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.

Stochastic Bifurcations of a Fractional-Order Vibro-Impact Oscillator Subjected to Colored Noise Excitation

2021 ◽  
Vol 31 (12) ◽  
pp. 2150177
Author(s):  
Ya-Hui Sun ◽  
Yong-Ge Yang ◽  
Ling Hong ◽  
Wei Xu
Keyword(s):  
Fractional Order ◽  
Colored Noise ◽  
Numerical Solutions ◽  
Gaussian White Noise ◽  
Research Work ◽  
Critical Conditions ◽  
Stochastic Bifurcation ◽  
Response Analysis ◽  
Impact Oscillator ◽  
Stochastic Bifurcations

A stochastic vibro-impact system has triggered a consistent body of research work aimed at understanding its complex dynamics involving noise and nonsmoothness. Among these works, most focus is on integer-order systems with Gaussian white noise. There is no report yet on response analysis for fractional-order vibro-impact systems subject to colored noise, which is presented in this paper. The biggest challenge for analyzing such systems is how to deal with the fractional derivative of absolute value functions after applying nonsmooth transformation. This problem is solved by introducing the Fourier transformation and deriving the approximate probabilistic solution of the fractional-order vibro-impact oscillator subject to colored noise. The reliability of the developed technique is assessed by numerical solutions. Based on the theoretical result, we also present the critical conditions of stochastic bifurcation induced by system parameters and show bifurcation diagrams in two-parameter planes. In addition, we provide a stochastic bifurcation with respect to joint probability density functions. We find that fractional order, coefficient of restitution factor and correlation time of colored noise excitation can induce stochastic bifurcations.

STOCHASTIC HOPF BIFURCATION OF QUASI-INTEGRABLE HAMILTONIAN SYSTEMS WITH FRACTIONAL DERIVATIVE DAMPING

2012 ◽  
Vol 22 (04) ◽  
pp. 1250083 ◽  
Author(s):  
F. HU ◽  
W. Q. ZHU ◽  
L. C. CHEN
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Derivative ◽  
Hamiltonian Systems ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Bifurcation Parameter ◽  
Integrable Hamiltonian Systems ◽  
Fractional Derivative Damping ◽  
Stochastic Hopf Bifurcation

The stochastic Hopf bifurcation of multi-degree-of-freedom (MDOF) quasi-integrable Hamiltonian systems with fractional derivative damping is investigated. First, the averaged Itô stochastic differential equations for n motion integrals are obtained by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, an expression for the average bifurcation parameter of the averaged system is obtained and a criterion for determining the stochastic Hopf bifurcation of the system by using the average bifurcation parameter is proposed. An example is given to illustrate the proposed procedure in detail and the numerical results show the effect of fractional derivative order on the stochastic Hopf bifurcation.

On the Global Analysis of a Piecewise Linear System that is excited by a Gaussian White Noise

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Chen Kong ◽  
Xue Gao ◽  
Xianbin Liu
Keyword(s):  
Dynamical System ◽  
Averaging Method ◽  
Global Analysis ◽  
Piecewise Linear ◽  
Gaussian White Noise ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Random Dynamical System ◽  
Piecewise Linear System ◽  
Dynamical Behaviors

The global analysis is very important for a nonlinear dynamical system which possesses a chaotic saddle and a nonchaotic attractor, especially for the one that is driven by a noise. For a random dynamical system, within which, chaotic saddles exist, it is found that if the noise intensity exceeds a critical value, the so called “noise-induced chaos” is observed. Meanwhile, the exit behavior is also found to be influenced significantly by the existence of chaotic saddles. In the present paper, based on the generalized cell-mapping digraph (GCMD) method, the global dynamical behaviors of a piecewise linear system, wherein a chaotic saddle exists and consists of subharmonic solutions in a wide frequency range, are investigated numerically. Further, in order to simplify the system that is driven by a Gaussian white noise excitation, the stochastic averaging method is applied and through which, a five-dimensional Itô system is obtained. Some of the global dynamical behaviors of the original system are retained in the averaged one and then are analyzed. The researches in this paper show that GCMD method is a good numerical tool to investigate the global behaviors of a nonlinear random dynamical system, and the stochastic averaging method is effective for solving the global problems.

scholarly journals Stochastic Response of Dynamical Systems with Fractional Derivative Term under Wide-Band Excitation

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Jun-Feng Zhao
Keyword(s):  
Fractional Derivative ◽  
Galerkin Method ◽  
Wide Band ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Transient Solution ◽  
Restoring Force ◽  
Stochastic Dynamical System ◽  
Dependent Form ◽  
Derivative Term

Transient solution of a fractional stochastic dynamical system under wide-band noise excitation is investigated. Generalized Harmonic Balance technique is firstly used to approximate restoring force of the given system as an amplitude-dependent form. In this way, stochastic averaging method then can be applied to transform the system into an Ito differential equation. Furthermore, the fractional derivative in the integral-differential form can be equivalent to a combination of periodic functions after the averaging procedure. As the following, Galerkin method therein is utilized to obtain the transient probability density functions by solving associated Fokker-Planck-Kolmogorov (FPK) equation. As an example, the Rayleigh oscillator is studied to illustrate the efficiency and accuracy of the proposed approaches. Numerical results show that exact stationary solution and transient solution derived from Galerkin method are in good agreement with those from Monte Carlo Simulation.

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.

Stability of SDOF Nonlinear Viscoelastic System Under the Excitation of Wide-Band Noise

2007 ◽  
Author(s):  
Qinghua Huang ◽  
Wei-Chau Xie
Keyword(s):  
Averaging Method ◽  
Compressive Load ◽  
Wide Band ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Stochastic Bifurcation ◽  
Viscoelastic System ◽  
Band Noise ◽  
Nonlinear Viscoelastic ◽  
Wide Band Noise

The stochastic stability of a single degree-of-freedom (SDOF) nonlinear viscoelastic system under the excitation of wide-band noise is studied in this paper. An example of such a system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The equation of motion is an integro-differential equation with parametric excitation. The stochastic averaging method and averaging method for integro-differential equations are applied to reduce the system. The largest Lyapunov exponents and stochastic bifurcation are studied after the averaged sytem is obtained.

Stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Lincong Chen ◽  
Fang Hu ◽  
Weiqiu Zhu
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Hamiltonian Systems ◽  
Stochastic Dynamics ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Integrable Hamiltonian Systems ◽  
Fractional Optimal Control ◽  
Fractional Derivative Damping

AbstractIn the present survey, some progress in the stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping is reviewed. First, the stochastic averaging method for quasi integrable Hamiltonian systems with fractional derivative damping under various random excitations is briefly introduced. Then, the stochastic stability, stochastic bifurcation, first passage time and reliability, and stochastic fractional optimal control of the systems studied by using the stochastic averaging method are summarized. The focus is placed on the effects of fractional derivative order on the dynamics and control of the systems. Finally, some possible extensions are pointed out.

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.

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