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

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.

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.

scholarly journals Stochastic Bifurcations of a Fractional-Order Vibro-Impact System Driven by Additive and Multiplicative Gaussian White Noises

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Yong-Ge Yang ◽  
Ya-Hui Sun ◽  
Wei Xu
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Stochastic Systems ◽  
Original System ◽  
Stochastic Averaging ◽  
Impact Factors ◽  
Stochastic Bifurcation ◽  
Fractional Order Systems ◽  
Impact System ◽  
White Noises

Stochastic fractional-order systems or stochastic vibro-impact systems can present rich dynamical behaviors, and lots of studies dealing with stochastic fractional-order systems or stochastic vibro-impact systems are available now, while the discussion on the stochastic systems with both vibro-impact factors and fractional derivative element is rare. This paper is concerned with the stochastic bifurcation of a fractional-order vibro-impact system driven by additive and multiplicative Gaussian white noises. Firstly, we can remove the discontinuity of the original system with the help of nonsmooth transformation and obtain the equivalent stochastic system. Then, we adopt the stochastic averaging method to get the approximately analytical solutions. At last, an example is discussed in detail to assess the reliability of the developed approach. We also find that the coefficient of restitution factor, fractional derivative coefficient, and fractional derivative order can induce the stochastic bifurcation.

scholarly journals Response of Duffing Oscillator with Time Delay Subjected to Combined Harmonic and Random Excitations

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
D. N. Hao ◽  
N. D. Anh
Keyword(s):  
Time Delay ◽  
Probability Density ◽  
Duffing Oscillator ◽  
Original System ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Auxiliary Function ◽  
Equivalent Linearization ◽  
Stationary Probability ◽  
Stationary Probability Density

This paper aims to investigate the stationary probability density functions of the Duffing oscillator with time delay subjected to combined harmonic and white noise excitation by the method of stochastic averaging and equivalent linearization. By the transformation based on the fundamental matrix of the degenerate Duffing system, the paper shows that the displacement and the velocity with time delay in the Duffing oscillator can be computed approximately in non-time delay terms. Hence, the stochastic system with time delay is transformed into the corresponding stochastic non-time delay equation in Ito sense. The approximate stationary probability density function of the original system can be found by combining the stochastic averaging method, the equivalent linearization method, and the technique of auxiliary function. The response of Duffing oscillator is investigated. The analytical results are verified by numerical simulation results.

scholarly journals Stochastic bifurcation analysis in Brusselator system with white noise

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Changzhao Li ◽  
Juan Zhang
Keyword(s):  
White Noise ◽  
Averaging Method ◽  
Planck Equation ◽  
Original System ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Stochastic Bifurcation ◽  
Largest Lyapunov Exponent ◽  
Singular Boundary ◽  
Polar Coordinate

Abstract In this paper, we mainly study the stochastic stability and stochastic bifurcation of Brusselator system with multiplicative white noise. Firstly, by a polar coordinate transformation and a stochastic averaging method, the original system is transformed into an Itô averaging diffusion system. Secondly, we apply the largest Lyapunov exponent and the singular boundary theory to analyze the stochastic local and global stability. Thirdly, by means of the properties of invariant measures, the stochastic dynamical bifurcations of stochastic averaging Itô diffusion equation associated with the original system is considered. And we investigate the phenomenological bifurcation by analyzing the associated Fokker–Planck equation. We will show that, from the view point of random dynamical systems, the noise “destroys” the deterministic stability. Finally, an example is given to illustrate the effectiveness of our analyzing procedure.

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.

Bifurcation Analysis of a Vibro-Impact Viscoelastic Oscillator with Fractional Derivative Element

2018 ◽  
Vol 28 (14) ◽  
pp. 1850170 ◽  
Author(s):  
Yong-Ge Yang ◽  
Wei Xu ◽  
YangQuan Chen ◽  
Bingchang Zhou
Keyword(s):  
Fractional Derivative ◽  
Numerical Solutions ◽  
Gaussian White Noise ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Stochastic Bifurcation ◽  
Damping Force ◽  
Impact Oscillator ◽  
Viscoelastic Parameters ◽  
Viscoelastic Term

To the best of authors’ knowledge, little work has been focused on the noisy vibro-impact systems with fractional derivative element. In this paper, stochastic bifurcation of a vibro-impact oscillator with fractional derivative element and a viscoelastic term under Gaussian white noise excitation is investigated. First, the viscoelastic force is approximately replaced by damping force and stiffness force. Thus the original oscillator is converted to an equivalent oscillator without a viscoelastic term. Second, the nonsmooth transformation is introduced to remove the discontinuity of the vibro-impact oscillator. Third, the stochastic averaging method is utilized to obtain analytical solutions of which the effectiveness can be verified by numerical solutions. We also find that the viscoelastic parameters, fractional coefficient and fractional derivative order can induce stochastic bifurcation.

scholarly journals Stationary Response of a Class of Nonlinear Stochastic Systems Undergoing Markovian Jumps

2015 ◽  
Vol 82 (5) ◽  
Author(s):  
Rong-Hua Huan ◽  
Wei-qiu Zhu ◽  
Fai Ma ◽  
Zu-guang Ying
Keyword(s):  
Stochastic Systems ◽  
Original System ◽  
Stochastic Averaging ◽  
Kolmogorov Equation ◽  
Markovian Jump ◽  
Nonlinear Stochastic Systems ◽  
Stationary Response ◽  
Set Up ◽  
Jump Systems ◽  
Stationary Probabilities

Systems whose specifications change abruptly and statistically, referred to as Markovian-jump systems, are considered in this paper. An approximate method is presented to assess the stationary response of multidegree, nonlinear, Markovian-jump, quasi-nonintegrable Hamiltonian systems subjected to stochastic excitation. Using stochastic averaging, the quasi-nonintegrable Hamiltonian equations are first reduced to a one-dimensional Itô equation governing the energy envelope. The associated Fokker–Planck–Kolmogorov equation is then set up, from which approximate stationary probabilities of the original system are obtained for different jump rules. The validity of this technique is demonstrated by using a nonlinear two-degree oscillator that is stochastically driven and capable of Markovian jumps.

Nonlinear vibrations of suspended cables—Part III: Random excitation and interaction with fluid flow

2004 ◽  
Vol 57 (6) ◽  
pp. 515-549 ◽  
Author(s):  
Raouf A Ibrahim
Keyword(s):  
Fluid Flow ◽  
Spectral Density ◽  
Duffing Oscillator ◽  
Random Excitation ◽  
Review Article ◽  
Stochastic Bifurcation ◽  
Inertia Force ◽  
Flowing Fluid ◽  
Complex Response ◽  
Suspended Cables

This review article deals with the random excitation of nonlinear strings and suspended cables in air and fluid flow. For strings and 1D cables, the system dynamics is governed by different forms of Duffing oscillator. A brief review is devoted to the stochastic excitation of a Duffing oscillator. Under random excitation, this oscillator may or may not possess multiple solutions depending on the excitation bandwidth and level. One may be interested in estimating response statistics, first passage problem, and power spectral density. Particular attention is given to the complex response phenomena associated with increasing the spectral density level of excitation. The numerical results of the problem of nonlinear modal interaction in suspended cables will be discussed in the neighborhood of multiple internal resonance conditions. For a unimodal response, the linear theory fails to predict nonzero mean response and underestimates the mean square response under white noise excitation. Complex response phenomena such as “on-off” intermittency, energy transfer, and stochastic bifurcation are reviewed. The dynamic behavior of suspended cables in still air is different from that in flowing fluid or severe wind current due to the action of vortices, fluid normal forces, added fluid inertia force, and fluid drag force. Aeolian and galloping vibration of suspended cables in air and their dynamics in fluid flow are discussed, together with the influence of dynamic tension. In the absence of external excitation, the action of fluid forces induces vibration to the cable. The dynamics of cables subjected to steady and random fluid flow is reviewed for mooring systems. Depending on the flow speed, the cable may experience divergence or flutter similar to the case of aeroelastic structures. While the deterministic theory of strings and cables has reached an advanced stage, the reader will realize that these systems need further investigations under random excitations. There are 297 references cited in this review article.

Data assimilation for a multiscale stochastic dynamical system with Gaussian noise

2019 ◽  
Vol 19 (03) ◽  
pp. 1950019
Author(s):  
Yanjie Zhang ◽  
Jian Ren
Keyword(s):  
Dynamical System ◽  
Data Assimilation ◽  
Energy Method ◽  
Gaussian Noise ◽  
Original System ◽  
Stochastic Averaging ◽  
Dimensional System ◽  
Mean Square ◽  
Stochastic Dynamical System ◽  
Lower Dimensional

This paper is devoted to studying dimensional reduction for slow-fast data assimilation driven by Gaussian noise via stochastic averaging. We apply an energy method to show that the probability density for the reduced lower-dimensional system approximates that for the original system in mean square. In other words, the reduced system filter thus effectively captures the filter of the original system.

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