Primary Resonance of van der Pol Oscillator under Fractional-Order Delayed Feedback and Forced Excitation

Chen, Jufeng; Li, Xianghong; Tang, Jianhua; Liu, Yafeng

doi:10.1155/2017/5975329

Cited by 23 publications

(8 citation statements)

References 36 publications

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“…Huang ( 2018) designed a nonlinear time delayed feedback controller for the proposed Van der Pol oscillator to control the dynamics and the obtained stability. Chen et al (2017) presented the fractional-order delayed negative velocity feedback for Van der Pol oscillator with primary resonance and forced excitation. This study was illustrated analytically using the averaging method.…”

Section: Introductionmentioning

confidence: 99%

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Resonance behavior in coupled Van der Pol harmonic oscillators with controllers and delayed feedback

EL-Sayed

2020

Journal of Vibration and Control

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It has been revealed in the proposed work that a pair of delay positive position feedback control can lessen the vibration response of double Van der Pol oscillators with external forces. We also studied the effects of both the control and the delayed feedback signal gains to illustrate the low vibration amplitudes. The averaging perturbation process has been used to consider the frequency-response equations of amplitudes and modulation phases at the primary resonance and one-to-one internal resonances. According to the perturbation solutions for the four-degrees-of-freedom system, we presented the frequency response curves that were periodic in the time delays. The stability analysis presented in this study has shown optimum stable ranges. If the time delays increase, the steady-state amplitudes of the oscillator’s system will periodically result in few stable regions and more unstable ones. The numerical simulation has been introduced to check the analytical approximation. It was also found to be almost identical after presenting the comparison of the results.

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Section: Introductionmentioning

confidence: 99%

“…Chen et al (2017) presented the fractional-order delayed negative velocity feedback for Van der Pol oscillator with primary resonance and forced excitation. This study was illustrated analytically using the averaging method.…”

Section: Introductionmentioning

confidence: 99%

Resonance behavior in coupled Van der Pol harmonic oscillators with controllers and delayed feedback

EL-Sayed

2020

Journal of Vibration and Control

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“…Eqs. (19) and (20) show that da doesn't depend on  , the averaged Itô equation of () at is independent of () t  and that the random process () at is a one-dimensional diffusion process. Thus, the correspondingly reduced Fokker-Planck-Kolmogorov (FPK) equation of () at can be written as:…”

Section: Stationary Pdf Of the System Amplitudementioning

confidence: 99%

“…Liu et al investigated a Duffing oscillator system with fractional damping under combined harmonic and Poisson white noise parametric excitation, and then the asymptotic Lyapunov stability with probability of the original system is analyzed based on the largest Lyapunov exponent [18]. Chen et al studied the primary resonance response of a Van der Pol system under fractional-order delayed negative feedback and forced excitation, and obtained the approximate analytical solution based on the averaging method [19]. Chen et al proposed a stochastic averaging technique which can be used to study the randomly excited strongly nonlinear system with delayed feedback fractional-order proportional-derivative controller, and obtained the stationary PDF of the system [20].…”

Section: Introductionmentioning

confidence: 99%

Stochastic transition behaviors in a generalized Van der Pol oscillator with fractional damping under Gaussian white noise

Lan

et al. 2021

THERM SCI

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The stochastic P-bifurcation behavior of bi-stability in a generalized Van der Pol oscillator with a fractional damping under multiplicative Gaussian white noise excitation is investigated. Firstly, using the principle of minimal mean square error, the nonlinear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order Van der Pol system. Secondly, the system amplitude?s stationary Probability Density Function (PDF) is obtained by stochastic averaging. And then according to the singularity theory, the critical parametric conditions for the system amplitude?s stochastic P-bifurcation are found. Finally, the types of the system?s stationary PDF curves of amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical results and the numerical results obtained from Monte Carlo simulation verifies the theoretical analysis in this paper and the method used in this paper can directly guide the design of the fractional order controller to adjust the response of the system.

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“…Zhou et al studied the dynamic stability and the Hopf bifurcation of a paddy ecosystem, obtained the necessary stability conditions for the system by analyzing its characteristic equation [38]. Chen et al studied the primary resonance response of a Van der Pol system under fractional-order delayed negative feedback and forced excitation, obtained the approximate analytical solution for the system based on the averaging method [39]. Leung et al analyzed a Van der Pol-Duffing oscillator with fractional derivatives and time delays based on the residue harmonic method and examined its periodic bifurcations using the fractional order, time delay, and feedback gain as continuation parameters [40].…”

Section: Introductionmentioning

confidence: 99%

Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

Zhang

et al. 2019

Adv Differ Equ

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The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

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Primary Resonance of van der Pol Oscillator under Fractional-Order Delayed Feedback and Forced Excitation

Cited by 23 publications

References 36 publications

Resonance behavior in coupled Van der Pol harmonic oscillators with controllers and delayed feedback

Resonance behavior in coupled Van der Pol harmonic oscillators with controllers and delayed feedback

Stochastic transition behaviors in a generalized Van der Pol oscillator with fractional damping under Gaussian white noise

Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

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