation.

STOCHASTIC HOPF BIFURCATION OF QUASI-INTEGRABLE HAMILTONIAN SYSTEMS WITH FRACTIONAL DERIVATIVE DAMPING

2012 ◽  
Vol 22 (04) ◽  
pp. 1250083 ◽  
Author(s):  
F. HU ◽  
W. Q. ZHU ◽  
L. C. CHEN
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Derivative ◽  
Hamiltonian Systems ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Bifurcation Parameter ◽  
Integrable Hamiltonian Systems ◽  
Fractional Derivative Damping ◽  
Stochastic Hopf Bifurcation

The stochastic Hopf bifurcation of multi-degree-of-freedom (MDOF) quasi-integrable Hamiltonian systems with fractional derivative damping is investigated. First, the averaged Itô stochastic differential equations for n motion integrals are obtained by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, an expression for the average bifurcation parameter of the averaged system is obtained and a criterion for determining the stochastic Hopf bifurcation of the system by using the average bifurcation parameter is proposed. An example is given to illustrate the proposed procedure in detail and the numerical results show the effect of fractional derivative order on the stochastic Hopf bifurcation.

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