On the response of the Van der Pol oscillator to white noise excitation

1987 ◽  
Vol 117 (3) ◽  
pp. 421-431 ◽  
Author(s):  
W.-Q. Zhu ◽  
J.-S. Yu
Keyword(s):  
White Noise ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
White Noise Excitation

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

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Guoqi Zhang ◽  
Feng Wang ◽  
Yuancen Wang
Keyword(s):  
Time Delay ◽  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Van Der Pol Oscillator ◽  
Order System ◽  
Damping Force ◽  
Van Der Pol ◽  
Restoring Force ◽  
White Noise Excitation

Abstract 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.

On using block pulse transform to perform equivalent linearization for a nonlinear Van der Pol oscillator under stochastic excitation

2016 ◽  
Vol 0 (0) ◽  
Author(s):  
Amir Younespour ◽  
Hosein Ghaffarzadeh
Keyword(s):  
Nonlinear System ◽  
Present Method ◽  
Gaussian White Noise ◽  
Van Der Pol Oscillator ◽  
Equivalent Linearization ◽  
Van Der Pol ◽  
White Noise Excitation ◽  
Linearization Methods ◽  
The Mean ◽  
Methods Numerical

AbstractThis paper applied the idea of block pulse (BP) transform in the equivalent linearization of a nonlinear system. The BP transform gives effective tools to approximate complex problems. The main goal of this work is on using BP transform properties in process of linearization. The accuracy of the proposed method compared with the other equivalent linearization including the stochastic equivalent linearization and the regulation linearization methods. Numerical simulations are applied to the nonlinear Van der Pol oscillator system under Gaussian white noise excitation to demonstrate the feasibility of the present method. Different values of nonlinearity are considered to show the effectiveness of the present method. Besides, by comparing the mean-square responses for divers values of nonlinearity and excitation intensity depicted the present method is able to approximate the behavior of nonlinear system and is in agreement with other methods.

A New Method for Suppressing Periodic Narrowband Interference Based on the Chaotic van der Pol Oscillator

2015 ◽  
Vol 25 (09) ◽  
pp. 1550120 ◽  
Author(s):  
Jia Lu ◽  
Xiaoxing Zhang ◽  
Hao Xiong
Keyword(s):  
Wavelet Analysis ◽  
White Noise ◽  
High Sensitivity ◽  
Van Der Pol Oscillator ◽  
Chaotic Oscillator ◽  
Narrowband Interference ◽  
Van Der Pol ◽  
Noise Interference ◽  
Van Der Pol Oscillators ◽  
Interference Signals

The chaotic van der Pol oscillator is a powerful tool for detecting defects in electric systems by using online partial discharge (PD) monitoring. This paper focuses on realizing weak PD signal detection in the strong periodic narrowband interference by using high sensitivity to the periodic narrowband interference signals and immunity to white noise and PD signals of chaotic systems. A new approach to removing the periodic narrowband interference by using a van der Pol chaotic oscillator is described by analyzing the motion characteristic of the chaotic oscillator on the basis of the van der Pol equation. Furthermore, the Floquet index for measuring the amplitude of periodic narrowband signals is redefined. The denoising signal processed by the chaotic van der Pol oscillators is further processed by wavelet analysis. Finally, the denoising results verify that the periodic narrowband and white noise interference can be removed efficiently by combining the theory of the chaotic van der Pol oscillator and wavelet analysis.

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

2021 ◽  
pp. 125-125
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Qixun Lan ◽  
Yujie Cai ◽  
Huafeng Xu ◽  
...  
Keyword(s):  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Original System ◽  
Stochastic Averaging ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Fractional Damping ◽  
Transition Set ◽  
Fractional Order Controller

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.

Stochastic Response of a Vibro-Impact System by Path Integration Based on Generalized Cell Mapping Method

2014 ◽  
Vol 24 (10) ◽  
pp. 1450129 ◽  
Author(s):  
Chao Li ◽  
Wei Xu ◽  
Xiaole Yue
Keyword(s):  
White Noise ◽  
Mapping Method ◽  
Cell Mapping ◽  
Van Der Pol ◽  
Generalized Cell Mapping ◽  
White Noise Excitation ◽  
Impact System ◽  
Steady State Responses ◽  
State Responses ◽  
Transient And Steady State

The generalized cell mapping method is extended to study the response of a vibro-impact system with white noise excitation. The transient and steady-state responses of a Duffing–van der Pol vibro-impact system under white noise excitation are obtained by using this method. The accuracy of the method is verified by comparison with Monte Carlo simulation results. In addition, stochastic P-bifurcation for different parameters is considered, and several special forms are observed in this paper.

EFFECT OF GAUSSIAN WHITE NOISE ON THE DYNAMICAL BEHAVIORS OF AN EXTENDED DUFFING–VAN DER POL OSCILLATOR

2006 ◽  
Vol 16 (09) ◽  
pp. 2587-2600 ◽  
Author(s):  
XIAOLI YANG ◽  
WEI XU ◽  
ZHONGKUI SUN
Keyword(s):  
White Noise ◽  
Lyapunov Exponents ◽  
Random Noise ◽  
Gaussian White Noise ◽  
Harmonic Excitation ◽  
Basins Of Attraction ◽  
Van Der Pol Oscillator ◽  
Necessary Condition ◽  
Van Der Pol ◽  
Dynamical Behaviors

The influence induced by random noise on dynamical behaviors is a classical yet challenging subject. This paper discusses the influence of Gaussian white noise on the dynamics of a self-excited triple well extended Duffing–Van der Pol oscillator already subjected to harmonic excitation. Firstly, the condition for the rise of hom/heteroclinic chaos is derived by random Melnikov's technique under its corresponding mean-square criterion and the result indicates that the threshold amplitude of harmonic excitation is lowered by the appearance of Gaussian white noise. Moreover, the threshold is decreased as the noise intensity increases. Since the Melnikov's criterion is only a necessary condition for the occurrence of chaotic motion, this prediction is tested against numerical simulations of the basins of attraction and the Lyapunov exponents. By vanishing the largest Lyapunov exponents, another criterion for the onset of chaos is obtained which is accorded with the theoretical one. Finally, how the noise effects the structure of periodic or chaotic attractor is investigated by simulating Poincare maps of the original system and rich transition states displayed by the considered extended Duffing–Van der Pol oscillator are observed.

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