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

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

Advances in Difference Equations ◽

10.1186/s13662-019-2356-1 ◽

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.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418300434 ◽

2018 ◽

Vol 28 (13) ◽

pp. 1830043 ◽

Cited By ~ 3

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.

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Stochastic P-Bifurcation of a Bistable Viscoelastic Beam with Fractional Constitutive Relation under Gaussian White Noise

Discrete Dynamics in Nature and Society ◽

10.1155/2018/6935095 ◽

2018 ◽

Vol 2018 ◽

pp. 1-10

Author(s):

Yajie Li ◽

Zhiqiang Wu ◽

Guoqi Zhang ◽

Feng Wang

Keyword(s):

White Noise ◽

Minimum Mean Square Error ◽

Gaussian White Noise ◽

Singularity Theory ◽

Original System ◽

Viscoelastic Beam ◽

Damping Force ◽

Restoring Force ◽

Transition Set ◽

Axially Moving

In this paper, we study the stochastic P-bifurcation problem for axially moving of a bistable viscoelastic beam with fractional derivatives of high order nonlinear terms under Gaussian white noise excitation. First, using the principle for minimum mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be simplified to an equivalent system. Second, we obtain the stationary Probability Density Function (PDF) of the system’s amplitude by stochastic averaging, and using singularity theory, we find the critical parametric condition for stochastic P-bifurcation of amplitude of the system. Finally, we analyze the types of the stationary PDF curves of the system qualitatively by choosing parameters corresponding to each region within the transition set curve. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order viscoelastic material model to adjust the response of the system.

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Stochastic bifurcation analysis of a bistable Duffing oscillator with fractional damping under multiplicative noise excitation

Thermal Science ◽

10.2298/tsci200210040l ◽

2021 ◽

pp. 40-40

Author(s):

Yajie Li ◽

Zhiqiang Wu ◽

Qixun Lan ◽

Yujie Cai ◽

Huafeng Xu ◽

...

Keyword(s):

Multiplicative Noise ◽

Duffing Oscillator ◽

Singularity Theory ◽

Original System ◽

Stochastic Averaging ◽

Stochastic Bifurcation ◽

Fractional Damping ◽

Derivative Term ◽

Restoring Forces ◽

Fractional Order Controller

The stochastic P-bifurcation behavior of bi-stability in a Duffing oscillator with fractional damping under multiplicative noise excitation is investigated. Firstly, in order to consider the influence of Duffing term, the nonlinear stiffness can be equivalent to a linear stiffness which is a function of the system amplitude, and then, using the principle of minimal mean square error, the fractional derivative term can be equivalent to a linear combination of damping and restoring forces, thus, the original system is simplified to an equivalent integer order Duffing 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, and the method used in this paper can directly guide the design of the fractional order controller to adjust the behaviors of the system.

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Stochastic P-bifurcation in a generalized Van der Pol oscillator with fractional delayed feedback excited by combined Gaussian white noise excitations

Journal of Low Frequency Noise Vibration and Active Control ◽

10.1177/1461348419878534 ◽

2019 ◽

pp. 146134841987853 ◽

Cited By ~ 1

Author(s):

Yajie Li ◽

Zhiqiang Wu ◽

Feng Wang ◽

Guoqi Zhang ◽

Yuancen Wang

Keyword(s):

White Noise ◽

Gaussian White Noise ◽

Delayed Feedback ◽

Van Der Pol Oscillator ◽

Van Der Pol

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127406016276 ◽

2006 ◽

Vol 16 (09) ◽

pp. 2587-2600 ◽

Cited By ~ 1

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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