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.

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.

scholarly journals Homoclinic Bifurcation and Chaos in a Noise-InducedΦ6Potential

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Gen Ge ◽  
Zhi Wen Zhu ◽  
Jia Xu
Keyword(s):  
White Noise ◽  
Gaussian White Noise ◽  
Nonlinear Damping ◽  
Harmonic Excitation ◽  
Simple Zero ◽  
Necessary Condition ◽  
Bounded Noise ◽  
Basin Boundaries ◽  
Zero Points ◽  
Safe Basin

The present paper focuses on the noise-induced chaos in aΦ6oscillator with nonlinear damping. Based on the stochastic Melnikov approach, simple zero points of the stochastic Melnikov integral theoretically mean the necessary condition causing noise-induced chaotic responses in the system. To quantify the noise-induced chaos, the Poincare maps and fractal basin boundaries are constructed to show how the system's motions change from a periodic way to chaos or from random motions to random chaos as the amplitude of the noise increases. Three cases are considered in simulating the system; that is, the system is excited only by the harmonic excitation, by both the harmonic and the Gaussian white noise excitations, or by both the bounded noise and the Gaussian white noise excitations. The results show that chaotic attractor is diffused by the noises. The larger the noise intensity is, the more diffused attractor it results in. And the boundary of the safe basin can also be fractal if the system is excited by the noises. The erosion of the safe basin can be aggravated when the frequency disturbing parameter of the bounded noise or the amplitude of the Gaussian white noise excitation is increased.

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.

Response Analysis of van der Pol Vibro-Impact System with Coulomb Friction Under Gaussian White Noise

2018 ◽  
Vol 28 (13) ◽  
pp. 1830043 ◽  
Author(s):  
Meng Su ◽  
Wei Xu ◽  
Guidong Yang
Keyword(s):  
White Noise ◽  
Coulomb Friction ◽  
Gaussian White Noise ◽  
Original System ◽  
Stochastic Averaging ◽  
Response Analysis ◽  
State Variables ◽  
Van Der Pol ◽  
Impact System ◽  
The Impact

In this paper, the stationary response of a van der Pol vibro-impact system with Coulomb friction excited by Gaussian white noise is studied. The Zhuravlev nonsmooth transformation of the state variables is utilized to transform the original system to a new system without the impact term. Then, the stochastic averaging method is applied to the equivalent system to obtain the stationary probability density functions (pdfs). The accuracy of the analytical results obtained from the proposed procedure is verified by those from the Monte Carlo simulation based on the original system. Effects of different damping coefficients, restitution coefficients, amplitudes of friction and noise intensities on the response are discussed. Additionally, stochastic P-bifurcations are explored.

Noise and Contact in Dynamic AFM Operations

2011 ◽  
Author(s):  
Ishita Chakraborty ◽  
Balakumar Balachandran
Keyword(s):  
White Noise ◽  
Evolution Equations ◽  
Stochastic Dynamics ◽  
Gaussian White Noise ◽  
Excitation Frequency ◽  
Period Doubling ◽  
Harmonic Excitation ◽  
Dynamic Mode ◽  
Grazing Impacts ◽  
Representative System

In this article, the authors study the effects of Gaussian white noise on the dynamics of an atomic force microscope (AFM) cantilever operating in a dynamic mode by using a combination of numerical and analytical efforts. As a representative system, a combination of Si cantilever and HOPG sample is used. The focus of this study is on understanding the stochastic dynamics of a micro-cantilever, when the excitation frequencies are away from the first natural frequency of the system. In the previous efforts of the authors, period-doubling bifurcations close to grazing impacts have been reported for micro-cantilevers when the excitation frequency is in between the first and the second natural frequencies of the system. In the present study, it is observed that the addition of Gaussian white noise along with a harmonic excitation produces a near-grazing contact, when there was previously no contact between the tip and the sample with only the harmonic excitation. Moment evolution equations derived from a Fokker-Planck system are used to obtain numerical results, which support the statement that the addition of noise facilitates contact between the tip and the sample.

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